Robust Wald-type tests for non-homogeneous
observations based on minimum density power
divergence estimator

arXiv:1707.02333v1 [stat.ME] 7 Jul 2017

Ayanendranath Basu1∗ , Abhik Ghosh1 , Nirian Martin2 and Leandro Pardo2
1
Indian Statistical Institute, Kolkata, India
2
Complutense University, Madrid, Spain
∗
Corresponding author; Email: ayanbasu@isical.ac.in
Abstract
This paper considers the problem of robust hypothesis testing under non-identically distributed data. We
propose Wald-type tests for both simple and composite hypothesis for independent but non-homogeneous
observations based on the robust minimum density power divergence estimator of the common underlying
parameter. Asymptotic and theoretical robustness properties of the proposed tests have been discussed.
Application to the problem of testing the general linear hypothesis in a generalized linear model with fixeddesign has been considered in detail with specific illustrations for its special cases under normal and Poisson
distributions.

Keywords: Non-homogeneous Data; Robust Hypothesis Testing; Wald-Type Test; Minimum Density Power
Divergence Estimator; Power Influence Function; Linear Regression; Poisson Regression.

1

Introduction

Suppose that the parametric model for the sample Y1 , ..., Yn asserts that the distribution of Yi is Fi,θ , θ ∈ Θ
⊂ Rp . We shall denote by fi,θ the probability density function associated with Fi,θ with respect to a convenient
σ-finite measure, for i = 1, ..., n. This situation is very common for statistical modeling in many applications
and an important example is the generalized linear model (GLM) with fixed design set-up.
The maximum likelihood score equation for these independently and non identically distributed data,
Y1 , ..., Yn , is given by
n
X
ui,θ (yi ) = 0,
(1)
i=1
∂
∂θ

b obtained as the
with ui,θ (yi ) =
log fi,θ (yi ). It is well-known that the maximum likelihood estimator, θ,
solution of the system of equation (1), has serious problems of robustness. For this reason statistical solutions
for several special cases like linear regression with normal errors (Huber, 1983; Muller, 1998) as well as some
general cases (Beran, 1982; Ghosh and Basu, 2013) have been considered in the literature. In this paper, we
shall follow the approach presented by Ghosh and Basu (2013) in order to propose some robust Wald-type
tests. In the cited approach given by Ghosh and Basu (2013) a robust estimator was introduced based on the
density power divergence (DPD) measure; details about this family of divergence measures can be found in
Basu et al. (1998, 2011). This estimator, called the minimum density power divergence estimator (MDPDE)
for non-homogeneous observations, is obtained as the solution of the system of equations

Z
n 
X
τ +1
τ
fi,θ (Yi )ui,θ (Yi ) − fi,θ (y)ui,θ (y) dy = 0, τ > 0.

(2)

i=1

Note that in the limit as τ → 0, the system of equations given in (2) tends to the system given in (1).
In Ghosh and Basu (2013) it was established under some standard regularity conditions, that the asymptotic
bτ , at the true model distribution {fi,θ :
distribution of the MDPDE for non-homogeneous observations, say θ
0

1

i = 1, . . . , n}, is given by
Ω−1/2
n,τ (θ 0 )Ψn,τ (θ 0 )
or equivalently,

√

h√

i
L
bτ − θ 0 ) −→
n(θ
N (0, I p ) ,
n→∞

L

bτ − θ 0 ) −→ N (0, Στ (θ 0 )) ,
n(θ
n→∞

(3)

(4)

with
−1
Στ (θ 0 ) = lim Ψ−1
n,τ (θ 0 )Ωn,τ (θ 0 )Ψn,τ (θ 0 ),
n→∞

(5)

where we define
n

1X
J i,τ (θ),
n i=1
Z
τ +1
J i,τ (θ) = ui,θ (y) uTi,θ (y) fi,θ
(y)dy,

Ψn,τ (θ) =
with

(6)

and

n Z
1X
2τ +1
ui,θ (y) uTi,θ (y) fi,θ
(yi )dy − ξ i,τ (θ)ξ Ti,τ (θ) ,
n
Z i=1
τ
ξ i,τ (θ) = ui,θ (y) fi,θ
(y)dy.

Ωn,τ (θ) =
with

(7)
(8)

The required regularity conditions are listed in Appendix A for the sake of completeness and will be referred as
the “Ghosh-Basu Conditions” throughout the rest of the paper.
Motivated by the strong robustness properties of the Wald-type test statistics based on MDPDEs (Basu
et al., 2016, 2017a; Ghosh et al., 2016, 2017) in case of independently and identically distributed observations,
in this paper we shall introduce and study the corresponding MDPDE based Wald-type tests for independently
but non identically distributed data. In particular, we will develop the asymptotic and theoretical robustness
of these Wald-type tests for both simple and composite hypotheses, along with applications to the generalized
linear models (GLMs) and its important subclasses. It is important to note that there is no established robust
hypothesis testing procedure under such general non-homogeneous data set-up except for one recent attempt by
Ghosh and Basu (2017), who developed the divergence based test statistics with DPD measure. However, the
asymptotic null distribution of their proposed test statistics is a linear combination of chi-square distributions
occasionally limiting its application in complicated situations. On the contrary, our proposed Wald-type test
statistics in this paper will be shown to have an ordinary chi-square distribution along with all the other
competitive properties and hence easier to implement in any practical applications.
The rest of the paper is organized as follows: In Section 2, we shall introduce the Wald-type tests for testing
simple null hypothesis as well as composite null hypothesis and we study their asymptotic distributions under
the null hypotheses as well as alternative hypotheses. The robustness of these Wald-type tests will be studied in
Section 3. In Section 4 the results are particularized to the GLM model. Some examples are studied in Section
5 and the paper ends with some insightful discussions in Section 6.

