Efficiency of change point tests in high
dimensional settings
John A D Aston∗

Claudia Kirch

†

arXiv:1409.1771v2 [] 25 Jun 2016

June 28, 2016

Abstract
While there is considerable work on change point analysis in univariate time series, more and more data being collected comes from high dimensional multivariate settings. This paper introduces the asymptotic concept of high dimensional
efficiency which quantifies the detection power of different statistics in such situations. While being related to classic asymptotic relative efficiency, it is different
in that it provides the rate at which the change can get smaller with dimension
while still being detectable. This also allows for comparisons of different methods with different null asymptotics as is for example the case in high-dimensional
change point settings. Based on this new concept we investigate change point
detection procedures using projections and develop asymptotic theory for how
full panel (multivariate) tests compare with both oracle and random projections.
Furthermore, for each given projection we can quantify a cone such that the corresponding projection statistic yields better power behavior if the true change
direction is within this cone. The effect of misspecification of the covariance on
the power of the tests is investigated, because in many high dimensional situations estimation of the full dependency (covariance) between the multivariate
observations in the panel is often either computationally or even theoretically
infeasible. It turns out that the projection statistic is much more robust in this
respect in terms of size and somewhat more robust in terms of power. The theoretic quantification by the theory is accompanied by simulation results which
confirm the theoretic (asymptotic) findings for surprisingly small samples. This
shows in particular that the concept of high dimensional efficiency is indeed suitable to describe small sample power, and this is demonstrated in a multivariate
example of market index data.

Keywords: CUSUM; High Dimensional Efficiency; Model Misspecification;
Panel Data; Projections
AMS Subject Classification 2000: 62M10;

1 Introduction
There has recently been a renaissance in research for statistical methods for change
point problems [Horváth and Rice, 2014]. This has been driven by applications where
non-stationarities in the data can often be best described as change points in the data
∗ Statistical

Laboratory, DPMMS, University of Cambridge, Cambridge, CB3 9HD, UK;
j.aston@statslab.cam.ac.uk
† Otto-von-Guericke University Magdeburg, Department of Mathematics, Institute of Mathematical
Stochastics, Postfach 4120, 39106 Magdeburg, Germany; claudia.kirch@ovgu.de

1

1 Introduction
generating process [Eckley et al., 2011, Frick et al., 2014, Aston and Kirch, 2012b].
However, data sets are now routinely considerably more complex than univariate time
series classically studied in change point problems [Page, 1954, Robbins et al., 2011,
Aue and Horváth, 2013, Horváth and Rice, 2014], and as such methodology for detecting and estimating change points in a wide variety of settings, such as multivariate
[Horváth et al., 1999, Ombao et al., 2005, Aue et al., 2009b, Kirch et al., 2015] functional [Berkes et al., 2009, Aue et al., 2009a, Hörmann and Kokoszka, 2010, Aston and
Kirch, 2012a] and high dimensional settings [Bai, 2010, Horváth and Hušková, 2012,
Chan et al., 2012, Cho and Fryzlewicz, 2015] have recently been proposed. In panel
data settings, these include methods based on taking maxima statistics across panels
coordinate-wise [Jirak, 2015], using sparsified binary segmentation for multiple change
point detection [Cho and Fryzlewicz, 2015], uses of double CUSUM procedures [Cho,
2015], as well as those based on structural assumptions such as sparsity [Wang and
Samworth, 2016].
Instead of looking at more and more complicated models, this paper uses a simple
mean change setting to illustrate how the power is influenced in high dimensional
settings. The results and techniques can subsequently be extended to more complex
change point settings as well as different statistical frameworks, such as two sample
tests. We make use of the following two key concepts: Firstly, we consider contiguous
changes where the size of the change tends to zero as the sample size and with it the
number of dimensions increases leading to the notion of high dimensional efficiency.
This concept is closely related to Asymptotic Relative Efficiency (ARE) (see Lehmann
[1999, Sec. 3.4] and Lopes et al. [2011] where ARE is used in a high dimensional
setting). Secondly, as a benchmark we investigate a class of tests based on projections,
where the optimal (oracle) projection test is closely related to the likelihood ratio test
under the knowledge of the direction of the change. Such tests can also be used in
order to include a priori information about the expected change direction, where we
can quantify how wrong the assumed direction can be and still yield better results than
a full multivariate statistic which uses no information about the change direction.
The aims of the paper are threefold: Firstly, we will investigate the asymptotic properties of tests based on projections as a plausible way to include prior information into
the tests. Secondly, by using high dimensional efficiency, we consider several projection tests (including oracle and random projections as benchmarks) and compare them
with the efficiency of existing tests that take the full covariance structure into account.
Finally, as in all high dimensional settings, the dependency between the components of
the series can typically neither be effectively estimated nor even uniquely determined
(for example if the sample size is less than the multivariate dimension) unless restrictions on the covariance are enforced. By considering the effect of misspecification of
the model covariance on the size as well as efficiency, we can quantify the implications
of this for different tests.
Somewhat obviously, highest efficiency can only be achieved under knowledge of the
direction of the change. However, data practitioners, in many cases, explicitly have
prior knowledge in which direction changes are likely to occur. It should be noted at
this point, that changes in mean are equivalent to changes of direction in multivariate
time series. In frequentist testing situations, practitioners’ main interest is in test
statistics which have power against a range of related alternatives while still controlling
the size. For example, an economist may check the performance of several companies
looking for changes caused by a recession. There will often be a general idea as to
which sectors of the economy will gain or lose by the recession and therefore a good
idea, at least qualitatively, as to what a change will approximately look like (downward
resp. upward shift depending on which sector a particular company is in) if there is a
change present. Similarly, in medical studies, it will often be known a-priori whether
genes are likely to be co-regulated causing changes to be in similar directions for groups
of genes in genetic time series.

2

1 Introduction
Incorporating this a-priori information about how the change affects the components
by using corresponding projections can lead to a considerable power improvement if the
change is indeed in the expected direction. It is also important that, as in many cases
the a-priori knowledge is qualitative, the test has higher power than standard tests not
only for that particular direction but also for other directions close by. Additionally,
these projections lead to tests where the size is better controlled if no change is present.
Using the concept of high dimensional efficiency allows the specification of a cone
around a given projection such that the projection statistic has better power than
the multivariate/panel statistic if the true change is within this cone. In addition,
while the prior information itself might be reliable, inherent misspecification in other
parts of the model, such as the covariance structure, will have a detrimental effect on
detection, and it is of interest to quantify the effect of these as well.
The results in this paper will be benchmarked against taking the simple approach of
using a random projection in a single direction to reduce the dimension of the data.
Random projections are becoming increasingly popular in high dimensional statistics
with applications in Linear Discriminant Analysis [Durrant and Kabán, 2010] and two
sample testing [Lopes et al., 2011, Srivastava et al., 2014]. This is primarily based on
the insight from the Johnson-Lindenstrauss lemma that an optimal projection in the
sense that the distances are preserved for a given set of data is independent of the
dimension of the data [Johnson and Lindenstrauss, 1984] and thus random projections
can often be a useful way to perform a dimension reduction for the data [Baraniuk
et al., 2008]. However, in our context, we will see that a random projection will not
work as well as truly multivariate methods, let alone projections with prior knowledge,
but can only serve as a lower benchmark.
We will consider a simple setup for our analysis, although one which is inherently the
base for most other procedures, and one which can easily be extended to complex
time dependencies and change point definitions using corresponding results from the
literature [Kirch and Tadjuidje Kamgaing, 2014a, Kirch and Tajduidje Kamgaing,
2014b]. For a set of observations Xi,t , 1 6 i 6 d = dT , 1 6 t 6 T , the change point
model is defined to be
Xi,t = µi + δi,T g(t/T ) + ei,t ,

1 6 i 6 d = dT , 1 6 t 6 T,

(1.1)

where E ei,t = 0 for all i and t with 0 < σi2 = var ei,t < ∞ and g : [0, 1] → R is
a Riemann-integrable function. Here δi,T indicates the size of the change for each
component. This setup incorporates a wide variety of possible changes by the suitable
selection of the function g, as will be seen below. For simplicity, for now it is assumed
that {ei,t : t ∈ Z} are independent, i.e. we assume independence across time but
not location. If the number of dimensions d is fixed, the results readily generalise to
situations where a multivariate functional limit theorem exists as is the case for many
weak dependent time series. If d can increase to infinity with T , then generalizations
are possible if the {ei,t : 1 6 t 6 T } form a linear process in time but the errors are
independent between components (dependency between components will be discussed
in detail in the next section). Existence of moments strictly larger than two is needed
in all cases. Furthermore, the developed theory applies equally to one- and two-sample
testing and can be seen as somewhat analogous to methods for multivariate adaptive
design [Minas et al., 2014].
The change (direction) is given by ∆d = (δ1,T , . . . , δd,T )T and the type of alternative
is given by the function g in rescaled time. While g is defined in a general way, it
includes as special cases most of the usual change point alternatives, for example,

0 0≤u≤θ
• At most one change (AMOC): g(u) =
1 θ<u≤1

3

2 Change Points and Projections

 0 0 ≤ u ≤ θ1
1 θ1 < u < θ 2
• Epidemic change (AMOC): g(u) =

0 θ2 < u ≤ 1
The form of g will influence the choice of test statistic to detect the change point. As
in the above two examples in the typical definition of change points the function g is
modelled by a step function (which can approximate many smooth functions well). In
such situations, test statistics based on partial sums of the observations have been well
studied [Csörgő and Horváth, 1997]. It will be shown that statistics based on partial
sums are robust (in the sense of still having non-zero power) to a wide variety of g.
The model in (1.1) is defined for univariate (d = 1), multivariate (d fixed) or panel
data (d → ∞). The panel data (also known as “small n large p” or “high dimensional
low sample size”) setting is able to capture the small sample properties very well in
situations where d is comparable or even larger than T using asymptotic considerations.
In this asymptotic framework the detection ability or efficiency of various tests can be
defined by the rates at which vanishing alternatives can still be detected. However,
many of our results, particularly for the proposed projection tests, are also qualitatively
valid in the multivariate or d fixed setting.
The paper proceeds as follows. In Section 2, the concept of high dimensional efficiency
as a way of comparing the power of high dimensional tests is introduced. This is
done using projection statistics, which will also act as benchmarks. In Section 3,
the projection based statistics will be compared with the panel based change point
statistics already suggested in Horváth and Hušková [2012], both in terms of control of
size and efficiency, particular with relation to the (mis)specification of the dependence
structure. Section 4 provides a short illustrative example with respect to multivariate
market index data. Section 5 concludes with some discussion of the different statistics
proposed, while Section 6 gives the proofs of the results in the paper. In addition,
rather than a separate simulation section, simulations will be interspersed throughout
the theory. They complement the theoretic results, confirming that the conclusion are
already valid for small samples, thus verifying that the concept of high-dimensional
efficiency is indeed suitable to understand the power behavior of different test statistics.
In all cases the simulations are based on 1000 repetitions of i.i.d. normally distributed
data for each set of situations, and unless otherwise stated the number of time points
is T = 100 with the change (if present) occurring half way through the series. Except
in the simulations concerning size itself, all results are empirically size corrected to
account for the size issues for the multivariate (panel) statistic that will be seen in
Figure 2.1.

2 Change Points and Projections
2.1 High dimensional efficiency
As the main focus of this paper is to compare several test statistics with respect
to their detection power, we introduce a new asymptotic concept that allows us to
understand this detection power in a high dimensional context. In the subsequent
sections, simulations accompanying the theoretic results will show that this concept is
indeed able to give insight into the small sample detection power.
Consider a typical testing situation, where (possibly after reparametrization) we test
H0 : a = 0,

against

H1 : a 6= 0.

