NON-ASYMPTOTIC RESULTS FOR CORNISH-FISHER
EXPANSIONS

arXiv:1604.00539v1 [] 2 Apr 2016

V.V. ULYANOV, M. AOSHIMA, AND Y. FUJIKOSHI
Abstract. We get the computable error bounds for generalized
Cornish-Fisher expansions for quantiles of statistics provided that
the computable error bounds for Edgeworth-Chebyshev type expansions for distributions of these statistics are known. The results
are illustrated by examples.

1. Introduction and main results
In statistical inference it is of fundamental importance to obtain
the sampling distribution of statistics. However, we often encounter
situations, where the exact distribution cannot be obtained in closed
form, or even if it is obtained, it might be of little use because of its
complexity. One practical way of getting around the problem is to
provide reasonable approximations of the distribution function and its
quantiles, along with extra information on their possible errors. It can
be made with help of Edgeworth–Chebyshev and Cornish–Fisher type
expansions. Recently the interest for Cornish–Fisher type expansions
stirred up because of intensive study of VaR (Value at Risk) models
in financial mathematics and financial risk management (see, e.g. [14]
and [15]).
Mainly, it is studied the asymptotic behavior of the expansions mentioned above. It means that accuracy of approximation for distribution
of statistics or its quantiles is given as O(·), that is in the form of order
with respect to some parameter(s) (usually, w.r.t. n as a number of
observations and/or p as dimension of observations). In this paper we
construct non-asymptotic error bounds, in other words – computable
error bounds, for Cornish–Fisher type expansions, that is for an error of approximation we prove upper bounds with closed-form dependence on n and/or p and, perhaps, on some moment characteristics of
observations. We get these bounds under condition that similar nonasymptotic results are already known for accuracy of approximation of
distributions of statistics by Edgeworth–Chebyshev type expansions.
Let X be a univariate random variable with a continuous distribution
function F . For α : 0 < α < 1, there exists x such that F (x) = α,
Key words and phrases. computable bounds, non-asymptotic results, CornishFisher expansions.
This work was supported by RSCF grant No. 141100364.
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V.V. ULYANOV, M. AOSHIMA, AND Y. FUJIKOSHI

which is called the (lower) 100α% point of F . If F is strictly increasing,
the inverse function F −1 (·) is well defined and the 100α% point is
uniquely determined. We also speak of “quantiles” without reference
to particular values of α meaning the values given by F −1 (·). Even
in the general case, when F (x) is not necessarily continuous nor is it
strictly increasing, we can define its inverse function by formula
F −1 (u) = inf{x; F (x) > u}.
This is a right-continuous nondecreasing function defined on the interval (0, 1) and F (x0 ) ≥ u0 if x0 = F −1 (u0 ).
Let Fn (x) be a sequence of distribution functions and let each Fn
admit the Edgeworth-Chebyshev type expansion (ECE) in the powers
of ǫ = n−1/2 or n−1 :
(1)

Fn (x) = Gk,n (x) + Rk (x) with Rk (x) = O(ǫk ) and

Gk,n (x) = G(x) + ǫa1 (x) + · · · + ǫk−1 ak−1 (x) g(x),

where g(x) is a density function of the limiting distribution function
G(x). An important approach to the problem of approximating the
quantiles of Fn is to use their asymptotic relation to those of G’s. Let
x and u be the corresponding quantiles of Fn and G, respectively. Then
we have
(2)

Fn (x) = G(u).

Write x(u) and u(x) to denote the solutions of (2) for x in terms of u
and u in terms of x, respectively [i.e. u(x) = G−1 (Fn (x)) and x(u) =
Fn−1 (G(u))]. Then we can use the ECE (1) to obtain formal solutions
x(u) and u(x) in the form
(3)
and

x(u) = u + ǫb1 (u) + ǫ2 b2 (u) + · · ·

(4)

u(x) = x + ǫc1 (x) + ǫ2 c2 (x) + · · · .

Cornish and Fisher in [3] and [6] obtained the first few terms of these
expansions when G is the standard normal distribution function (i.e.,
G = Φ). Both (3) and (4) are called the Cornish–Fisher expansions,
(CFE). Concerning CFE for random variables obeying limit laws from
the family of Pearson distributions see, e.g., [1]. Hill and Davis in [13]
gave a general algorithm for obtaining each term of CFE when G is an
analytic function.
Usually the CFE are applied in the following form with k = 1, 2 or
3:
(5)

xk (u) = u +

k−1
X
j=1

ǫj bj (u) + R̂k (u) with R̂k (u) = O(ǫk ).