2

Wald-type tests under independent but non-homogeneous data

In the following two sections we shall consider the simple null hypothesis as well as composite null hypothesis
versions of the Wald-type test statistics for independently and non identically distributed data.

2.1

Wald-type tests for simple null hypotheses

Let Y1 , ..., Yn independently and non identically distributed data according to the probability density function
fi,θ , where θ ∈ Θ ⊂ Rp . In this section we define a family of Wald-type test statistics based on MDPDE for
testing the hypothesis
H0 : θ = θ 0 against H1 : θ 6= θ 0 ,
(9)
for a given θ 0 ∈ Θ, which will henceforth be referred to as the proposed Wald-type statistics.
2

bτ be the MDPDE of θ. The family of proposed Wald-type test statistics for testing the null
Definition 1 Let θ
hypothesis (9) is given by
bτ − θ 0 )T Σ−1 (θ 0 )(θ
bτ − θ 0 ),
Wn0 (θ 0 ) = n(θ
(10)
τ
where Στ (θ 0 ) is as defined in (5).
The asymptotic distribution of Wn0 (θ 0 ) is presented in the next theorem. The result follows easily from
the asymptotic distribution of the MDPDE considered in (4) and so we omit the proof. Throughout the rest
of the paper, for all the theoretical results, we will assume that the Ghosh-Basu conditions hold and Στ (θ) is
continuous in θ ∈ Θ.
Theorem 2 The asymptotic distribution, under the null hypothesis considered in (9), of the proposed Wald-type
test statistics given in (10) is χ2p , a chi-square distribution with p degrees of freedom.
In the next Theorem we are going to present a result that will be important in order to get an approximation
to the power function of the proposed Wald-type test statistic given in (10) because in many practical situation
is not possible to get a simple expression for the exact power function.
Theorem 3 Let θ ∗ be the the true value of parameter with θ ∗ 6= θ 0 . Then, we have


√
L
bτ ) − s(θ ∗ )) −→
n(s(θ
N 0, σ 2Wn0 (θ0 ) (θ ∗ )) ,
n→∞

T

T

∗
∗
2
where s (θ) = (θ − θ 0 ) Σ−1
τ (θ 0 ) (θ − θ 0 ) and σ W 0 (θ 0 ) (θ ) = 4 (θ − θ 0 )
n

 −1
 ∗
Στ (θ 0 )Στ (θ ∗ )Σ−1
τ (θ 0 ) (θ − θ 0 ) .

bτ is given by
Proof. A first-order Taylor expansion of s (θ) around θ ∗ at θ
bτ ) − s(θ ∗ ) =
s(θ

∂s (θ)
∂θ T

bτ − θ ∗ ) + op (n−1/2 ).
(θ
θ=θ ∗

√
bτ )−s(θ ∗ )) matches the asymptotic distribution
Then the asymptotic distribution of therandom variable n(s(θ
√
bτ − θ ∗ and the desired result follows.
of the random variable ∂s(θ)
n θ
∂θ T
∗
θ=θ

Based on Theorem 2 we shall reject the null hypothesis given in (9) if
Wn0 (θ 0 ) > χ2p,α ,

(11)

and Theorem 3 makes it possible to have an approximation of the power function for the test given in (11).
This is given by

π τWn0 (θ0 ) (θ ∗ ) = Pr (Rejecting H0 |θ = θ ∗ ) = Pr Wn0 (θ 0 ) > χ2p,α |θ = θ ∗
!
χ2p,α
∗
∗
bτ ) − s(θ ) >
= Pr s(θ
− s(θ )
n
!!
χ2p,α
n1/2
− s(θ ∗ )
,
(12)
= 1 − Φn
σ Wn0 (θ0 ) (θ ∗ )
n
where χ2p,α denote the (1 − α)-th quantile of χ2p distribution, and Φn (·) is a sequence of distribution functions
tending uniformly to the standard normal distribution function Φ(·). We can observe that the Wald-type tests
are consistent in the Fraser sense since
lim π τWn0 (θ0 ) (θ ∗ ) = 1 ∀τ ≥ 0.

n→∞

(13)

This result can be applied in the sense of getting the necessary sample size for the Wald-type tests to have a
predetermined power, π τW 0 (θ0 ) (θ ∗ ) ≈ π ∗ and size α. The necessary sample size is given by
n

"

#
p
A + B + A(A + 2B)
n=
+1
2s2 (θ ∗ )
3

2
where [z] denotes the largest integer less than or equal to z, A = σ 2W 0 (θ0 ) (θ ∗ ) Φ−1 (1 − π ∗ ) , and B =
n
2s(θ ∗ )χ2p,α .
In order to produce a non-trivial asymptotic power, see (13), Cochran (1952) suggested using a set of local
alternatives contiguous to the null hypothesis given in (9) as n increases. In the next theorem we shall present
the asymptotic distribution of the Wald-type test statistics under contiguous alternative hypotheses.
Theorem 4 Under the contiguous alternative hypothesis
H1,n : θ n = θ 0 + n−1/2 d,

(14)

where d is a fixed vector in Rp such that θ n ∈ Θ ⊂ Rp , the asymptotic distribution of the Wald-type test
statistics given in (10) is χ2p (δ), a non-central chi-square distribution with p degrees of freedom and non-centrality
parameter
δ = dT Σ−1
(15)
τ (θ 0 )d.
Proof. We can write
bτ − θ 0 = θ
bτ − θ n + θ n − θ 0 = (θ
bτ − θ n ) + n−1/2 d.
θ
Therefore, under H1,n given in (14), we have from (4),
√