(2.1)

4

2 Change Points and Projections
Typically, large enough alternatives, will be detected by all reasonable statistics for
a given problem. In asymptotic theory this corresponds to fixed alternatives, where
a = c 6= 0, for which tests typically have asymptotic power one.
To understand the small sample power of different statistics such asymptotics are
therefore not suitable. Instead the asymptotics for local or contiguous alternatives
with a = aT → 0 are considered. For a panel setting we define:
Definition 2.1. Consider the testing situation (2.1) with sample size T → ∞ and
sample dimension d = dT → ∞. A test with statistic T (X1 , . . . , XT ) has (absolute)
high dimensional efficiency E(ad ) for a sequence of alternatives ad if
L

(i) T (X1 , . . . , XT ) −→ L for some non-degenerate limit distribution L under H0 ,
P

(ii) T (X1 , . . . , XT ) −→ ∞ if
L

(iii) T (X1 , . . . , XT ) −→ L if

√
√

T E(ad ) → ∞,

T E(ad ) → 0.

Note that the E(ad ) is only defined up to multiplicative constants, and has to be understood as a representative of a class.
√
In particular this shows
√ that the asymptotic power is√one if T E(ad ) → ∞, but
equal to the level if T E(ad ) → 0. Typically, for T E(ad ) → α 6= 0 it holds
L

D

T (X1 , . . . , XT ) −→ L(α) 6= L, usually resulting in an asymptotic power strictly between the level and one. In the classic notion (with d constant) of absolute relative
efficiency (ARE, or Pitman Efficiency) for test statistics with a standard normal limit
it is the additive shift between L(α) and L [see Lehmann, 1999, Sec 3.4]) that shows
power differences for different statistics. Consequently, this shift has been used to
define asymptotic efficiency. For different null asymptotics the comparison becomes
much more cumbersome as the quantiles of the different limit distributions had to be
taken into account as well. In our definition above, we concentrate on the efficiency in
terms of the asymptotic rates with respect to the increasing dimension (as two estimators are equivalent up to a dimension free constant). Should the rates be equivalent,
classic notions of ARE then apply, although with the usual difficulties should the limit
distributions be different.
In the asymptotic panel setup, on the other hand, the differences with respect to
the dimension d are now visible in the rates, with which contiguous alternatives can
disappear and still be asymptotically detectable. Therefore, we chose this rate to
define asymptotic high dimensional efficiency. Additionally, it is no longer required
that different test statistics have the same limit distribution under the null hypothesis
(which would be a problem in this paper).

2.2 Projections
We now describe how projections can be used to obtain change point statistics in
high dimensional settings, which will be used as both benchmark statistics for a truly
multivariate statistic as well as a reasonable alternative if some knowledge about the
direction of the change is present.
In model (1.1), the change ∆d = (δ1,T , . . . , δd,T )T (as a direction) is always a rank
one (vector) object no matter the number of components d. This observation suggests
that knowing the direction of the change ∆d in addition to the underlying covariance
structure can significantly increase the signal-to-noise ratio. Furthermore, for µ and
∆d /k∆d k (but not k∆d k) known with i.i.d. normal errors, one can easily verify that

5

2 Change Points and Projections
the corresponding likelihood ratio statistic is obtained as a projection statistic with
projection vector Σ−1 ∆d , which can also be viewed as an oracle projection. Under
(1.1) it holds
hXd (t), pd i = hµ, pd i + h∆d , pd ig(t/T ) + het , pd i,
where Xd (t) = (X1,t , . . . , Xd,T )T , µ = (µ1 , . . . , µd )T and et = (e1,t , . . . , ed,t )T . The
projection vector pd plays a crucial role in the following analysis and will be called
the search direction. This representation shows that the projected time series exhibits
the same behavior as before as long as the change is not orthogonal to the projection
vector. Furthermore, the power is the better the larger h∆d , pd i and the smaller the
variance of het , pd i is. Consequently, an optimal projection in terms of power depends
on ∆d as well as Σ = var e1 . In applications, certain changes are either expected or of
particular interest e.g. an economist looking at the performance of several companies
expecting changes caused by a recession will have a good idea which companies will
profit or lose. This knowledge can be used to increase the power in directions close
to the search direction pd while decreasing it for changes that are close to orthogonal
to it. Using projections can furthermore robustify the size of the test under the null
hypothesis with respect to misspecification and estimation error.
In order to qualify this informal statement, we will consider contiguous changes for
several change point tests, where k∆d k → 0 but with such a rate that the power of the
corresponding test is strictly between the size and one, as indicated in the previous
subsection.
In order to be able to prove asymptotic results for change point statistics based on
projections even if d → ∞, we need to make the following assumptions on the underlying error structure. This is much weaker than the independence assumption as
considered by Horváth and Hušková [2012]. Furthermore, we do not need to restrict
the rate with which d grows. If we do have restrictions on the growth rate in particular
for the multivariate setting with d fixed, these assumptions can be relaxed and more
general error sequences can be allowed.
Assumption A. 1. Let η1,t (d), η2,t (d), . . . be independent with E ηi,t (d) = 0, var ηi,t (d) =
1 and E |ηi,t (d)|ν 6 C < ∞ for some ν > 2 and all i and d. For t = 1, . . . , T we additionally assume for simplicity that (η1,t (d), η2,t (d), . . .) are identically distributed
(leading to data which is identically distributed across time). The errors within the
components are then given as linear processes of these innovations:
X
X
el,t (d) =
al,j (d)ηj,t (d), l = 1, . . . , d,
al,j (d)2 < ∞
j>1

j>1

or equivalently in vector notation et (d) = (e1,t (d), . . . , ed,t (d))T and aj (d) = (a1,j (d), . . . , ad,j (d))T
X
et (d) =
aj (d)ηj,t (d).
j>1

These assumptions allow us to considered many varied dependency relationships between the components (and we will concentrate on within the component dependency
at this point, as temporal dependency adds multiple layers of notational difficulties,
but little in the way of insight as almost all results generalise simply for weakly dependent and linear processes).
The following three cases of different dependency structures are very helpful in understanding different effects that can occur and will be used as examples throughout the
paper:

6

2 Change Points and Projections
Case C. 1 (Independent Components). The components are independent, i.e. aj =
(0, . . . , sj , . . . , 0)T the vector which is sj > 0 at point j and zero everywhere else, j 6 d,
and aj = 0 for j > d + 1. In particular, each channel has variance
σj2 = s2j .
Case C. 2 (Fully Dependent Components). There is one common factor to all components, leading to completely dependent components, i.e. a1 = Φd = (Φ1 , . . . , Φd )T ,
aj = 0 for j > 2. In this case,
σj2 = Φ2j .
This case, while being somewhat pathological, is useful for gaining intuition into the
effects of possible dependence and also helps with understanding the next case.
Case C. 3 (Mixed Components). The components contain both an independent and
dependent term. Let aj = (0, . . . , sj , . . . , 0)T the vector which is sj > 0 at point j and
zero everywhere else, and ad+1 = Φd = (Φ1 , . . . , Φd )T , aj = 0 for j > d + 2. Then
σj2 = s2j + Φ2j
This mixed case allows consideration of dependency structures between cases C.1 and
C.2. It is used in the simulations with Φd = Φ(1, . . . , 1)T , where Φ = 0 corresponds to
C.1 and Φ → ∞ corresponds to C.3. We also use this particular example for the panel
statistic in Section 3.2 to quantify the effect of misspecification.
Of course, many other dependency structures are possible, but these three cases give
insight into the cases of no, complete and some dependency respectively. In particular,
as the change is always rank one, taking a rank one form of dependency, as in cases
C.2 and as part of C.3, still allows somewhat general conclusions to be drawn.

2.3 Change point statistics
Standard statistics such as the CUSUM statistic are based on partial sum processes, so
in order to quantify the possible power gain by the use of projections we will consider
the partial sum process of the projections, i.e.


bT xc
T
X
X
1
hXd (t), pd i − 1
hXd (j), pd i ,
(2.2)
Ud,T (x) = hZT (x), pd i = √
T j=1
T t=1


bT xc
T
X
X
1
bT xc
ZT,i (x) = 1/2 
Xi,t −
Xi,t  ,
(2.3)
T t=1
T
t=1
where Xd (t) = (X1,1 , . . . , Xd,T )T .
Different test statistics can be defined for a range of g in (1.1), however, assuming that
g 6≡ 0, the hypothesis of interest is
H0 : ∆d = 0
versus the alternative
H1 : ∆d 6= 0.

7

2 Change Points and Projections
Test statistics are now defined in order to give good power characteristics for a particular g function. For example, the classic AMOC statistic for univariate and multivariate
change point detection is based on Ud,T (x)/τ (pd ), with
τ 2 (pd ) = pTd var (e1 (d)) pd .

(2.4)

Typically, either the following max or sum type statistics are used:

max w(k/T )

16k6T

Ud,T (k/T )
,
τ (pd )

T
1 X
Ud,T (k/T )
,
w(k/T )
T
τ (pd )
k=1

where w > 0 is continuous (which can be relaxed) and fulfills (2.10) (confer e.g. the
book by Csörgő and Horváth [1997]). The choice of weight function w(·) can increase
power for certain locations of the change points [Kirch et al., 2015].
For the epidemic change, typical test statistics are given by
max

16k1 <k2 6T

or
1
T2

1
|Ud,T (k2 /T ) − Ud,T (k1 /T ) |,
τ (pd )

X
16k1 <k2 6T

1
|Ud,T (k2 /T ) − Ud,T (k1 /T ) |.
τ (pd )

In the next section we first derive a functional central limit theorem for the process
Ud,T (x), which implies the asymptotic null behavior for the above tests. Then, we
derive the asymptotic behavior of the partial sum process under contiguous alternatives
to obtain the high dimensional efficiency for projection statistics.

2.4 Efficiency of Change point tests based on projections
In this section, we derive the efficiency of change point tests based on projections under
rather general assumptions. Furthermore, we will see that the size behavior is very
robust with respect to deviations from the assumed underlying covariance structure.
The power on the other hand turns out to be less robust but more so than statistics
taking the full multivariate information into account.

2.4.1 Null Asymptotics
As a first step towards the efficiency of projection statistics, we derive the null asymptotics. This is also of independent interest if projection statistics are applied to a given
data set in order to find appropriate critical values. In the following theorem d can
be fixed but it is also allowed that d = dT → ∞, where no restrictions on the rate of
convergence are necessary.
Theorem 2.1. Let model (1.1) hold. Let pd be a possibly random projection independent of {ei,t : 1 6 t 6 T, 1 6 i 6 d}. Furthermore, let pTd cov(e1 (d))pd 6= 0 (almost
surely), which means that the projected data is not degenerate with probability one.
a) Under Assumption A.1 and if {pd } is independent of {ηi,t (d) : i > 1, 1 6 t 6 T },
then it holds under the null hypothesis


Ud,T (x)
D[0,1]
: 0 6 x 6 1 | pd −→ {B(x) : 0 6 x 6 1}
a.s.,
(2.5)
τ (pd )
where B(·) is a standard Brownian bridge.

8

2 Change Points and Projections
b) For i.i.d. error sequences {et (d) : t = 1, . . . , d}, et (d) = (e1,t (d), . . . , ed,t (d))T with
an arbitrary dependency structure across components, and if E |e1,t (d)|ν 6 C < ∞
for all t and d as well as
pTd

kpd k21
= o(T 1−2/ν )
cov(et )pTd

where kak1 =

Pd

j=1

a.s.,

(2.6)

|aj |, then (2.5) holds.

2
The assertions remain true if τ 2 (pd ) is replaced by τbd,T
such that for all  > 0

P

!
2
τbd,T
−1 > →0
τ 2 (pd )

a.s.