NON-ASYMPTOTIC RESULTS FOR CORNISH-FISHER EXPANSIONS

3

It is known (see, e.g., [15]) how to find the explicit expressions for b1 (u)
and b2 (u) as soon as we have (1). By Taylor’s expansions for G, g, and
a1 , we obtain
b1 = −a1 (u),
(6)
1
b2 = {g ′(u)/g(u)}a21(u) − a2 (u) + a′1 (u)a1 (u),
2
provided that g and a1 are smooth enough functions.
In the following Theorems we show how xk (u) from (5) could be
expressed in terms of u. Moreover, we show what kind of bounds we
can get for R̂k (x) as soon as we have some bounds for Rk (x) from (1).
Theorem 1. Suppose that for the distribution function of a statistic
U we have
(7)

F (x) ≡ Pr{U ≤ x} = G(x) + R1 (x),

where for remainder term R1 (x) there exists a constant c1 such that
|R1 (x)| ≤ d1 ≡ c1 ǫ.

Let xα and uα be the upper 100α% points of F and G respectively, that
is
(8)

Pr{U ≤ xα } = G(uα ) = 1 − α.

Then for any α such that 1 − c1 ǫ > α > c1 ǫ > 0 we have
(i): uα+d1 ≤ xα ≤ uα−d1 .
(ii): |xα − uα | ≤ c1 ǫ/g(u(1) ), where g is the density function of
the limiting distribution G and
g(u(1) ) =
min
g(u).
u∈[uα+d1 ,uα−d1 ]

Theorem 2. In the notation of Theorem 1 we assume that
F (x) ≡ Pr{U ≤ x} = G(x) + ǫg(x)a(x) + R2 (x),

where for remainder term R2 (x) there exists a constant c2 such that
|R2 (x)| ≤ d2 ≡ c2 ǫ2 .

Let T = T (u) be a monotone increasing transform such that
Pr{T (U) ≤ x} = G(x) + R̃2 (x) with |R̃2 (x)| ≤ d˜2 ≡ c̃2 ǫ2 .

Let x̃α and uα be the upper 100α% points of Pr{T (U) ≤ x} and G,
respectively. Then for any α such that
we have
(9)
where

1 − c̃2 ǫ2 > α > c̃2 ǫ2 > 0,
|x̃α − uα | ≤ c̃2 ǫ2 /g(u(2) ),
g(u(2) ) =

min

u∈[uα+d̃ ,uα−d̃ ]
2

2

g(u).

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V.V. ULYANOV, M. AOSHIMA, AND Y. FUJIKOSHI

Theorem 3. We use the notation of Theorem 2. Let b(x) be a function
inverse to T , i.e. b(T (x)) = x. Then xα = b(x̃α ) and for α such that
1 − c̃2 ǫ2 > α > c̃2 ǫ2 we have
|b′ (u∗ )| 2
(10)
|xα − b(uα )| ≤ c̃2
ǫ,
g(u(2) )
where
|b′ (u∗ )| =

max

u∈[uα+d̃ ,uα−d̃ ]
2

2

|b′ (u)|.

Moreover,
(11)

b(x) = x − ǫa(x) + O(ǫ2 ).

Remark 1. The main assumption of the Theorems is that for distributions of statistics and for distributions of transformed statistics we
have some approximations with computable error bounds. There are
not many papers with this kind of non-asymptotic results because it
requires technique which is different from the asymptotic results methods (cf., e.g., [10] and [20]). In series of papers [7], [8], [10], [11], [2],
[16], [18], [19] we got non-asymptotic results for wide class of statistics
including multivariate scale mixtures and MANOVA tests. We considered as well the case of high dimensions, that is the case when the
dimension of observations and sample size are comparable. The results
were included in the book [9]. See also [5].
Remark 2. The results of Theorems 1–3 could not be extended
to the whole range of α ∈ (0, 1). It follows from the fact that the
Cornish-Fisher expansion does not converge uniformly in 0 < α < 1.
See corresponding example in Section 2.5 of [12].
Remark 3. In Theorem 2 we required the existence of a monotone
increasing transform T (z) such that distribution of transformed statistic T (U) is approximated by some limit distribution G(x) in better way
than the distribution of original statistic U. We call this transformation T (z) the Bartlett type correction. See corresponding examples in
Section 3.
Remark 4. According to (10) and (11) the function b(uα ) in Theorem 3 could be considered as an ”asymptotic expansion” for xα up to
order O(ǫ2 ).
2. Proofs of main results
Proof of Theorem 1. By the mean value theorem,
|G(xα ) − G(uα )| ≥ |xα − uα | min g(uα + θ(xα − uα )).
0<θ<1

From (7) and the definition of xα and uα in (8), we get
|G(xα ) − G(uα )| = |G(xα ) − Pr{U ≤ xα }|
= |R1 (xα )| ≤ d1 .

NON-ASYMPTOTIC RESULTS FOR CORNISH-FISHER EXPANSIONS

5

Therefore,
(12)

|xα − uα | ≤

d1
.
min0<θ<1 g(uα + θ(xα − uα )

On the other hand, it follows from (7) that
G(xα ) = 1 − α − R1 (α)

≤ 1 − (α − d1 ) = G(uα−d1 ).