L

bτ − θ n ) −→ N (0, Στ (θ 0 ))
n(θ
n→∞

and hence

√

L

bτ − θ 0 ) −→ N (d, Στ (θ 0 )) .
n(θ
n→∞

Note that Wn0 (θ 0 ) can be written by

T 

bτ − θ 0 )
bτ − θ 0 )
(θ 0 )(θ
(θ 0 )(θ
n1/2 Σ−1/2
Wn (θ 0 ) = n1/2 Σ−1/2
τ
τ
and under H1,n given in (14), we have,


L
−1/2
bτ − θ 0 ) −→
n1/2 Σ−1/2
(θ
)(
θ
N
Σ
(θ
)d,
I
.
0
0
p×p
τ
τ
n→∞

We apply the following result concerning quadratic forms. “If Z ∼ N (µ, Σ), Σ is a symmetric projection
of rank k and Σµ = µ, then Z T Z is a chi-square distribution with k degrees of freedom and non-centrality
parameter µT µ”. Therefore
L
Wn0 (θ 0 ) −→ χ2p (δ) ,
n→∞

where δ was defined in (15).
The last theorem permits us to get an approximation to the power function at θ n by

π τWn0 (θ0 ) (θ n ) = 1 − Gχ2p (δ) χ2p,α ,

(16)

where Gχ2p (δ) (z) is the distribution function of χ2p (δ) evaluated at the point z.
Based on this result we can also obtain an approximation of the power function at a generic point θ ∗ , because
we can consider d = n1/2 (θ ∗ − θ 0 ) and then θ n = θ ∗ .

2.2

Wald-type tests for composite null hypotheses

In many practical hypothesis testing problems, the restricted parameter space Θ0 ⊂ Θ is defined by a set of
r < p non-redundant restrictions of the form
h(θ) = 0
(17)
on Θ, where h : Rp → Rr is a vector-valued function such that the full rank p × r matrix
H (θ) =

4

∂h(θ)
∂θ T

(18)

exists and is continuous in θ.
Our interest will be in testing
H0 : θ ∈ Θ0 ⊂ Rp−r against H1 : θ ∈ Θ − Θ0

(19)

using Wald-type test statistics.
bτ be the MDPDE of θ. The family of proposed Wald-type test statistics for testing the
Definition 5 Let θ
composite null hypothesis (19) is given by
−1

bτ ) = nhT (θ
bτ ) H T (θ
bτ )Στ (θ
bτ )H(θ
bτ )
bτ ),
Wn ( θ
h(θ

(20)

bτ ).
In the next theorem we are going to present the asymptotic distribution of Wn (θ
bτ ), under the null hypothesis
Theorem 6 The asymptotic distribution of the Wald-type test statistics Wn (θ
given in (19), is chi-squared with r degrees of freedom.
Proof. Let θ 0 ∈ Θ0 be the true value of the parameter. A Taylor expansion gives
bτ ) = h(θ 0 ) +H T (θ 0 )(θ
bτ − θ 0 ) + op (n−1/2 1)
h(θ
bτ − θ 0 ) + op (n−1/2 1).
= H T (θ 0 )(θ
Under H0

√

L

bτ − θ 0 ) −→ N (0p , Στ (θ 0 ))
n(θ
n→∞

and so

√



L
bτ ) −→
nh(θ
N 0p , H T (θ 0 )Στ (θ 0 )H(θ 0 ) .
n→∞

Taking into account that rank (H (θ)) = r, we get

−1
L
bτ ) H T (θ 0 )Στ (θ 0 )H(θ 0 )
bτ ) −→
nhT (θ
h(θ
χ2r .
n→∞

bτ )Στ (θ
bτ )H(θ
bτ ) is a consistent estimator of H T (θ 0 )Στ (θ 0 )H(θ 0 ) by continuity of the matrices H(θ)
But H T (θ
and Στ (θ) at θ = θ 0 . Then, it holds that
L
bτ ) −→
Wn (θ
χ2r .
n→∞

Based on the previous result, we shall reject the null hypothesis given in (19) if
bτ ) > χ2 .
Wn (θ
r,α

(21)

It is not easy to get an exact expression for the power function of the test given in (21). For that reason we are
going to present a theorem that will be important in order to get an approximation of the power function for
the test statistic presented in (21).
P
bτ −→
Theorem 7 Let θ ∗ ∈
/ Θ0 the true value of the parameter with θ
θ ∗ . Define
n→∞


−1
s∗ (θ 1 , θ 2 ) = hT (θ 1 ) H T (θ 2 )Στ (θ 2 )H(θ 2 )
h(θ 1 ).
Then, we have



√  ∗
L
bτ , θ
bτ ) − s∗ (θ ∗ , θ ∗ ) −→
n s (θ
N 0, σ 2 (θ ∗ ) ,
n→∞

where


−1
σ 2 (θ ∗ ) = 4hT (θ ∗ ) H T (θ ∗ )Στ (θ ∗ )H(θ ∗ )
h(θ ∗ ).