(2.7)

Assumption (2.6) is always fulfilled for the multivariate situation with d fixed or if
d is growing sufficiently
slowly with respect to T as the left hand side of (2.6) is
√
always bounded by d if pTd cov(e)pd /kpd k2 is bounded away from zero. Otherwise,
the assumption may hold for certain projections but not others. However, in this case,
it is possible to put stronger assumptions on the error sequence such as in a), which
are still much weaker than the usual assumption for panel data, that components
are independent. In these cases projection methods hold the size asymptotically, no
matter what the dependency structure between components is and without having to
estimate this dependency structure.
This is in contrast to the multivariate statistic which suffers from considerable size
distortions if this underlying covariance structure is estimated incorrectly. The estimation of the covariance structure is a difficult problem in higher dimensions in
particular since an estimator for the inverse is needed with additional numerical problems arising. The problem becomes even harder if time series errors are present, in
which case the long-run covariance rather than the covariance matrix needs to be estimated [Hörmann and Kokoszka, 2010, Aston and Kirch, 2012b, Kirch et al., 2015].
While the size of the projection procedure is unaffected by the underlying dependency
across components, we will see in the next section that for optimal efficiency hence
power we need not only to know the change ∆d but also the inverse of the covariance
matrix. Nevertheless the power of projection procedures turns out to be more robust
with respect to misspecification than a size-corrected panel statistic, that takes the
full multivariate information into account.
The following lemma shows that the following two different estimators for τ (pd ) under
the null hypothesis are both consistent. The second one is typically still consistent in
the presence of one mean change which usually leads to a power improvement in the
test for small samples. An analogous version can be defined for the epidemic change
situation. However, it is much harder to get an equivalent correction in the multivariate setting because the covariance matrix determines how different components are
weighted, which in turn has an effect on the location of the maximum. This problem
does not arise in the univariate situation, because the location of the maximum does
not depend on the variance estimate.
Lemma 2.2. Consider
!2
T
T
X
X
1
1
2
τb1,d,T
(pd ) =
pT et (d)
pTd et (d) −
T j=1
T i=1 d

9

(2.8)

2 Change Points and Projections
as well as


b
kd,T

2
τb2,d,T
(pd ) =



2

b
kd,T

1 X  T
1 X T
pd ej (d) −
p ei (d) +

T j=1
T i=1 d

T
X
j=b
kd,T +1



T
X

pTd et (d) − 1
T

i=b
kd,T +1

(2.9)
where

b
kd,T = arg max Ud,T (t/T ).
t=1,...,T

a) Under the assumptions of Theorem 2.1 a) both estimators (2.8) as well as (2.9)
fulfill (2.7).
b) Under the assumptions of Theorem 2.1 b), then estimator (2.8) fulfills (2.7) under
the assumption
pTd

kpd k21
= o(T 1−2/ min(ν,4) )
cov(et )pTd

a.s.,

while estimator (2.9) fulfills it under the assumption
kpd k21
= o(T 1−2/ min(ν,4) (log T )−1 )
pTd cov(et )pTd

a.s.,

The following theorem gives the null asymptotic for the simple CUSUM statistic for
the at most one change, other statistics as given in Section 2.3 can be dealt with along
the same lines.
Corollary 2.3. Let the assumptions of Theorem 2.1 be fulfilled and τb(pd ) fulfill (2.7)
under the null hypothesis, then for all x ∈ R it holds under the null hypothesis
!


2
U
(k/T
)
d,T
2
2
2
P
max w (k/T ) 2
6 x pd → P max w (t)B (t) 6 x
a.s.
06t61
16k6T
τb (pd )


Z 1

2
X
U
(k/T
)
1
d,T
2
2
2


w (k/T ) 2
6 x pd → P
w (t)B (t) dt 6 x
a.s.
P
T
τb (pd )
0
16k6T

for any continuous weight function w(·) with
lim tα w(t) < ∞,

t→0

sup

w(t) < ∞

lim (1 − t)α w(t)

t→1

for some 0 6 α < 1/2,

for all 0 < η 6

η6t61−η

1
.
2

(2.10)

As can be seen in Figure 2.1, regardless of whether the variance is known or estimated,
the projection methods all maintain the correct size even when there is a high degree
of dependence between the different components (the specific projection methods and
indeed the non-projection methods will be characterised in Section 2.5 below). The
full tests, where size is not controlled, will be discussed in Section 3.

2.4.2 Absolute high dimensional efficiency
We are now ready to derive the high dimensional efficiency of projection statistics.
Furthermore, we show that a related estimator for the location of the change is asymptotically consistent.

10

2 

pTd ei (d)  ,

2 Change Points and Projections
1
Oracle
Random
Quasi-Oracle
Pre-Oracle

0.9
0.8
0.7

H&H 2i

0.6

H&H i + i

size

2

2

0.5
0.4
0.3
0.2
0.1
0
0

0.2

0.4

0.6



0.8

1

(a) Known variance
0.9
0.8

0.8
0.7

0.6

0.6

0.5

0.5

size

size

0.7

0.9
Oracle
Random
Quasi-Oracle
Pre-Oracle
H&H Var

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0
0

0.2

0.4



0.6

0.8

1

(b) Estimated variance as in (2.8)

0
0

Oracle
Random
Quasi-Oracle
Pre-Oracle
H&H Var

0.2

0.4



0.6

0.8

1

(c) Estimated variance as in (2.9)

Figure 2.1: Size of tests as the degree of dependency between the components increases.
As can be seen, all the projection methods, Oracle, Quasi-Oracle, PreOracle and Random projections defined in Section 2.5 maintain the size
of the tests. Those based on using the full information as described in
Section 3 have size problems as the degree of dependency increases. The
simulations correspond to Case C.3 with sj = 1, Φj = φ, j = 1, . . . , d with
d = 200, where φ is given on the x-axis).

Theorem 2.4. Let the assumptions of Theorem 2.1 either a) or b) on the errors
respectively pd be fulfilled. Furthermore, consider a weight function w(·) as in CorolR
2
R1
x
lary 2.3 fulfilling w2 (x) 0 g(t) dt − x 0 g(t) dt 6= 0. Then, both the max and sum
statistic from Corollary 2.3 have the following absolute high dimensional efficiency:
E1 (∆d , pd ) :=

|h∆d , pd i|
k∆d kkpd k cos(α∆d ,pd )
=
,
τ (pd )
τ (pd )

(2.11)

where τ 2 (pd ) is as in (2.4) and αu,v is the (smallest) angle between u and v. In
addition, the asymptotic power increases with increasing multiplicative constant.
In the following, E1 (∆d , pd ) is fixed to the above representative of the class, so that
different projection procedures with the same rate but with different constants can be
compared.
Remark 2.1. For random projections the high dimensional efficiency is a random
variable. The convergences in Definition 2.1 is understood given the projection vector
pd , where
√ we get either a.s.-convergence or P -stochastic convergence depending on
whether T E1 (∆d , pd ) converges a.s. or in a P -stochastic sense (in the latter case the
assertion follows from the subsequence-principle).

11

2 Change Points and Projections
Remark 2.2. In particular all deviations from a stationary
mean are
Rx
R 1detectable with
asymptotic power one for weight functions w > 0 as 0 g(t) dt − x 0 g(t) dt 6= 0 for
non-constant g. It is this g function which determines which weight function gives
best power.
We derive the high dimensional efficiency for a given g and disappearing magnitude of
the change k∆d k. For an epidemic change situation with g(x) = 1{ϑ1 <x6ϑ2 } for some
0 < ϑ1 < ϑ2 < 1, this means that the duration of the change is relatively large but the
magnitude relatively small with respect to the sample size. Alternatively, one could
also consider the situation, where the duration gets smaller asymptotically (see e.g.
[Frick et al., 2014]) resulting in a different high dimensional efficiency, which is equal
for both the projection as well as the multivariate or panel statistic, as long as the same
weight function and the same type of statistic (sum/max) is used. Some preliminary
investigations suggest that in this case using projections based on principle component
analysis similar to Aston and Kirch [2012a] can be advantageous, however this is not
true for the setting discussed in this paper.
The above result shows in particular that sufficiently large changes (as defined by the
high dimensional efficiency) are detected asymptotically with power one. For such
changes in the at-most-one-change situation, the following corollary shows that the
corresponding change point estimator is consistent in rescaled time.
√
Corollary 2.5. Let the assumptions of Theorem 2.4 hold and additionally T E1 (∆d , pd ) →
∞ a.s. Under the alternative of one abrupt change, i.e. g(x) = 1{x>ϑ} for some
0 < ϑ < 1, the estimator
%
$
2
arg maxk Ud,T
(k/T )
b
ϑT =
T
is consistent for the change point in rescaled time, i.e.


P ϑbT − ϑ >  | pd → 0
a.s.
2
An analogous statement holds, if the arg max of w2 (k/T )Ud,T
(k/T ) is used instead
2
2
and w (x) ((x − ϑ)+ − x(1 − ϑ)) has a unique maximum at ϑ, which is the case for
many standard weight functions such as w(t) = (t(1 − t))−β for some 0 6 β < 1/2.

In the next section we will further investigate the high dimensional efficiency and see
that the power depends essentially on the angle between Σ1/2 pd and the ’standardized’
change Σ−1/2 ∆ if Σ is invertible. In fact, the smaller the angle the larger the power.
Some interesting insight will also come from the situation where Σ is not invertible by
considering case C.2 above.

2.5 High dimensional efficiency of oracle and random projections
In this section, we will further investigate the high dimensional efficiency of certain
particular projections that can be viewed as benchmark projections. In particular, we
will see that the efficiency depends only on the angle between the projection and the
change both properly scaled with the underlying covariance structure.
The highest efficiency is obtained by o = Σ−1 ∆d as the next theorem shows, which
will be called the oracle projection. This oracle is equivalent to a projection after first
standardizing the data on the ’new’ change Σ−1/2 ∆d . The corresponding test is related
to the likelihood ratio statistic for i.i.d. normal innovations, where both the original

12

2 Change Points and Projections
mean and the direction (but not magnitude) of the change are known. As a lower (worst
case) benchmark we consider a scaled random projection rd,Σ = Σ−1/2 rd , where rd is
a random projection on the d-dimensional unit sphere. This is equivalent to a random
projection onto the unit sphere after standardizing the data. Both projections depend
on Σ which is usually not known so that it needs to be estimated. The latter is rather
problematic in particular in high dimensional settings without additional parametric
or sparsity assumptions (see Zou et al. [2006], Bickel and Levina [2008] and Fan et al.
[2013] including related discussion). Furthermore, it is actually the inverse that needs
to be estimated which results in additional numerical problems if d is large. For this
reason we check the robustness of the procedure with respect to not knowing or
misspecifying Σ in a second part of this section
In Section 3 we will compare the efficiency of the above projections with a procedure
taking the full information into account. We will show that we lose an order d1/4 in
terms of high dimensional efficiency between the oracle and the full panel data statistic
and another d1/4 between the panel and the random projection.

2.5.1 Correctly scaled projections
The following proposition characterizes which projection yields an optimal high dimensional efficiency associated with the highest power.
Proposition 2.6. If Σ is invertible, then
E1 (∆, pd ) = kΣ−1/2 ∆d k cos(αΣ−1/2 ∆d ,Σ1/2 pd ).