This implies that xα ≤ uα−d1 . Similarly, we have uα+d1 ≤ xα . Therefore, we proved Theorem 1 (i).
It follows from Theorem 1 (i) that
min

u∈[uα+d1 ,uα−d1 ]

g(u) ≤ min g(uα + θ(xα − uα )).
0<θ<1

Thus, using (12) we get statement of Theorem 1 (ii).
Proof of Theorem 2. It is easy to see that it is sufficient to apply Theorem 1 (ii) to the transformed statistic T (U).
Proof of Theorem 3.
we obtain
(13)

Using now (9) and the mean value theorem


x̃α − uα = b−1 (xα ) − b−1 (b(uα )) = (b−1 )′ (x∗ ) xα − b(uα ) ,


where x∗ is a point on the interval min{xα , b(uα )} , max{xα , b(uα )} .
By Theorem 1 (i) we have
uα+d˜2 ≤ x̃α ≤ uα−d˜2 .

Therefore, for xα = b(x̃α ) we get
(14)



min{b−1 (xα ), uα } , max{b−1 (xα ), uα } ⊆ uα+d˜2 , uα−d˜2 .

Since by properties of derivatives of inverse functions
(b−1 )′ (z) = 1/ b′ (b−1 (z)) = 1/b′ (y)

for z = b(y), the relations (13) and (14) imply (10).
Representation (11) for b(x) follows from (6) and (10).
3. Examples
In [17] we gave sufficient conditions for transformation T (x) to be
the Bartlett type correction (see Remark 3 above) for wide class of
statistics U allowing the following represantion
(15)

Pr{U ≤ x} = Gq (x) +

k
1 X
aj Gq+2j (x) + R2k ,
n j=0

where R2k = O(n−2 ) and Gq (x) is the distribution function of chisquared distribution with q degrees of freedom and coefficients aj ’s
P
satisfy the relation kj=0 aj = 0. Some examples of the statistic U are

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V.V. ULYANOV, M. AOSHIMA, AND Y. FUJIKOSHI

as follows: for k = 1, the likelihood ratio test statistic; for k = 2, the
Lawley-Hotelling trace criterion and the Bartlett-Nanda-Pillai trace
criterion, which are test statistics for multivariate linear hypothesis
under normality; for k = 3, the score test statistic and Hotelling’s T 2 statistic under nonnormality. The results of [17] were extended in [4]
and [5].
In [10] we were interested in the null distribution of Hotelling’s generalized T02 statistic defined by
T02 = n trSh Se−1 ,
where Sh and Se are independently distributed as Wishart distributions
Wp (q, Ip ) and Wp (n, Ip ) with identity operator Ip in Rp , respectively.
In Theorem 4.1 (ii) in [10] we proved (15) for all n ≥ p with k = 3 and
computable error bound:
r
|Pr(T02 ≤ x) − Gr (x) −
{(q − p − 1)Gr (x)
4n
−2qGr+2 (x) + (q + p + 1)Gr+4 (x)}|
cp,q
≤ 2,
n
where r = pq and for constant cp,q we gave expicit formula with dependence on p and q.
Therefore, according to [17] we can take in this case the Bartlett
type correction T (z) as
s
2
z
a−1
a−1
+
T (z) =
+ ,
2b
2b
b
where
a=

1
p(q − p − 1),
2n

1
p(q + p + 1)(q + 2) −1.
2n
It is clear that T (z) is invertable and we can apply Theorem 3.
Other examples and numerical calculations and comparisons of approximation accuracy see in [4] and [5].
One more example is connected with sample correlation coefficient.
~ = (X1 , ..., Xn )T , and Y~ = (Y1 , ..., Yn )T be two vectors from an
Let X
n-dimensional normal distribution N(0, In ) with zero mean, identity
covariance matrix In and the sample correlation coefficient
Pn
Xk Y k
~
~
.
R = R(X, Y ) = pPn k=1 2 Pn
2
k=1 Yk
k=1 Xk
b=

In [2] it was proved for n ≥ 7 and N = n − 2.5:
√

Bn
x3 ϕ(x)
supx Pr
≤ 2,
N R ≤ x − Φ(x) −
4N
N

NON-ASYMPTOTIC RESULTS FOR CORNISH-FISHER EXPANSIONS

7

with Bn ≤ 2.2. It is easy to see that we can take T (z) as the Bartlett
type correction in the form T (z) = z + z 3 /(4N). Then the inverse
function b(z) = T −1 (z) is defined by formula
1/3

p
2
3
b(z) = 2 N z + (2Nz) + (4N/3)
1/3

p
− − 2 N z + (2Nz)2 + (4N/3)3
z3
3 z5
+
+ O(N −3 ).
4N
16 N 2
Now we can apply Theorem 3.
= z−

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V.V. Ulyanov, Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119991, Russia, and
National Research University Higher School of Economics (HSE),
Moscow, 101000, Russia
E-mail address: vulyanov@cs.msu.ru
M. Aoshima, Institute of Mathematics, University of Tsukuba,
Tsukuba, Ibaraki 305-8571, Japan
E-mail address: aoshima@math.tsukuba.ac.jp
Y. Fujikoshi, Department of Mathematics, Hiroshima University,
Higashi-Hiroshima, 739-8526, Japan
E-mail address: fujikoshi y@yahoo.co.jp