5

P
bτ , θ
bτ ) and s∗ (θ
bτ , θ ∗ ) have the same asymptotic distribution because θ
bτ −→
Proof. We can observe that s∗ (θ

n→∞

bτ around θ ∗ gives
θ ∗ . A first-order Talyor expansion of s∗ (θ, θ ∗ ) at θ
bτ , θ ∗ ) − s∗ (θ ∗ , θ ∗ ) =
s∗ (θ

∂s∗ (θ, θ ∗ )
∂θ T

bτ − θ ∗ ) + op (n−1/2 ).
(θ
θ=θ ∗

Now the result follows, because
σ 2 (θ ∗ ) =
and

∂s∗ (θ, θ ∗ )
∂θ T

Στ (θ ∗ )
θ=θ ∗

∂s∗ (θ, θ ∗ )
∂θ

.
θ=θ ∗


−1
∂s∗ (θ, θ ∗ )
T
∗
∗
∗
T
=
2h
(θ)
H
(θ
)Σ
(θ
)H(θ
)
H T (θ).
τ
∂θ T

Using the above theorem we can derive an approximation to the power of the proposed Wald-type tests of
composite null hypothesis at any θ ∗ ∈
/ Θ0 using an argument similar to that of the derivation of the expression
in (12) for the case of simple null hypothesis. This further indicates the consistency of our proposal at any fixed
alternatives even for the composite hypotheses.
bτ ) at an alternative close to the null hypothesis.
We may also find an approximation of the power of Wn (θ
Let θ n ∈ Θ − Θ0 be a given alternative and let θ 0 be the element in Θ0 closest to θ n in the Euclidean distance
sense. A first possibility to introduce contiguous alternative hypotheses is to consider a fixed d ∈ Rp and to
permit θ n to move towards θ 0 as n increases through the relation
H1,n : θ n = θ 0 + n−1/2 d.

(22)

A second approach is to relax the condition h (θ) = 0 defining Θ0 . Let d∗ ∈ Rr and consider the following
sequence, {θ n }, of parameters moving towards θ 0 according to
∗
H1,n
: h(θ n ) = n−1/2 d∗ .

(23)

Note that a Taylor series expansion of h(θ n ) around θ 0 yields
h(θ n ) = h(θ 0 ) + H T (θ 0 ) (θ n − θ 0 ) + o (kθ n − θ 0 k 1) .

(24)

By substituting θ n = θ 0 + n−1/2 d in (24) and taking into account that h(θ 0 ) = 0, we get
h(θ n ) = n−1/2 H T (θ 0 )d + o (kθ n − θ 0 k 1) ,
so that the equivalence in the limit is obtained for d∗ = H T (θ 0 )d.
Theorem 8 Under the contiguous alternative hypotheses given in (22) and (23), we have
L

bτ ) −→ χ2 (a) under H1,n given in (22), where the non-centrality parameter “a” is given by
i) Wn (θ
r
n→∞


−1
a = dT H(θ 0 ) H T (θ 0 )Στ (θ 0 )H(θ 0 )
H T (θ 0 )d.
L

bτ ) −→ χ2 (b) under H ∗ given in (23), where the non-centrality parameter “b” is given by
ii) Wn (θ
r
1,n
n→∞


−1
b = d∗T H T (θ 0 )Στ (θ 0 )H(θ 0 )
d∗ .
bτ ) around θ n yields
Proof. A Taylor series expansion of h(θ
bτ ) = h(θ n ) + H T (θ n )(θ
bτ − θ n ) + o
h(θ

6




bτ − θ n 1 .
θ

(25)

From (25), we have
bτ ) = n−1/2 H T (θ 0 )d + H T (θ n )(θ
bτ − θ n ) + o
h(θ
As

√


bτ − θ n 1 + o (kθ n − θ 0 k 1) .
θ



√   b
n o θ τ − θ n 1 + o (kθ n − θ 0 k 1) = op (1), we have

L

bτ − θ n ) −→ N (0, Στ (θ 0 )) and
n(θ
n→∞

√



L

bτ ) −→ N (H T (θ 0 )d, H T (θ 0 )Στ (θ 0 )H(θ 0 )).
nh(θ
n→∞

We can observe by the relationship d∗ = H T (θ 0 )d, if h(θ n ) = n−1/2 d∗ that
√

L

bτ ) −→ N (d∗ , H T (θ 0 )Στ (θ 0 )H(θ 0 )).
nh(θ
n→∞

In our case, the quadratic form is Wn = Z T Z with Z =
L

Z −→ N



n→∞

√


−1/2
bτ ) H T (θ 0 )Στ (θ 0 )H(θ 0 )
nh(θ
and


−1/2
H T (θ 0 )Στ (θ 0 )H(θ 0 )
H T (θ 0 )d, I ,

where I is the identity r × r matrix. Hence, the application of the result is immediate and the non-centrality
parameter is

−1

−1
dT H(θ 0 ) H T (θ 0 )Στ (θ 0 )H(θ 0 )
H T (θ 0 )d = d∗T H T (θ 0 )Στ (θ 0 )H(θ 0 )
d∗ .

3
3.1

Robustness of the Wald-type tests for Non-homogeneous Observations
Influence functions of the Wald-type test statistics

In order to study the robustness of a testing procedure, the first measure to consider is Hampel’s influence
function (IF) of the test statistics, introduced by Rousseeuw and Ronchetti (1979) for i.i.d. data; see also
Rousseeuw and Ronchetti (1981) and Hampel et al. (1986) for detail. In case of non-homogeneous data, the
concept of IF has been extended suitably by Huber (1983) and Ghosh and Basu (2013, 2016) for the estimators
and by Ghosh and Basu (2017) and Aerts and Haesbroeck (2016) for test statistics. Here, we will follow these
extended definitions of IF to study the robustness of our proposed Wald-type test statistics for non-homogeneous
observations.
In order to define and study the IF for the Wald-type test statistics, we first need the same for the MDPDE
used in constructing the Wald-type test statistics; we will briefly recall the IF of the MDPDE under nonhomogeneous observations for the sake of completeness. Suppose Gi denote the true distribution of Yi having
corresponding density gi for each i = 1, . . . , n; under the model null distribution with true parameter value θ 0
we have Gi = Fi,θ and gi = fi,θ for each i. Denote G = (G1 , · · · , Gn ) and Fθ0 = (F1,θ0 , · · · , Fn,θ0 ). Then the
minimum DPD functional T τ (G) for independent but non-homogeneous observations at the true distribution
n
X
G is defined as the minimizer, with respect to θ ∈ Θ, of the average DPD measure n1
dτ (gi , fi,θ ) with
i=1

Z 
dτ (gi , fi,θ ) =



fi,θ (y)1+τ − 1 +

1
τ



fi,θ (y)τ gi (y) +

1
gi (y)1+τ
τ


dy.