(2.12)

Proposition 2.6 shows in particular, that after standardizing the data, i.e. for Σ =
Id , the power depends solely on the cosine of the angle between the oracle and the
projection (see Figure 2.2).
From the representation in this proposition it follows immediately that the ’oracle’
choice for the projection to maximize the high dimensional efficiency is o = Σ−1 ∆d as
it maximizes the only term which involves the projection namely cos(αΣ−1/2 ∆d ,Σ1/2 pd ).
Therefore, we define:
Definition 2.2. The projection o = Σ−1 ∆d is called oracle if Σ−1 exists. Since the
projection procedure is invariant under multiplications with non-zero constants of the
projected vector, all non-zero multiples of the oracle have the same properties, so that
they correspond to a class of projections.
By Proposition 2.6 the oracle choice leads to a high dimensional efficiency of E1 (∆d , o) =
kΣ−1/2 ∆d k.
Another way of understanding the Oracle projection is the following: If we first
standardize the data, then for a projection on a unit (w.l.o.g.) vector the variance
of the noise is constant and the signal is given by the scalar product of Σ−1/2 ∆
and the (unit) projection vector, which is obviously maximized by a projection with
Σ−1/2 ∆/kΣ−1/2 ∆k which is equivalent to using pd = Σ−1 ∆ as a projection vector
for the original non-standardized version.
So, if we know Σ and want to maximize the efficiency respectively power close to
a particular search direction sd of our interest, we should use the scaled search
direction sΣ,d = Σ−1 sd as a projection.
Because the cosine falls very slowly close to zero, the efficiency will be good if the search
direction is not too far off the true change. From this, one could get the impression

13

2 Change Points and Projections
1
0.9

Random
Search
H&H

0.8
0.7

Power

0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.2

0.4

0.6

0.8
Radians

1

1.2

1.4

1.6

Figure 2.2: Power of tests as the angle between the search direction and the oracle
increases. As can be seen, the search projection method decreases similarly
to cosine of the angle, while the random projection and Horváth -Hušková
tests as introduced in Section 3 are given for comparison. (Here Σd = Id ,
d = 200, and ∆d = 0.05 1d , corresponding to Case C.1).

that even a scaled random projection rΣ,d = Σ−1/2 rd may not do too badly, where
rd is a uniform random projection on the unit sphere. This is equivalent to using a
random projection on the unit sphere after standardizing the data, which also explains
the different scaling as compared to the oracle or the scaled search direction, where
the change ∆d is also transformed to Σ−1/2 ∆d by the standardization. However, since
for increasing d the space covered by the far away angles is also increasing, the high
dimensional√efficiency of the scaled random projection is not only worse than the oracle
by a factor d but also by a factor d1/4 than a full multivariate or panel statistic which
will be investigated in detail in Section 3.
The following theorem shows the high dimensional efficiency of the scaled random
projection.
Theorem 2.7. Let the alternative hold, i.e. k∆d k 6= 0. Let rd be a random uniform
projection on the d-dimensional unit sphere and rΣ,d = Σ−1/2 rd , then for all  > 0
there exist constants c, C > 0, such that


d
P c 6 E12 (∆d , rΣ,d )
6
C
> 1 − .
kΣ−1/2 ∆d k2
Such a random projection on the unit sphere can be obtained as follows: Let X1 , . . . , Xd
be i.i.d. N(0,1), then rd = (X1 , . . . , Xd )T /k(X1 , . . . , Xd )T k is uniform on the d-dimensional
unit sphere [Marsaglia, 1972].
Comparing the high dimensional efficiency of the scaled random projection with the
one obtained for the oracle
projection (confer Proposition 2.6) it becomes apparent
√
that we lose an order d. In Section 3 we will see that the panel statistic taking the
full multivariate information into account has a contiguous rate just between those
two losing d1/4 in comparison to the oracle but gaining d1/4 in comparison to a scaled
random projection. From these results one obtains a cone around the search direction
such that the projection statistic has higher power than the panel statistic, if the true
change falls within this cone.
Figure 2.3 shows the results of some simulations showing that a change that can be
detected for the oracle with constant power as d increases rapidly loses power for the

14

2 Change Points and Projections
1
0.9
0.8
0.7

power

0.6
0.5
random projection
/4 projection
Horvath & Huskova
oracle projection

0.4
0.3
0.2
0.1
0
10

1

10

2

10

3

d

Figure 2.3: Power of the tests as d increases with a fixed sample size (T = 100). Here
k∆d k = const.and Σd = Id , i.e. kΣ−1/2 ∆d k = const., corresponding to
Case C.1. This gives roughly constant power for fixed angle projection
tests (as k∆d k is constant), while results in decreasing power for both the
panel statistic test and random projections as predicted by theory.

panel statistic as predicted by its high dimensional efficiency in Section 3 as well as
for the random projection. This and the following simulations show clearly that the
concept of high dimensional efficiency is indeed capable of explaining the small sample
power of a statistic very well.

2.5.2 The oracle in the case of non-invertibility
Let us now have a look at the situation if Σ is not invertible hence the above oracle
does not exist. To this end, let us consider Case C.2 above – other non-invertible
dependent situations can essentially be viewed in a very similar fashion, but become
a combination of the two scenarios below.
Case C. 2 (Fully dependent Components). In this case Σ = Φd ΦTd is a rank 1 matrix
and not invertible. Consequently, the oracle as in Definition 2.2 does not exist. To
understand the situation better, we have to distinguish two scenarios:
(i) If Φd is not a multiple of ∆d we can transform the data into a noise-free sequence
that only contains the signal by projecting onto a vector that is orthogonal to Φd
(cancelling the noise term) but not to ∆d . All such projections are in principle
equivalent as they yield the same signal except for a different scaling which is not
important if there is no noise present. Consequently, all such transformations
could be called oracle projections.
(ii) On the other hand if ∆d is a multiple of Φd , then any projection cancelling the
noise will also cancel the signal. Projections that are orthogonal to Φd hence
by definition also to ∆d will lead to a constant deterministic sequence hence to
a degenerate situation. All other projections lead to the same (non-degenerate)

15

2 Change Points and Projections
time series except for multiplicative constants and different means (under which
the proposed change point statistics are invariant by definition) so all of them
could be called oracles.
The following interpretation also explains the above mathematical findings: In this
situation, all components are obtained from one common factor {ηt } with different
weights according to Φd i.e. they move in sync with those weights. If a change is
proportional to Φd it could either be attributed to the noise coming from {ηt } or from
a change, so it will be difficult to detect as we are essentially back in a duplicated
rank one situation and no additional information about the change can be obtained
from the multivariate situation. However, if it is not proportional to Φ, then it is
immediately clear (with probability one) that a change in mean must have occurred
(as the underlying time series no longer moves in sync). This can be seen to some extent
in Figure 2.4, where the different panels in the figure mimic the different scenarios as
outlined above (with a large value of φ being close to the non-invertible situation).

2.5.3 Misscaled projections with respect to the covariance structure
The analysis in the previous section requires the knowledge or a precise estimate of
the inverse (structure) of Σ. However, in many situations such an estimate may not be
feasible or too imprecise due to one or several of the below reasons, where the problems
get worse due to the necessity for inversion
• If d is large in comparison to T statistical estimation errors can accumulate and
identification may not even be possible [Bickel and Levina, 2008].
• The theory can be generalized to time series errors but in this case the covariance
matrix has to be replaced by the long-run covariance (which is proportional to
the spectrum at 0) and is much more difficult to estimate [Aston and Kirch,
2012b, Kirch and Tadjuidje Kamgaing, 2012].
• Standard covariance estimators will be inconsistent under alternatives as they are
contaminated by the change points. Consequently, possible changes have to be
taken into account, but even in a simple at most one change situation it is unclear
how best to generalize the standard univariate approach as in (2.9) as opposed
to (2.8) to a multivariate situation as the estimation of a joint location already
requires an initial weighting for the projection (or the multivariate statistic).
Alternatively, component-wise univariate estimation of the change points could
be done but require a careful asymptotic analysis in particular in a setting with
d → ∞.
• If d is large, additional numerical errors may arise when inverting the matrix
[Higham, 2002, Ch 14].
We will now investigate the influence of misspecification or estimation errors on the
high dimensional efficiency of a misscaled oracle oM = M−1 ∆d in comparison to
the misscaled random projection rM,d = M−1/2 rd , where we only assume that
the assumed covariance structure M is symmetric and positive definite and model A.1
is fulfilled.
Theorem 2.8. Let the alternative hold, i.e. k∆d k =
6 0. Let rd be a random projection on the d-dimensional unit sphere and rM,d = M−1/2 rd be the misscaled random
projection. Then, there exist for all  > 0 constants c, C > 0, such that
!

tr M−1/2 ΣM−1/2
2
6 C > 1 − ,
P c 6 E1 (∆d , rM,d )
kM−1/2 ∆d k2
where tr denotes the trace.

16

2 Change Points and Projections

1

1

0.8

0.8

0.6

0.4

0.4

0.2

0
0

Oracle
Random
Pre-Oracle
H&H Sigma
H&H Var

0.6

Oracle
Random
Pre-Oracle
H&H Sigma
H&H Var

0.2

0.2

0.4



0.6

0.8

0
0

1

0.2

0.4



0.6

0.8

1

(a) Angle between ∆d and Φ = 0 radians (b) Angle between ∆d and Φ = π/8 radians
1

1

0.8

0.8

0.6

0.6
Oracle
Random
Pre-Oracle
H&H Sigma
H&H Var

0.4

0.2

0
0

0.2

0.4



0.6

0.8

Oracle
Random
Pre-Oracle
H&H Sigma
H&H Var

0.4

0.2

1

0
0

0.2

0.4



0.6

0.8

1

(c) Angle between ∆d and Φ = π/4 radians (d) Angle between ∆d and Φ = π/2 radians

Figure 2.4: Power of tests as the angle between the change and the direction of dependency increases. As can be seen, if the change lies in the direction of
dependency, then all methods struggle, which is in line with the theory of
Section 2.5. However, if the change is orthogonal to the dependency structure the projection method works regardless of whether the dependency is
taken into account or not. H&H Sigma and Var as in Section 3 represent
the panel tests taking into account the true or estimated variances of the
components. All results are empirically size corrected to account for the
size issues seen
√ in Figure 2.1. (sj = 1, Φj = φ, j = 1, . . . , d with d = 200,
k∆d k = 0.05 d, corresponding to Case C.3), with φ as given on the x-axis.

We are now ready to prove the main result of this section stating that the high dimensional efficiency of a misscaled oracle can never be worse than the corresponding
misscaled random projection.
Theorem 2.9. Let Assumption A.1 hold. Denote the misscaled oracle by oM =
M−1 ∆d , then
E12 (∆d , oM ) >

kM−1/2 ∆d k2
tr(M−1/2 ΣM−1/2 )

where tr denotes the trace and equality holds iff there is only one common factor which
is weighted proportional to ∆d ,
Because it is often assumed that components are independent and it is usually feasible to estimate the variances of each component, we consider the correspondingly
misscaled oracles, which are scaled with the identity matrix (pre-oracle) respectively

17

2 Change Points and Projections
with the diagonal matrix of variances (quasi-oracle). The quasi-oracle is of particular
importance as it uses the same type of misspecification as the panel statistic discussed
in Section 3 below.
Definition 2.3.

(i) The projection po = ∆d is called pre-oracle.

2
2 T
2
2
(ii) The projection qo = Λ−1
d ∆d = (δ1 /σ1 , . . . , δd /σd ) , Λd = diag(σ1 , . . . , σd ) is
2
called quasi-oracle, if σj > 0, j = 1, . . . , d.

As with the oracle, these projections should be seen as representatives of a class of
projections.
The following proposition shows that in the important special case of uncorrelated
components, the (quasi-)oracle and pre-oracle have an efficiency of same order if the
variances in all components are bounded and bounded away from zero. The latter
assumption is also needed for the panel statistic below and means that all components
are on similar scales. In addition, the efficiency of the quasi-oracle is even in the
misspecified situation always better than an unscaled random projection.
Proposition 2.10. Assume that all variances are on the same scale, i.e. there exist
c, C such that 0 < c 6 σi2 < C < ∞ for i = 1, . . . , d.
a) Let Σ = diag(σ12 , . . . , σd2 ), then
c2 2
E (∆d , qo) 6 E12 (∆d , po) 6 E12 (∆, qo) = kΣ−1/2 ∆d k2 .
C2 1
b) Under Assumption A.1, it holds
E12 (∆d , qo) >

c2 k∆d k2
.
C 2 tr(Σ)

Figures 2.4 and 2.5 show the results of some small sample simulations of case C.3
confirming again that the theoretical findings based on the high dimensional efficiency
are indeed able to predict the small sample power for the test statistics. Essentially,
the following assertions are confirmed:
1) The power of the pre- and quasi-oracle is always better than the power of the
misscaled random projection (the random projection assumes an identity covariance
structure).
2) The power of the (correctly scaled) oracle can become as bad as the power of the
(misscaled) random projection but only if Φd ∼ ∆d . In this case the power of
the misscaled panel statistic (i.e. where the statistic but not the critical values are
constructed under the wrong assumption of independence between components) is
equally bad.
3) While the power of the (misscaled) panel statistic becomes as bad as the power
of the (misscaled) random projection for φ → ∞ irrespective of the angle between
∆d and Φd , it can be significantly better for the pre- and quasi-oracle. In fact, we
will see in Section 3 that the high dimensional efficiency of the misspecified panel
statistic will be of the same order as a random projection for any choice Φd with
ΦTd Φd ∼ d, irrespective of the direction of any change that might be present.
We will now have a closer look at the three standard examples in order to understand
the behavior in the simulations better (Case C.1 is included in the simulations for
Φ = 0, while C.3 is the limiting case for Φ → ∞).

18

2 Change Points and Projections
Case C. 1 (Independent components). If the components are uncorrelated, each with
variance σi2 , i.e. Σ1 = diag(σ12 , . . . , σd2 ), we get
tr(Σ1 ) =

d
X

σj2 ,

j=1

which is of order d if 0 < c 6 σj2 6 C < ∞. Proposition 2.10, Theorem 2.7 and
Theorem 2.8 show that in this situation
√ both the high dimensional efficiency of the
pre- and (quasi-)oracle are of an order d better than the correctly scaled and unscaled
random projection.
The second case shows that high dimensional efficiency of misscaled oracles can indeed
become as bad as a random projection and helps in the understanding of the mixed
case:
Case C. 2 (Fully dependent components). As already noted in 2.5.2 we have to
distinguish two cases:
(i) If ∆d is not a multiple of Φd , then the power depends on the angle of the
projection with Φd with maximal power for an orthogonal projection. So the
goodness of the oracles depends on their angle with the vector Φd .
(ii) If ∆d is a multiple of Φd , the pre- and quasi-oracle are not orthogonal to the
change, hence they share the same high dimensional efficiency with any scaled
random projection as all random projections are not orthogonal to Φd with probability 1.
We can now turn to the mixed case that is also used in the simulations.
Case C. 3 (Mixed case). Let aj = (0, . . . , sj , . . . , 0)T the vector which is sj > 0 at
point j and zero everywhere else, and ad+1 = Φd = (Φ1 , . . . , Φd )T , aj = 0 for j > d+2.
Then Σ3 = diag(s21 , . . . , s2d ) + Φd ΦTd and
tr(Σ3 ) =

d
X

s2j +

j=1

d
X

Φ2j .

(2.13)

j=1

The high dimensional efficiency of the pre-oracle can become as bad as for the random
projection if the change ∆d is a multiple of the common factor Φd and there is a
substantial common effect. This is similar to Case C.2 (which can be seen as a limiting
case for increasing kΦd k). Intuitively, the problem is the following: By projecting
onto the change, we want to maximize the signal i.e. the change in the projected
sequence while minimizing the noise. In this situation however, the common factor
dominates the noise in the projection as it essentially adds √
up in a linear manner, while
the uncorrelated components add up only in the order of d (CLT). Now, projecting
onto ∆d = Φd maximizes not only the signal but also the noise, which is why we
cannot gain anything (but this also holds true for competing procedures as in Section
3 below).
P
2
Pd
d
More precisely, in C.3 it holds τ 2 ( po) = j=1 s2j δj2 +
δ
Φ
. If additionally
j
j
j=1
∆d = kΦd , for some k > 0, we get the following high dimensional efficiency for the
pre-oracle by (2.11)
E1 (∆d , po) = r

Pd

2
i=1 si

k∆d k

2
δi
k∆d k

.
+ kΦd

19

k2

3 High dimensional efficiency of panel change point test
The high dimensional efficiency for the unscaled random projection is given by (confer
Theorem 2.8 and (2.13))
k∆d k
qP

d
2
j=1 sj

.

+ kΦd k2

As soon as sj , Φj are of the same order, i.e. 0 < c 6 sj , Φj 6 C < ∞ for all j, the
pre-oracle behaves as badly as the unscaled random projection. The same holds for the
quasi-oracle under the same assumptions. Interestingly, however, in this particular
situation, even the oracle is of the same order as the random projection if the sj are
of the same order, i.e. 0 < c 6 sj < C < ∞. More precisely we get (for a proof we
refer to the Section 6)
v
u Pd δ2
j
u
u j=1 s2j
k∆d k
t
.
(2.14)
E1 (∆d , o) = r
Pd
Pd Φ2j
δj2
j=1
1 + j=1 s2
j

Figure 2.4 shows simulations which confirm the underlying theory in finite samples.
On the other hand, if ∆d is orthogonal to Φd , then the noise from Φd cancels for the
pre-oracle projection and we get the rate
E1 (∆d , po) = r

k∆d k

2 ,
δi
2
s
i=1 i k∆d k

Pd

which is of the order k∆d k2 if the sj are all of the same order. Anything between
those two cases is possible and depends on the angle between ∆ and Φd (again see
Figures 2.4 and 2.5 for finite sample simulations).
The following interpretation also explains the above mathematical findings: In situation C.3, each component has a common factor {ηt } weighted according to Φd plus
some independent noise. If a change occurs in sync with the common factor it will be
difficult to detect as in order to get the correct size, we have to allow for the random
movements of {ηt } thus increasing the critical values in that direction. In directions
orthogonal to it, we only have to take the independent noise into account which yields
comparably smaller noise in the projection. In an economic setting, this driving factor
could for example be thought of as an economic factor behind certain companies (e.g.
ones in the same industry). If a change occurs in those companies proportional to this
driving factor it will be difficult to distinguish a different economic state of this driving
factor from a mean change that is proportional to the influence of this factor.

3 High dimensional efficiency of panel change point
test
In this section, we will compare the power of the above projection tests with corresponding CUSUM tests that take the full multivariate information into account. First
statistics of this type were developed for the multivariate setting with d fixed [Horváth
et al., 1999]. The multivariate change point statistic (using the full multivariate information and no additional knowledge about the change) for the at most one mean
change is given as a weighted maximum or sum of the following quadratic form
VdM (x) = ZT (x)T AZT (x)T ,

(3.1)

20

3 High dimensional efficiency of panel change point test

1
0.9
0.8
0.7

1
Oracle
Pre-Oracle
Quasi-Oracle
Random
H&H Sigma
H&H Var

0.9

0.7
0.6
Power

Power

0.6
0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1
0
0.01

Oracle
Pre-Oracle
Quasi-Oracle
Random
H&H Sigma
H&H Var

0.8

0.1
0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0
0.01

0.1

0.02

0.03

0.04

0.05

||

0.06

0.07

0.08

0.09

0.1

||

(a) No Dependency - φ = 0

(b) φ = 0.5

1
0.9
Oracle
Pre-Oracle
Quasi-Oracle
Random
H&H Sigma
H&H Var

0.8
0.7

Power

0.6
0.5
0.4
0.3
0.2
0.1
0
0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

||

(c) φ = 1

Figure 2.5: Power of tests as the dependency increases. The covariance structure becomes closer to degenerate across the three graphs, but in all cases the
pre-oracle and quasi-oracle still outperform random projections, although
they become closer as the degeneracy increases. Here different variances
are used across components, namely si = 0.5 + i/d, Φi = φi , i = 1, . . . , d,
d = 200, angle(Φ, ∆d )=π/4, corresponding
to Case C.3, and size of change
√
as given on the x-axis (multiplied by d).

where ZT (x) = (ZT,1 (x), . . . , ZT,d (x))T is defined as in (2.3). The usual choice is
A = Σ−1 , where Σ is the covariance matrix of the multivariate observations. The
weighting with Σ−1 has the advantages that it (a) leads to a pivotal limit and (b)
the statistic can detect all changes no matter what the direction. The second remains
true for any positive definite matrix A, the first also remains true for lower rank
matrices with a decorrelation property of the errors, where this latter approach is
essentially a projection (into a lower-dimensional space) as discussed in the previous
sections. For an extensive discussion of this issue for the example of changes in the
autoregressive structure of time series we refer to Kirch et al. [2015]. The choice
A = Σ−1 corresponds to the correctly scaled case, while the misscaled case corresponds
to the choice A = M −1 .
However, this multivariate setup is not very suitable for the theoretic power comparison we are interested in because the limit distribution (a sum of d squared Brownian
bridges with covariance matrix Σ1/2 AΣ1/2 ) still depends on d as well as the possible
misspecification. Therefore, a comparison needs to take both the rates, the additive
term and the noise level (which depends also on the misspecification of the covariance)
present in the limit distribution into account. For the panel data settings on the other
hand, where d → ∞, all the information about d is contained only in the rates rather
than the limit distribution as in the previous sections. This makes the results inter-

21

3 High dimensional efficiency of panel change point test
pretable in terms of the high dimensional efficiency. The panel null limit distribution
differs from the one obtained for the projections but they are at least on the same
scale, and not dependent on d nor the covariance structure Σ. Furthermore, the panel
statistic is strongly related to the multivariate statistic so that the same qualitative
statements can be expected, which is confirmed by simulations (results not shown).
We will now introduce the statistic for detecting changes in the mean introduced by
Horváth and Hušková [2012], termed the HH Panel Statistic in this paper. Unlike in
the above theory for projections, it is necessary to assume independence between components. Because the proofs are based on a central limit theorem across components,
they cannot be generalized to uncorrelated (but dependent) data. For this reason, we
cannot easily derive the asymptotic theory after standardization of the data. This is
different from the multivariate situation, where this can easily be achieved.
We are interested in a comparison of the high dimensional efficiency for correctly
specified covariance, i.e. A = Σ−1 , in addition to a comparison in the misspecified case,
A = M −1 . The latter has already been discussed by Horváth and Hušková [2012] to
some extent. To be precise, a common factor is introduced as in C.3 and the limit of the
statistic (with A = Λ−1 ) under the assumption that the components are independent
(i.e. Λ being a diagonal matrix) is considered. Because of the necessity to estimate
the unknown covariance structure for practical purposes, the same qualitative effects
as discussed here can be expected if a statistic and corresponding limit distribution
were available for the covariance matrix Σ.

3.1 Efficiency for panel change point tests for independent panels
The above multivariate statistics have been adapted to the panel data setup under the
assumption of independent components by Bai [2010] for estimation as well as Horváth
and Hušková [2012] for testing. Those statistics are obtained as weighted maxima or
sum of the following (univariate) partial sum process
d

1 X
Vd,T (x) = √
d i=1



1 2
bT xc(T − bT xc)
Z (x) −
σi2 T,i
T2


,

(3.2)

where ZT,i is as in (2.3) and σi2 = var ei,1 .
The following theorem gives a central limit theorem for this partial sum process (under
the null) from which null asymptotics of the corresponding statistics can be derived.
It was proven by Horváth and Hušková [2012, Theorem 1], under somewhat more
general assumptions allowing in particular for time series errors (in the form of linear processes). While this makes estimation of the covariances more difficult and less
precise as long-run covariances need to be estimated, it has no effect on the high dimensional efficiency. Therefore, we will concentrate on the i.i.d. (across time) situation
in this work to keep things simpler purely in terms of the calculations.
Theorem 3.1. Let Model (1.1) hold with {ei,t : i, t} independent (where the important
assumption is the independence across components) such that var ei,t > c > 0 for all
Pd
i and lim supd→∞ d1 i=1 E |ei,t |ν < ∞ for some ν > 4. Furthermore, let Td2 → 0.
Then, it holds under the null hypothesis of no change


x2
D[0,1] √
,
Vd,T (x) −→ 2(1 − x)2 W
(1 − x)2
where W (·) is a standard Wiener process.