Now, for each i = 1, . . . , n, let us denote by Gi, = (1 − )Gi + ∧ti the -contaminated distribution in the i-th
direction, where ∧ti denotes the degenerate distribution at the contamination point ti . Note that, in the case
of non-homogeneous data the contamination can be either in any fixed direction, say i0 -th direction, or in all

7

the n directions. The corresponding IF of the minimum DPD functional T τ (G) has been established in Ghosh
and Basu (2013); their forms at the model null distributions are given by
IFi0 (ti0 ; T τ , Fθ0 )
IF (t1 , . . . , tn ; T τ , Fθ0 )

1
= Ψ−1
n,τ (θ 0 ) D τ ,i0 (ti0 ; θ 0 ),
n
n
1X
= Ψ−1
D τ ,i (ti ; θ 0 ),
n,τ (θ 0 )
n i=1

(26)
(27)



where D τ ,i (t; θ) = fi,θ (t)τ ui,θ (t) − ξ i,τ with ξ i,τ being as defined in Equation (8). Note that these IFs are
bounded at τ > 0 and unbounded at τ = 0, implying the robustness of the MDPDEs with τ > 0 over the
classical MLE (at τ = 0).
Now, we can defined the IF for the proposed Wald-type test statistics. We define the associated statistical
functional, evaluated at G, as (ignoring the multiplier n)
Wτ0 (G)

=

(T τ (G) − θ 0 )T Σ−1
τ (θ 0 )(T τ (G) − θ 0 )

(28)

corresponding to (10) for the simple null hypothesis, and
Wτ (G)

=


−1
hT (T τ (G)) H T (T τ (G))Στ (T τ (G))H(T τ (G))
h(T τ (G))

(29)

corresponding to (20) for the composite null hypothesis.
First we consider the Wald-type test functional Wτ0 for the simple null hypothesis and contamination only
one direction, say i0 -th direction. The corresponding IF is then defined as
IFi0 (ti0 ; Wτ0 , G) =

∂ 0
W (G1 , · · · , Gi0 −1 , Gi0 , , Gi0 +1 , · · · , Gn )
∂ τ

= 2(T τ (G) − θ 0 )T Σ−1
τ (θ 0 )IFi0 (ti0 ; T τ , G),
=0

which, when evaluated at the null distribution G = Fθ0 , becomes identically zero as T τ (Fθ0 ) = θ 0 . So, one
need to consider the second order IF of the proposed Wald-type test functional Wτ0 defined as
(2)

IFi0 (ti0 ; Wτ0 , G)

=

∂2 0
W (G1 , · · · , Gi0 −1 , Gi0 , , Gi0 +1 , · · · , Gn )
∂2 τ

=0

.

When evaluated at the null model distribution G = Fθ0 , this second order IF has the simplified form
(2)

IFi0 (ti0 ; Wτ0 , Fθ0 )

=
=

2IFi0 (ti0 ; T τ , Fθ0 )T Σ−1
τ (θ 0 )IFi0 (ti0 ; T τ , Fθ 0 )

T


 −1
 1
1
−1
2
D τ ,i0 (ti0 ; θ 0 )
Ψn,τ (θ 0 )Σ−1
(θ
)Ψ
(θ
)
D
(t
;
θ
)
.
0
0
τ
,i
i
0
τ
n,τ
0
0
n
n

(30)

Similarly, we can derive the first and second order IF of Wτ0 for contamination in all directions at the point
t = (t1 , . . . , tn ) respectively defined as
IF (t; Wτ0 , G) =

∂ 0
W (G1, , · · · , Gn, )
∂ τ

, and IF (2) (t; Wτ0 , G) =
=0

∂2 0
W (G1, , · · · , Gn, )
∂2 τ

=0

.

A direct calculation shows that, at the simple null model distribution G = Fθ0 , these IFs simplifies to
IF (t; Wτ0 , Fθ0 )
IF

(2)

(t; Wτ0 , Fθ0 )

=

0,

=

2IF (t; T τ , Fθ0 )T Σ−1
τ (θ 0 )IF (t; T τ , Fθ 0 )
" n
#T
" n
#
 −1
 1X
1X
−1
−1
Ψn,τ (θ 0 )Στ (θ 0 )Ψn,τ (θ 0 )
2
D τ ,i (ti ; θ 0 )
D τ ,i (ti ; θ 0 ) .
n i=1
n i=1

=

(31)

Note that, both the second order IF in (30) and (31) of the Wald-type test functional Wτ0 for testing simple
null hypothesis under contamination in one or all directions are bounded, whenever the corresponding MDPDE
functional has bounded IF, i.e., for any τ > 0. This implies robustness of our proposed Wald-type tests for
simple null hypothesis with τ > 0.
8

Next we can similarly derive the first and second order IFs of the proposed Wald-type tests functional Wτ
in (29) for composite null hypotheses. For brevity, we will skip the details and present only the final results
under composite null G = Fθ0 with θ 0 ∈ Θ. In particular, the first order IF for contamination in either one or
all directions are both identically zero, i.e.,
IFi0 (ti0 ; Wτ0 , Fθ0 ) = 0,

IF (t; Wτ , Fθ0 ) = 0,

and the corresponding second order IF has the form
(2)

IFi0 (ti0 ; Wτ , Fθ0 )

=

IF (2) (t; Wτ , Fθ0 )

=

h
i−1
2IFi0 (ti0 ; T τ , Fθ0 )T H(θ 0 ) H T (θ 0 )Στ (θ 0 )H(θ 0 )
H T (θ 0 )IFi0 (ti0 ; T τ , Fθ0 )
h
i−1
2IF (t; T τ , Fθ0 )T H(θ 0 ) H T (θ 0 )Στ (θ 0 )H(θ 0 )
H T (θ 0 )IF (t; T τ , Fθ0 ).