22

3 High dimensional efficiency of panel change point test
The following theorem derives the high dimensional efficiency in this setting for HH
Panel statistics such as max06x61 Vd,T (x), which we use in simulations with both
R1
known and estimated standard deviations or 0 Vd,T (x).
Theorem 3.2. Let the assumptions of Theorem 3.1 on the errors be fulfilled, which
implies in particular that Σ = diag(σ12 , . . . , σd2 ). Then, the high dimensional efficiency
of HH Panel statistic tests is given by
E2 (∆d ) =

1
kΣ−1/2 ∆d k.
d1/4

Equivalent assertions to Corollary 2.5 can be obtained analogously.
Comparing this high dimensional efficiency with the ones given in Theorem 2.4, Proposition 2.6 as well as Theorem 2.7, we note that the high dimensional efficiency of the
above HH Panel Statistic is an order d1/4 worse than for the oracle but a d1/4 better
than the scaled random projection (also see Figure 2.3). By Theorem 2.4 we also get
an indication of how wrong our assumption on ∆d can be while still obtaining a better
efficiency than with the full multivariate information. More precisely, the theorems
define a cone around the search direction such that the projection statistic has higher
efficiency than the panel statistic if the true change direction is in this cone. We can
see the finite sample nature of this phenomena in Figure 2.2.

3.2 Efficiency of panel change point tests under dependence
between Components
We now turn again to the misspecified situation, where we use the above statistic in
a situation where components are not uncorrelated. Following Horváth and Hušková
[2012], we consider the mixed case C.3 for illustration. The next proposition derives
the null limit distribution for that special case. It turns out that the limit as well as
convergence rates depend on the strength of the contamination by the common factor.
Lemma 3.3. Let Case C.3 hold with ν > 4, 0 < c 6 si 6 C < ∞ and Φ2i 6 C < ∞
for all i and some constants c, C and consider Vd,T (x) defined as in (3.2), where
σi2 = var ei,1 but the rest of the dependency structure is not taken into account. The
asymptotic behavior of Vd,T (x) then depends on the behavior of
Ad :=

d
X
Φ2
i

i=1

σi2

.

√
a) If Ad / d → 0, then the dependency is negligible, i.e.


x2
D[0,1] √
2
Vd,T (x) −→ 2(1 − x) W
,
(1 − x)2
where W (·) is a standard Wiener process.
√
b) If Ad / d → ξ, 0 < ξ < 1, then


x2
D[0,1] √
Vd,T (x) −→ 2(1 − x)2 W
+ ξ (B 2 (x) − x(1 − x)),
(1 − x)2
where W (·) is a standard Wiener process and B(·) is a standard Brownian bridge.
√
c) If Ad / d → ∞, then
√
d Vd,T (x) D[0,1] 2
−→ B (x) − x(1 − x),
Ad
where {B(x) : 0 6 x 6 1} is a standard Brownian bridge.

23

3 High dimensional efficiency of panel change point test
Because Ad in the above theorem cannot feasibly be estimated, this result cannot be
used to derive critical values for panel test statistics. Consequently, the exact shape
of the limit distribution in the above lemma is not important. However, the lemma
is necessary to derive the high dimensional efficiency of the panel statistics in this
misspecified case. Furthermore, it indicates that using the limit distribution from the
previous section to derive critical values will result in asymptotically wrong sizes if a
stronger contamination by a common factor is present. The simulations in Figure 2.1
also confirm this fact and show that the size distortion can be enormous. It does not
matter whether the variance of the components in the panel statistic takes into account
the dependency or simply uses the noise variance (Figure 2.1(a)), or whether a change
is accounted for or not in the estimation (Figure 2.1(b)-(c)). This illustrates, that
the full panel statistic is very sensitive with respect to deviations from the assumed
underlying covariance structure in terms of size.
In the situation of a) and b) above, the dependency structure introduced by the common factor is still small enough asymptotically to not change the high dimensional
efficiency as given in Theorem 3.2, which is analogous to the proof of Theorem 3.2.
Therefore, we will now concentrate on situation c) in the below proposition, which is
the case where the noise coming from the common factor does not disappear asymptotically.
Theorem
3.4. Let the assumptions of Lemma 3.3 on the errors be fulfilled and
√
Ad / d → ∞, then the corresponding panel statistics have high dimensional efficiency
s


1
1
1
T
√
,..., 2
∆d .
E3 (∆d ) =
∆d diag 2
s1 + Φ21
sd + Φ2d
Ad
The 
next corollary shows
 that the efficiency of the quasi oracle (which is scaled with
1
1
diag s2 +Φ2 , . . . , s2 +Φ2 analogously to the panel statistic) is always at least as good
1
1
d
d
as the efficiency of the panel statistic. Additionally, the efficiency of the panel statistic
becomes as bad as the efficiency of the corresponding (diagonally) scaled random
projection if Ad /d → A > 0, which is typically the case if the dependency is nonsparse and non-negligible.
Corollary 3.5. Let the assumptions of Lemma 3.3 on the errors be fulfilled, then the
following assertions hold:
a) The high dimensional efficiency of the quasi-oracle is always at least as good as
the one of the misspecified panel statistic, i.e. with Σ = diag(σ12 , . . . , σd2 ) + ΦΦT ,
Λd = diag(σ12 , . . . , σd2 ), it holds
E12 (∆d , qo) >

∆Td Λ−1
d ∆d
,
1 + Ad

where equality holds iff ∆d ∼ Φ.
b) If Ad /d → A > 0, then the high dimensional efficiency of the panel statistic is as
bad as a randomly scaled projection, i.e.
E32 (∆d ) =

∆Td Λ−1
d ∆d
(Ad + o(1)).
d

In particular, for Ad /d → A > 0 the efficiency of the misscaled panel statistic is always
as bad as the efficiency of the random projection, this only holds for the misscaled
(quasi-) projection if ∆d ∼ Φ. This effect can be clearly see in Figures 2.4 and 2.5,
where in all cases H&H Sigma refers to the panel statistic using known variance, and
H&H Var uses an estimated variance, showing again that this concept of efficiency is
very well suited to understand the small sample power behavior of the corresponding
statistics.

24

4 Data Example
Multivariate Binary Segmentation using Diagonal Covariance

0

-2
FTSE
NASDAQ
DAX
NIKKEI
Hang Seng
CAC

-4
-6

Fuller Log Returns

Fuller Log Returns

Multivariate Binary Segmentation using Diagonal Covariance

0

-2

-8
-10

FTSE
NASDAQ
DAX
NIKKEI
Hang Seng
CAC

-4
-6
-8
-10

0

50

100

150

200

250

0

50

100

Jan-Nov 2015
Multivariate Binary Segmentation using Full Covariance

0

200

-2

-6

Fuller Log Returns

FTSE
NASDAQ
DAX
NIKKEI
Hang Seng
CAC

-4

-8
-10

FTSE
NASDAQ
DAX
NIKKEI
Hang Seng
CAC

-4
-6
-8
-10

0

50

100

150

200

250

0

50

100

Jan-Nov 2015

150

200

250

2015

Projection Binary Segmentation

0

Projection Binary Segmentation

0
Projection

Projection

-10

Fuller Log Returns

Fuller Log Returns

250

Multivariate Binary Segmentation using Full Covariance

0

-2

Fuller Log Returns

150

2015

-20

-30

-40

-10

-20

-30

-40
0

50

100

150

200

250

0

Jan-Nov 2015

50

100

150

200

250

2015

Figure 4.1: Estimated change point locations for the market indices from binary segmentation based on different test statistics and different spans of data.
First Column: Data from Jan-Nov 2015, Second Column: Data from all
of 2015. First Row: Multivariate statistic with full covariance estimation; Second Row: Multivariate Statistic with Diagonal Variance Estimate;
Third Row: Projection Statistic in direction [1,1,1,1,1,1]. Red vertical lines
indicate changes deemed to be significant at 5% level.

4 Data Example
As an illustrative example which shows the small sample behaviour of the statistics
illustrated above also apply in real data, we examine the stability of change points
detected by different methods in several world stock market indices. More specifically,
the Fuller Log Squared Returns [Fuller, 1996, p 496] of the FTSE, NASDAQ, DAX,
NIKKEI, Hang Seng and CAC 1 indices for the year 2015 were examined for change
points. Tests based on the multivariate statistics using full covariance estimates, a multivariate statistic using only variance estimates (i.e., a diagonal covariance structure),
a projection statistic in the average direction [1,1,1,1,1,1], and a projection statistic in
the direction of European countries vs non-European countries [1,-1,1,-1,-1,1] (orthogonal to the average direction) were carried out. Given the considerable dependence
between the different components, we would expect economies to likely rise and fall
together, justifying the use of the former projection direction. However, we think it
unlikely that there will be changes of the kind that when European markets goes up,
non-European markets go down, and visa versa, so take this projection as an example of direction where no change is likely. It should be noted at this point that the
multivariate statistic treats both of these alternatives as equally likely. As there are
possible multiple changes points in this data, we examine stability by performing binary segmentation using the proposed tests, firstly on data from January to November
2015, and then subsequently adding in the data from December 2015.
As can be seen in Figure 4.1, the multivariate test statistic is considerably less robust
than the average projection based statistic, both to the length of the data, as well as
to the choice of the covariance estimate. The major cause of this instability was that
the CUSUM statistic over time had two peaks, but the location of the maximal peak
differed from one to the other when further data was added. This caused knock-on
1 We

only use a small number of series to allow reliable estimates for the covariance to be used in
the full multivariate statistic.

25

5 Conclusions

Table 4.1: Location, Statistic and p-value for the changes found in the 2015 market
index data. (Limit Distributions: Multivariate: sum of six independent
Brownian Bridges; Projection: Single Brownian Bridge)
Multivariate: Full Covariance
Day
143
169
Statistic Value
6.8541
5.1581
p
0.0012
0.0173
Multivariate: Diagonal Covariance
Day
144
169
Statistic Val
9.9995
11.7030
p-value
< 10−5 < 10−5
Projection: scaled [1 1 1 1 1 1]
Day
12
110
121
144
169
Statistic Value 2.1307 3.5390 2.9518 3.3173
2.0900
p
0.0285 0.0017 0.0057 0.0027
0.0307
effects in the entire binary segmentation. Here, in all cases, independence in time was
assumed as once the changes were accounted for, there was little evidence of temporal
dependence in the data. However, even if time series dependence is accounted for by
using an estimate of the long run covariance in place of the independent covariance
estimate, there is no difference in the qualitative conclusions (although the change
points themselves varied considerably in all cases depending on the parameters chosen
in the long run covariance estimation procedure [Politis, D.N., 2005]). In addition,
the projection estimate was robust to whether the direction was scaled by the full
covariance, the diagonal of the covariance or not scaled at all, as well as to increasing
the length of the data.
The p-values for the changes on full year’s data are given in Table 4.1. While it can be
seen that the projection p-vals are larger for the two common change points than in the
multivariate case, the same changes are detected with all methods. However, additional
changes are found with the projection method, and the p-vals are well below the critical
value of 5%. This shows that having knowledge of the likely direction of change can
allow further changes to be found beyond those in an unrestricted multivariate search.
As expected though, using an unlikely direction does not find change points, with
the hypothesis that there are no changes which affect European markets differently to
non-European markets being accepted (p=0.18).