Again these second order IFs are bounded for any τ > 0 and unbounded at τ = 0 implying the robustness of
our proposed Wald-type tests for composite hypothesis testing also.

3.2

Level and Power Influence Functions

We will now study the robustness of the level and power of the proposed Wald-type tests through the corresponding influence functions for their asymptotic level and powers (Hampel et al., 1986; Heritier and Ronchetti, 1994;
Toma and Broniatowski, 2010; Ghosh and Basu, 2017). Noting the consistency of these proposed Wald-type
tests, we consider their asymptotic power under the contiguous alternatives in (14) and (22) respectively for
the simple and composite hypotheses. Additionally, considering suitable contamination over these alternatives
and the null hypothesis, we define the contaminated distributions, for each i = 1, . . . , n,








P
L
√
√
√
∧
,
and
F
=
1
−
F
+
Fi,θn + √ ∧ti ,
Fi,n,,t
=
1
−
t
i,θ 0
i,n,,ti
i
n
n i
n
n
P
P
respectively for the analysis of level and power stability. Denote t = (t1 , · · · , tn )T , FP
n,,t = (F1,n,,ti , · · · , Fn,n,,ti )
L
L
L
and Fn,,t = (F1,n,,ti , · · · , Fn,n,,ti ). Then the level influence function (LIF) and the power influence function
(PIF) for the proposed Wald-type test statistics Wn0 (θ 0 ) for the simple null hypothesis (9) are defined, assuming
the nominal level of significance to be α, as

LIF (t; Wn0 , Fθ0 )

=

P IF (t; Wn0 , Fθ0 )

=

∂
P L (W 0 (θ 0 ) > χ2p,α )
∂ Fn,,t n
∂
lim
PFPn,,t (Wn0 (θ 0 ) > χ2p,α )
n→∞ ∂
lim

n→∞

=0

,

=0

.

bτ ) for the composite null hypothesis
Similarly the LIF and PIF of the proposed Wald-type test statistics Wn (θ
0
bτ ) and χ2 respectively,
(19) can be defined through above expressions by replacing Wn (θ 0 ) and χ2p,α by Wn (θ
r,α
where θ 0 is now the true null parameter in Θ0 .
Let us first consider the case of simple null hypothesis and derive the asymptotic power under the contiguous
contaminated distribution FP
n,,t in the following theorem.
Theorem 9 Consider the problem of testing the simple null hypothesis (9) by the proposed Wald-type test
statistics Wn0 (θ 0 ) at α-level of significance and consider the contiguous alternative hypotheses given by (14).
Then the following results hold.
2
1. The asymptotic distribution of Wn0 (θ 0 ) under FP
n,,t is χp (δ  ) with

eT (θ 0 )Σ−1 (θ 0 )d
e,t,τ (θ 0 ),
δ = d
,t,τ
τ
e,t,τ (θ 0 ) = d + IF (t; T τ , F ) with IF (t; T τ , F ) being given by (27).
where d
θ0
θ0
2. The corresponding asymptotic power under contiguous contaminated distribution FP
n,,t is given by
π Wn0 (θ n , , t) = lim PFPn,,t (Wn0 (θ 0 ) > χ2p,α ) = 1 − Gχ2p (δ ) (χ2p,α )
=

n→∞
∞
X




2
e,t,τ (θ 0 ), Σ−1 (θ 0 ) P χ2
Cv d
τ
p+2v > χp,α ,

v=0

9

(32)

v

where Cv (s, A) =

(sT As)
v!2v

1 T

e− 2 s

As

.

Proof. Denote θ ∗n = T τ (FP
n,,t ). Then, we can express our Wald-type test statistics in (10) as

=

bτ − θ 0 )T Σ−1 (θ 0 )(θ
bτ − θ 0 )
Wn0 (θ 0 ) = n(θ
τ
∗
∗
bτ − θ )T Σ−1 (θ 0 )(θ
bτ − θ ) + 2n(θ
bτ − θ ∗ )T Σ−1 (θ 0 )(θ ∗ − θ 0 ) + n(θ ∗ − θ 0 )T Σ−1 (θ 0 )(θ ∗ − θ 0 ). (33)
n(θ
n
τ
n
n
τ
n
n
τ
n

A suitable Taylor series expansion of θ ∗n , as a function of  at n = 0 yields (Ghosh et al., 2016)
θ ∗n = θ n +

√ IF (t; T τ , F )
θ0
n

1
+ o p ( √ 1p )
n

and hence
√
√

n(θ ∗n − θ n ) = IF (t; T τ , Fθ0 ) + op (1p ),

n(θ ∗n − θ 0 − n−1/2 d) = IF (t; T τ , Fθ0 ) + op (1p ).

Writing θ ∗n in terms of θ 0 , we get
√

n(θ ∗n − θ 0 ) = d + IF (t; T τ , Fθ0 ) + op (1p )
e,t,τ (θ 0 ) + op (1p ).
=d

(34)

Using these, we can rewrite (33) as
√
T
√

bτ − θ ∗ ) + d
e,x,τ (θ 0 ) Σ−1 (θ 0 )
bτ − θ ∗ ) + d
e,x,τ (θ 0 ) + op (1).
Wn0 (θ 0 ) = n
n(θ
n(θ
n
τ
n

(35)

But, under FP
n,,t , the asymptotic distribution of MDPDE and continuity of Στ (θ) implies that
√
L
bτ − θ ∗ ) −→
N (0p , Στ (θ 0 )).
n(θ
n

(36)

n→∞

Hence combining (35) and (36), we finally get
L
bτ ) −→
Wn0 (θ
χ2p (δ  )
n→∞

with δ  as defined in the statement of the theorem. This completes the proof of Part 1 of the theorem.
Next, Part 2 of the theorem follows directly from the infinite series expansion of non-central distribution
functions in terms of that of the central chi-square variables as follows:
π Wn0 (θ n , , t) = lim PFPn,,t (Wn0 (θ 0 ) > χ2p,α )
n→∞


= P (χ2p (δ  ) > χ2p,α ) = 1 − Gχ2p (δ ) χ2p,α
∞


X

2
e,t,τ (θ 0 ), Σ−1 (θ 0 ) P χ2
=
Cv d
τ
p+2v > χp,α .
v=0

Note that, substituting  = 0 in Expression (32) of the above theorem, we have an infinite series expression
for the asymptotic power of the proposed Wald-type tests under the contiguous alternative hypotheses (14) as
given by
∞
X


2
2
π Wn0 (θ n ) = π Wn0 (θ n , 0, t) =
Cv d, Σ−1
τ (θ 0 ) P χp+2v > χp,α ,
v=0

which has been previously obtained in terms of suitable distribution function in (16).
Further, substituting d = 0p in Expression (32), we get the asymptotic level of the proposed Wald-type tests
under the contaminated distribution FL
n,,t as given by
αWn0 (, t) = π Wn0 (θ 0 , , t)

d=0p

=

∞
X



2
2
Cv IF (t; T τ , Fθ0 ), Σ−1
τ (θ 0 ) P χp+2v > χp,α .

v=0

Finally, using the expression of the asymptotic power π Wn0 (θ 0 , , t) from (32) and differentiating it suitably,
we get the required PIF and then get the LIF by substituting d = 0p as presented in the following theorem.
10

Theorem 10 Under the assumptions of Theorem 9, the power and level influence functions of the proposed
Wald-type tests for the simple null hypothesis (9) are given by
 T −1
P IF (t; Wn0 , Fθ0 ) = Kp∗ dT Σ−1
τ (θ 0 )d d Στ (θ 0 )IF (t; T τ , Fθ 0 ),
s

with Kp∗ (s) = e− 2

∞
P
v=0

sv−1
v!2v (2v


− s)P χ2p+2v > χ2p,τ and
LIF (t; Wn0 , Fθ0 ) = 0.

Proof. Considering the expression of π Wn0 (θ 0 , , t) from (32) in Theorem 9 and using the definition of the PIF
along with the chain rule of derivatives, we get
P IF (t; Wn0 , Fθ0 )

∂
π Wn0 (θ 0 , , t) =0
∂
∞


X

∂
e,t,τ (θ 0 ), Σ−1 (θ 0 )
=
P χ2p+2v > χ2p,α
Cv d
τ
∂
=0
v=0
T 
∞ 
X

∂
∂ e
−1
=
Cv s, Στ (θ 0 ) s=de0,t,τ (θ0 )
d,t,τ (θ 0 )
∂s
∂
v=0
=

e0,x,τ (θ 0 ) = d,
Now, one can check that d


=0


P χ2p+2v > χ2p,α .

(37)

∂ e
∂ d,t,τ (θ 0 )

= IF (t; T τ , Fθ0 ) and

v−1

tT At
1 T
∂
Cv (t, A) =
2v − tT At Ate− 2 t At .
v
∂t
v!2
Substituting these expressions in (37) and simplifying, we obtain the required expression of the PIF as given in
the theorem.
Finally, the LIF follows from the PIF by substituting d = 0p .
Note that the above PIF is clearly bounded if and only if the IF of the MDPDE functional T τ is bounded,
i.e., whenever τ > 0. This again implies the robustness of the asymptotic power of the proposed Wald-type
tests with τ > 0 under contiguous contamination, over the classical MLE based Wald test (at τ = 0) that has an
unbounded PIF. Further, the asymptotic level of the Wald-type tests will not also be affected by a contiguous
contamination as suggested by its zero LIF.
bτ ) for composite
Next we can similarly derive the PIF and LIF for the proposed Wald-type test statistics Wn (θ
null hypotheses also. For brevity, we only present the main results corresponding to Theorem 9 and 10 for
the composite hypotheses case in the following theorems; proofs are similar and hence omitted. The main
implications are again the same proving the claimed robustness of our proposal with τ > 0 in terms of its
asymptotic level and power under contiguous contamination through zero LIF and bounded PIF.
Theorem 11 Consider the problem of testing the composite null hypothesis (19) by the proposed Wald-type test
bτ ) at α-level of significance and consider the contiguous alternative hypotheses given by (22).
statistics Wn (θ
Then the following results hold.
bτ ) under FP
1. The asymptotic distribution of Wn (θ
is χ2 (δ ∗ ) with
n,,t

δ ∗ =

eT (θ 0 )H(θ 0 )
d
,t,τ

r



h
i−1
e,t,τ (θ 0 ),
H T (θ 0 )Στ (θ 0 )H(θ 0 )
H T (θ 0 )d

2. The corresponding asymptotic power under contiguous contaminated distribution FP
n,,t is given by
bτ ) > χ2 ) = 1 − Gχ2 (δ∗ ) (χ2 )
π Wn (θ n , , t) lim PFPn,,t (Wn (θ
r,α
r,α
r

n→∞

∞
h
i−1 
X

e,t,τ (θ 0 ), H T (θ 0 )Στ (θ 0 )H(θ 0 )
=
Cv H T (θ 0 )d
P χ2r+2v > χ2r,α .