5 Conclusions
The primary aims of this paper were to introduce a theoretic method to compare
the small sample behavior of different high dimensional tests by asymptotic methods.
Furthermore, projection based statistics are introduced into the analysis of change
points in high dimensions and compared and contrasted with the panel based statistics
that are currently available. The new concept of high dimensional efficiency allows
a comparison of the magnitude of changes that can be detected asymptotically as
the number of dimensions increases. All the tests in the paper are benchmarked
against random projections. Because the space covered by far away angles increases
rapidly with the dimension, the power of these becomes very poor in higher dimensions
rendering random projections useless in practice for detecting change points.
In summary, the following two assertions were proven: First, a suitable projection will
substantially increase the power of detection but at the cost of a loss in power if the
change is at a large angle away from the projection vector. Second, projections are

26

6 Proofs
more robust compared to the panel based statistic with respect to misspecification in
the covariance structure both in terms of size and power.
The panel statistic [Bai, 2010, Horváth and Hušková, 2012] test works well in situations
where the panels are independent across dimension, and there is little to no information
about the direction of the change. However, as soon as dependency is present, the size
properties of these statistics become difficult and their high dimensional efficiencies
mimic those of random projections. Misspecification of the covariance structure can
be problematic for all tests. However, if the direction of the likely change is known, then
it is always preferable to use the corresponding projection (scaled with the assumed
covariance structure), rather than either the panel statistic or a random projection,
regardless of whether the covariance is misspecified or not.
This results in this paper raise many questions for future work. It would be of considerable interest to determine whether projections can be derived using data driven
techniques, such as sparse PCA, for example, and whether such projections would be
better than random projections. Preliminary work suggests that this may be so in
some situations but not others, and a nice procedure by Wang and Samworth [2016]
investigates a related technique. Further many multiple change point procedures use
binary segmentation or related methods to find the multiple change points, so much
of the work here would apply equally in suitably defined sub intervals which are then
assumed to contain at most one change. This was the approach taken in the data
example here. In addition, all the results here have been assessed with respect to
choosing a single projection for the test which is optimal if the direction of the change
is known. However, in some situations only qualitative information is known or several
change scenarios are of interest. Then, it could be very beneficial to determine how
best to combine this information into testing procedures based on several projections,
where a standard subspace approach may not be ideal as the information about the
likely direction of changes is lost. Finally, while the framework in this paper concentrates on tests with a given size, as soon as a-priori information is considered, then it is
natural to ask whether related Bayesian approaches are of use, and indeed quantifying
not only the a-priori direction of change, but also its uncertainty, prior to conducting
the test is a natural line of further research.

6 Proofs
Proof of Theorem 2.1. We need to prove the following functional
Pd central limit
theorem for the triangular array of projected random variables Yt,d = j=1 pj (d)ej,t (d)
given the (possibly random) projection pd = (p1 (d), . . . , pd (d))T :


bT xc
 D[0,1]

X
1
p
Yt,d : 0 6 x 6 1 | pd −→ {W (x) : 0 6 x 6 1}
a.s.,

 T τ 2 (pd )
t=1

(6.1)
where {W (·)} denotes a standard Wiener process.
The proof for tightness is analogous to the one given in Theorem 16.1 of Billingsley
[1968] as it only depends on the independence across time (which also holds conditionally given pd due to the independence of pd and {et (d)}). Similarly, the proof for the
convergence of the finite dimensional distributions follows the proof of Theorem 10.1
in Billingsley [1968], where we need to use the Lindeberg-Levy-version of the univariate central limit theorem for triangular arrays. More precisely, we need to prove the

27

6 Proofs
Lindeberg condition given by
E

2
Y1,d
√
1
| pd
τ 2 (pd ) {Y1,d /τ (pd )> T }

!
→0

a.s.

for any  > 0. The following Lyapunov-type condition implies the above Lindeberg
condition:

 T


ν
ν
pd e1 (d)
Y1,d
| pd = E
| pd = o(T ν/2−1 )
a.s.,
(6.2)
E
τ (pd )
τ (pd )
where ν > 2 as given in the theorem. Let
p̃d = q

pd

,

pTd cov e1 (d)pd

then the above Lyapunov condition is equal to


ν
E p̃Td e1 (d) | pd = o(T ν/2−1 )
a.s.
P
T
In the situation of a) cov e1 (d) =
j>1 aj (d)aj (d) and we get by the Rosenthal
inequality (confer e.g. Lin and Bai [2010, 9.7c])


ν
n
X
E
p̃Td aj (d)ηj,1 (d) | pd 
j=m

6 O(1)

n
X


p̃Td aj (d)

ν

E |ηj,1 (d)|ν + O(1) 

n
X

ν/2
p̃Td aj (d)

2

var ηj,1 (d)

,

j=m

j=m

where the right-hand side is bounded for any m, n with a bound that does not depend
on T or d and converges to zero forPm, n → ∞ as E |ηj (d)|ν 6 C hence var ηj (d) 6 1+C
n
and by definition of p̃d it holds j=m |p̃Td aj (d)|2 6 p̃Td cov e1 (d) p̃d 6 1, hence also
P
n
|p̃Td aj (d)|ν 6 |p̃Td aj (d)|2 and j=m |p̃Td aj (d)|ν 6 1.
Consequently, the infinite series exists in an Lν -sense with the following uniform (in
T and d) moment bound


ν
E p̃Td e1 (d) | pd = O(1) = o(T ν/2−1 )
a.s.
(6.3)
To prove the Lyapunov-condition under the assumptions of b) we use the Jensseninequality which yields
!ν
!
d


X
|p̃i,d |
ν
T
ν
|ei,1 (d)|
| pd
E p̃d e1 (d) | pd = kp̃d k1 E
kp̃d k1
i=1

ν
d
X
|p̃i,d |
kpd k1
 = o(T ν/2−1 )
6 kp̃d kν1
E |ei,1 (d)|ν 6 C  q
a.s.
kp̃
k
T
T
d
1
p cov(e (d))p
i=1
d

1

d

(6.4)

Proof of Lemma 2.2. With the notation of the proof of Theorem 2.1 both estimators (as functions of pd ) fulfill (j = 1, 2)
2
τbj,d,T
(pd )
2
= τbj,d,T
(p̃d ).
τ 2 (pd )

28

6 Proofs
First by the independence across time we get by the van Bahr-Esseen inequality (confer
e.g. Lin and Bai [2010, 9.3 and 9.4]) for some constant C > 0, which may differ from
line to line,
b

X

Epd

ν/2

p̃Td ej (d)

2


−1

6 C(b − a)max(1,ν/4) Epd

2
p̃Td e1 (d) − 1

ν/2

j=a+1



ν
6 C (b − a)max(1,ν/4) max 1, Epd p̃Td e1 (d)

a.s.,
in a),
C(b − a)max(1,ν/4)
ν 
 
6
kp
k
1
d
C(b − a)max(1,ν/4) max 1, √ T
, in b),

(6.5)

pd cov e1 (d)pd

by (6.3) resp. (6.4), where Epd denotes the conditional expectation given pd . An
application of the Markov-inequality now yields for any  > 0


T 

X

1
2
P
p̃Td ej (d) − 1 >  pd 
T j=1
(
C
T −ν/2+max(1,ν/4)
a.s.,
in a),
6 ν/2
C
−ν/2+max(1,ν/4)
ν/2−ν/ min(ν,4)
T
o(T
)
a.s., in b),
ν/2
→0

a.s.

Similar arguments yield


T
X
1
P
p̃T ej (d) >  pd  → 0
T j=1 d

a.s.

2
proving a) and b) for τb1,d,T
(pd ).

From (6.5) it follows by Theorem B.1 resp. B.4 in Kirch [2006]

Epd max

16k6T

6

k 
X

ν/2


2
p̃Td ej (d) − 1

j=1


(4−ν)+ ν

CT max(1,ν/4) (log T ) 2(4−ν)

CT max(1,ν/4) (log T )

→0

(4−ν)+ ν
2(4−ν)

a.s.,
 
max 1, √

in a),
kpd k1
pT
d cov e1 (d)pd

ν 
, in b),

a.s.

An application of the Markov inequality now yields for any  > 0


k 

X

1
2
P  max
( p̃Td ej (d) − 1 >  pd  → 0
a.s.
16k6T T
j=1
By the independence across time it holds

 

T
−k 
 X
 TX






2
2
L
p̃Td ej (d) − 1 : 1 6 k 6 T =
p̃Td ej (d) − 1 : 1 6 k 6 T ,

 

j=1

j=k+1

which implies

1
P  max
16k6T T

T

X




2
p̃Td ej (d) − 1 >  pd  → 0

j=k+1

29

a.s.

6 Proofs
Pk
T
Similar assertions can be obtained along the same lines for max16k6T T1
j=1 p̃d ej (d)
PT
2
T
b2,d,T
(pd ).
as well as max16k6T T1
j=k+1 p̃d ej (d) , which imply the assertion for τ
Proof of Corollary 2.3. By an application of the continuous mapping theorem
and Theorem 2.1 we get the assertions for the truncated maxima resp. the sums over
[τ T, (1−τ )T ] for any τ > 0 towards equivalently truncated limit distributions. Because
we assume independence across time (with existing second moments) the Hájek-Rényi
inequality yields for all  > 0
!
k
X
T
P
p̃d et (d) >  pd → 0 a.s.
max w(k/T )
16k6τ T

P

t=1

max

w(k/T )

(1−τ )T 6k6

T
X

!
p̃Td et (d)

>  pd

→0

a.s.

t=k+1

as τ → 0 uniformly in T , where the notation of the proof of Theorem 2.1 has been
used. This in addition to an equivalent argument for the limit process shows that the
truncation is asymptotically negligible proving the desired results.
√
Proof of Theorem 2.4. We consider the situation where T E1 (∆d , pd ) converges
a.s. Under alternatives it holds


bT xc
T
X
X
√
Ud,T (x; e)
bT
xc
1
Ud,T (x)
=
+ sgn(∆Td pd ) T E1 (∆d , pd ) 
g(i/T ) −
g(j/T ) ,
τ (pd )
τ (pd )
T i=1
T 2 j=1
where Ud,T (x; e) is the corresponding functional of the error process. By Theorem 2.1
it holds


Ud,T (x; e)
D[0,1]
: 0 6 x 6 1 | pd −→ {B(x) : 0 6 x 6 1}
a.s.
τ (pd )
Furthermore, by the Riemann-integrability of g(·) it follows
sup
06x61

Z x

Z 1
bT xc
T
1 X
bT xc X
g(i/T ) −
g(j/T
)
−
g(t)
dt
−
x
g(t)
dt
→ 0.
T i=1
T 2 j=1
0
0

For any τ > 0
max

w2 (k/T )

τ 6k/T 61−τ

=T

E12 (∆d , pd )

2
Ud,T
(k/T )
τ 2 (pd )

sup
τ 6x61−τ

2

Z

x

Z
g(t) dt − x

w (x)
0

0

1

!
2
g(t) dt + oPpd (1)

a.s.,

where Ppd denotes the conditional probability given pd . Because by assumption
R
2
R1
x
supτ 6x61−τ w2 (x) 0 g(t) dt − x 0 g(t) dt > 0 for some τ > 0, so that the above
term becomes unbounded asymptotically. This gives the assertion for the max statistics, similar arguments give the assertion for the sum statistic.
Proof of Corollary 2.5. Similarly to the proof of Theorem 2.4 it follows (where the
uniformity at 0 and 1 follows by the assumptions on the rate of divergence for w(·) at
0 or 1)
sup w2 (x)
0<x<1

2
Ud,T
(x)
2
− ((x − ϑ)+ − x(1 − ϑ)) = oPpd (1)
2
2
τ (pd )T E1 (∆d , pd )

30

a.s.,

6 Proofs
which implies the assertion by standard arguments on noting that
ϑbT = arg max w2 (x)
06x61

2
Ud,T
(x)
,
τ 2 (pd )T E12 (∆d , pd )

2

ϑ = arg max w2 (x) ((x − ϑ)+ − x(1 − ϑ)) .
06x61

Proof of Proposition 2.6. The assertion follows from
τ 2 (pd ) = pTd Σpd = kΣ1/2 pd k2 ,
|h∆d , pd i| = (Σ−1/2 ∆d )T (Σ1/2 pd ) = kΣ−1/2 ∆d k kΣ1/2 pd k cos(αΣ−1/2 ∆d ,Σ1/2 pd ).

Proof of Theorem 2.7. Let X d = (X1 , . . . , Xd )T be N(0,Id ), then by Marsaglia
L
[1972] it holds rd = (X1 , . . . , Xd )T /k(X1 , . . . , Xd )T k and it follows by (2.11)

E12 (∆d , Σ−1/2 rd )

d
kΣ−1/2 ∆d k2

L

=

−1/2
XT
∆d
dΣ
kΣ−1/2 ∆d k

2

XT
d Xd
E XT
d Xd

Since the numerator has a χ21 distribution (not depending on d), there exist for any
 > 0 constants 0 < c1 < C1 < ∞ such that


2
T −1/2
Xd Σ
∆d
sup P c1 6
6 C1  > 1 − .
kΣ−1/2 ∆d k
d>1
Furthermore, the denominator has a χ2d -distribution divided by its expectation, consequently an application of the Markov-inequality yields for any  > 0 the existence of
0 < C2 < ∞ such that
!
X Td X d
sup P
> C2 6 .
E X Td X d
d>1

−1
By integration by parts we get E X Td X d
6 2/d for d > 3 so that another application of the Markov-inequality yields that for any  > 0 there exists c2 > 0 such
that
!
X Td X d
lim sup P
6 c2 6 ,
E X Td X d
d→∞
completing the proof of the theorem by standard arguments.
Proof of Theorem 2.8. Let X d = (X1 , . . . , Xd )T be N(0,Id ), then as in the proof
of Theorem 2.7 it holds
tr(M−1/2 ΣM−1/2 ) L
=
E12 (∆, M−1/2 rd )
kM−1/2 ∆d k2

−1/2
XT
∆d
dM
kM−1/2 ∆d k

2

−1/2 ΣM−1/2 X
XT
d
dM
tr(M−1/2 ΣM−1/2 )

.

The proof of the lower bound is analogous to the proof of Theorem 2.7 by noting that
(A = M−1/2 ΣM−1/2 )
E XT AX = E

d
X
i,j=1

ai,j Xi Xj =

d
X

ai,j δi,j =

i,j=1

d
X
i=1

31

ai,i = tr(A).

6 Proofs
For the proof of the upper bound, first note that by a spectral decomposition it holds
d

XT M−1/2 ΣM−1/2 X L X
=
αj Xj2 ,
tr(M−1/2 ΣM−1/2 )
j=1

for some 0 < αd 6 . . . 6 α1 ,

d
X

αj = 1.

j=1

From this we get on the one hand by the Markov inequality


 1/4
d
X
c
E(|X12 |−1/4 ),
P
αj Xj2 6 c 6 P (α1 X12 6 c) 6
α
1
j=1
√
where E(|X12 |−1/4 ) = Γ(1/4)/(21/4 π) exists (as can be seen using the density for a
χ21 -distribution). On the other hand it holds for any c 6 1/2 by another application
of the Markov inequality




d
d
d
X
X
X
P
αj Xj2 6 c 6 P 
αj Xj2 − 1 > 1/2 6 8
αi2 6 8α1 .
j=1

j=1

i=1

By chosing c = min(1/2, (E(|X12 |−1/4 ))−4 /8 5 ) we finally get


d
X
supP
P
αj Xj2 6 c
d
i=1

0<αd 6...6α1 ,

αi =1

j=1


sup

6

min 

P
0<αd 6...6α1 , d
i=1 αi =1


8α1

!

1/4
, 8α1

6 ,

completing the proof.
Proof of Theorem 2.9. By the Cauchy-Schwarz inequality
X
X
X
aj aTj M−1 ∆d =
(aTj M−1 ∆d )2 6
aTj M−1 aj ∆Td M−1 ∆d
τ 2 (M−1 ∆d ) = ∆Td M−1
j>1

j>1



j>1



= tr M−1/2

X

aj aTj M−1/2  ∆Td M−1 ∆d ,

j>1

which implies the assertion by (2.11).
Proof of Proposition 2.10. Assertion a) follows from
|h∆d , poi|2 =

d
2
X
δi,T
i=1

T

τ 2 ( po) = po Σ po =

σi2

!2
σi2

i=1

d
2
X
δi,T
i=1

> c2

d
2
X
δi,T

σi2

!2
2
= c2 |h∆d , qoi| ,

σi2

σi4 6 C 2 |h∆d , qoi| .

Concerning b) first note that by the Cauchy-Schwarz inequality with Λ = diag(σ12 , . . . , σd2 )
τ 2 ( qo) =

X

(∆Td Λ−1 aj )2 6 ∆Td Λ−2 ∆d

j>1

X
j>1

This implies assertion b) by (2.11) on noting that
|∆Td Λ−1 ∆d |2 >

|∆Td ∆d |2
.
C2

32

aTj aj 6

∆Td ∆d
tr(Σ).
c2

6 Proofs

Proof of Equation 2.14. By Proposition 2.6 it holds for ∆d = k Φd
E12 (∆d , o) = kΣ−1/2 ∆d k2 = ∆Td (D + Φd ΦTd )−1 ∆d ,
where D = diag(s21 , . . . , s2d )T . Hence
−1

D −1/2 ∆d
∆Td (D + Φd ΦTd )−1 ∆d = (D −1/2 ∆d )T Id + (D −1/2 Φd )(D −1/2 Φd )T
=

(D −1/2 ∆d )T D −1/2 ∆d
1 + D −1/2 ΦTd D −1/2 Φd

,

where the last line follows from the fact that D −1/2 ∆d = kD −1/2 Φd is an eigenvector
of Id + (D −1/2 Φd )(D −1/2 Φd )T with eigenvalue 1 + (D −1/2 Φd )T D −1/2 Φd hence also
for the inverse of the matrix with inverse eigenvalue.
Proof of Theorem 3.2. Similarly as in the proof of Theorem 2.4 it holds


bT xc
T
X
X
√
1
bT
xc
g(j/T ) +
g(j/T ) ,
ZT,i (x) = ZT,i (x; e) + δi,T T 
T j=1
T 2 j=1
where ZT,i (x; e) is the corresponding functional for the error sequence (rather than
the actual observations). From this it follows
Z x

Z 1
2
Vd,T (x) = Vd,T (x; e) + T E2 (∆d )
g(t) dt − x
g(t) dt + o(1) + RT (x),
0

0

where RT (x) is the mixed term given by
√
Z x

Z 1
d
2 T X δi,T
Z
(x;
e)
g(t)
dt
−
x
RT (x) = √
g(t)
dt
+
o(1)
T,i
d i=1 σi2
0
0
which by an application of the Hájek -Rényi inequality (across time) yields

P

d


sup |RT (x)| > c
06x61

= O (1)

1
1 1 X δi2
T
= OP (1) √ T E2 (∆d ).
2
2
2
c d i=1 σi
c d

From this the assertion follows by an application of Theorem 3.1.
Proof of Lemma 3.3. The proof follows closely the proof of (28) – (30) in Horváth
and Hušková [2012] but where we scale diagonally with the true variances. We will give
a short sketch for the sake of completeness. The key is the following decomposition
Vd,T (x)



2
bT xc
d
T
2
X
X
X
1
bT xc
bT xc (T − bT xc) 
1
 si
=√
ηi,t (d) −
ηi,T (d) −
 2

2 T
s
+
Φ
T
T2
d i=1
i
i
t=1
t=1





bT xc
bT xc
d
T
T
X
X
X
X
X
Φi si
1
bT xc
1
bT xc
2
√ 
+√ 
ηi,t (d) −
ηi,T (d) √ 
ηd+1,t (d) −
ηd+1,t (d)
2 + Φ2
s
T
T t=1
T
T
d i=1 i
i
t=1
t=1
t=1

2
bT xc
T
1
1 X
bT xc X
+
ηd+1,t (d) −
ηd+1,t (d) √ Ad .
T
T t=1
d
t=1

33

6 Proofs
The first term converges to the limit given in a). To see this, note that the proof of
the Lyapunov condition in Horváth and Hušková [2012] following equation (39) still
holds because s2i /(s2i + Φ2i ) is uniformly bounded from above by assumption (showing
that the numerator is bounded) while again by assumption
d

1X
s4i
> D > 0,
d i=1 (s2i + φ2i )2
showing that the denominator is bounded. Similarly, the proof of tightness in Horváth
and Hušková [2012] (equations (43) and following) remains valid. The asymptotic
variance remains the same under a) and b) because by assumption
d

3
1X
s4i
− 1 6 Ad → 0.
d i=1 (s2i + Φ2i )2
d
The middle term in the above decomposition is bounded by an application of the Hájek
-Rényi inequality




bT xc
d
T
X
X
X
Φ
s
1
bT
xc
1
i i
√ 
ηi,t (d) −
ηi,T (d) > D
P  sup √
T
0<x<1
d i=1 s2i + Φ2i T
t=1
t=1
d

= O(1)

1X
φ2i s2i
1
= O(1) Ad ,
2
d j=1 (si + φ2i )2
d

√
which converges to 0 for a) and b) – for c) we multiply the original statistic by d/Ad ,
which means this term is multiplied with d/A2d leaving us with 1/Ad which converges to
√
PbT xc
bT xc PT
0 if Ad / d → ∞. Similarly, we can bound √1T
t=1 ηd+1,t (d) ,
t=1 ηd+1,t (d) − T
showing that the middle term is asymptotically negligible. The assertions now follow
by an application of the functional central limit theorem for
P
2
bT xc
bT xc PT
1
η
(d)
−
η
(d)
.
d+1,t
d+1,t
t=1
t=1
T
T
Proof of Theorem
3.4. The proof is analogous to the one of Theorem 3.2 on noting
√
that E32 (∆d ) = Add E22 (∆d ) and σi2 = s2i +Φ2i by using Lemma 3.3 c) above. Concerning
eT (x) note that ei,t = si ηi,t +Φi ηd+1,t , so that the remainder term
the remainder term R
can be split into two terms. The first
term can be dealt
q
 with analogously to the proof
1
of Theorem 3.2 and is of order OP
Ad T E3 (∆d ) , while for the second summand
we get by an application of the Cauchy-Schwarz-inequality
v


u Pd δ2
bT xc
i
d
T
X
X
X
√ u
t i=1 σi2
1
δ i φi 
bT xc
 = OP ( T )
sup
η
−
η
d+1,t
d+1,t
2
T t=1
Ad
06x61 Ad i=1 σi
t=1
q

=O
T E32 (∆d ) .

Proof of Corollary 3.5.
holds
−1
∆Td Λ−1
d ΣΛd ∆d

=

By an application of the Cauchy-Schwarz inequality it

d
X

2
δi,T

i=1

6

d
2
X
δi,T
i=1

σi2

1+

d
X
Φ2
i

i=1

σi2

s2i
+
(s2i + Φ2i )2

d
X
δi,T Φi
s2 + Φ2i
i=1 i

!
= ∆Td Λ−1
d ∆d (1 + Ad ),

34

!2

References
which implies assertion a) on noting that
E12 (∆d , qo) =

2
(∆Td Λ−1
d ∆d )
.
−1
∆Td Λ−1
d ΣΛd ∆d

b) This follows immediately from Theorem 2.8 since by 0 < c 6 s2j 6 C < ∞ as well
as as Φ2i 6 C, it follows that


1
1
,
.
.
.
,
∆d .
k∆d k2 ∼ ∆Td diag 2
s1 + Φ21
s2d + Φ2d

Acknowledgements
The first author was supported by the Engineering and Physical Sciences Research
Council (UK) grants : EP/K021672/2 & EP/N031938/1. Some of this work was
done while the second author was at KIT where her position was financed by the
Stifterverband für die Deutsche Wissenschaft by funds of the Claussen-Simon-trust.
Furthermore, this work was supported by the Ministry of Science, Research and Arts,
Baden-Württemberg, Germany. Finally, the authors would like to thank the Isaac
Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality
during the programme ’Inference for Change-Point and Related Processes’, where part
of the work on this paper was undertaken.

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