(38)

v=0

Theorem 12 Under the assumptions of Theorem 11, the power and level influence functions of the proposed
Wald-type test of the composite null hypothesis are given by
h
i−1
H T (θ 0 )IF (t; T τ , Fθ0 ),
P IF (t; Wn , Fθ0 ) = Kr∗ (δ ∗0 ) dT H(θ 0 ) H T (θ 0 )Στ (θ 0 )H(θ 0 )
LIF (t; Wn , Fθ0 )

=

0.

(39)
11

4

Application: Testing for Linear Hypotheses in Generalized Linear
Models (GLMs) with fixed design

In this Section we apply the theoretical results obtained in this paper for non-homogeneous observations to
the generalized linear model (GLM). Therefore now the density function associated to the independent random
variables Yi , 1 ≤ i ≤ n, is given by


yi η i − b(η i )
fi,θ (yi ) = f (yi , η i , φ) = exp
+ c (yi , φ) , 1 ≤ i ≤ n,
(40)
a(φ)
where the canonical parameter, η i , is an unknown measure of localization depending on the given fixed design
points xi ∈ Rk , 1 ≤ i ≤ n and φ is a known or unknown nuisance scale or dispersion parameter typically required
to produce standard errors following Gaussian, Gamma or inverse Gaussian distributions. The functions a(φ),
b(η i ) and c (yi , φ) are known. In particular, a(φ) is set to 1 for binomial, Poisson, and negative binomial
distribution (known φ) and it does not enter into the calculations for standard errors. The mean µi of Yi is
given by µi = µ (η i ) = E [Yi ] = b0 (η i ) and the variance by σ 2i = σ 2 (η i , φ) = Var [Yi ] = a(φ)b00 (η i ). The mean
response is assumed, according to GLMs, to be modeled linearly with respect to xi through a known link
function, g, i.e., g(µi ) = xTi β, where g is a monotone
and differentiable function and β ∈ Rk is an unknown

T
parameter. In this setting, since η i = η i xi β , we shall also denote (40) by f yi , xTi β, φ and the common
 T
T
bτ = β
b ,φ
b
parameters of the GLM by θ = (β T , φ)T , p = k + 1. At the same time we denote by θ
the
τ
τ
minimum density power divergence estimator of θ with tuning parameter τ . The estimating equations, based
bτ in this present case are given by
on (2), to get θ
n
X


γ 1,τ (xi ) − K1 (yi , xTi β, φ)f τ (yi , xTi β, φ) xi = 0,

(41)

i=1

and

n
X


γ 2,τ (xi ) − K2 (yi , xTi β, φ)f τ (yi , xTi β, φ) = 0.

(42)

i=1

where
K1 (yi , xTi β, φ) =

yi
2
σ (η

yi η − b (η i ) 0
∂c (yi , φ)
− µ(η i )
, K2 (yi , xTi β, φ) = − i 2
a (φ) +
.
0
a (φ)
∂φ
i )g (µ(θ i ))

and

Z
γ j,τ (xi ) =

Kj (y, xTi β, φ)f 1+τ (y, xTi β, φ)dy,

for j = 1, 2.

(43)

If we want to ignore the parameter φ and to estimate β taking φ fixed (or, substituted suitably), it is enough
to consider only the set of estimating equations in (41). Further, τ = 0, we have γ 1,0 (xi ) = 0 and the estimating
equations for β are
n
X
yi − µ(η i )
xi = 0.
2
σ (η i )g 0 (µ(η i ))
i=1
bτ is then given by (4), where we now have
The asymptotic distribution of θ

 n
n


P
P
2
T
γ
(x
)
−
γ
(x
)γ
(x
)
x
γ
(x
)
−
γ
(x
)
x
x
i
i
i
i
i
i
i
12,2τ
1,τ
1,τ
1,τ
i

 i=1 11,2τ
i=1
,
Ωn,τ (θ) = 
n
n

 T
P

P
γ 12,2τ (xi ) − γ 1,τ (xi )γ 1,τ (x) xi
γ 22,2τ (xi ) − γ 22,τ (xi )
i=1

and

i=1



n
P

γ 11,τ (xi )xi xTi


i=1
Ψn,τ (θ) = 
n
 P

γ 12,τ (xi )xi

i=1

n
P


γ 12,τ (xi )xi 
,

γ 22,τ (xi )

i=1
n
P

i=1

12

with γ j,τ (x), j = 1, 2, being given by (43) and
Z



γ jh,τ (x) = Kj y, xT β, φ Kh y, xT β, φ f 1+τ y, xT β, φ dy, for j, h = 1, 2.
Notice that for the case where φ is known we get
Ωn,τ (θ) =

n
X

n
X

γ 11,τ (xi ) − γ 21,τ (xi ) xi xTi , and Ψn,τ (θ) =
γ 11,τ (xi )xi xTi .

i=1

(44)

i=1

bτ under this fixed-design GLM.
See Ghosh and Basu (2016) for more details on the properties of the MDPDE θ
Here, we consider the most important hypothesis testing problem in the context of GLM, namely testing
the linear hypothesis on regression coefficient β as given by
H0 : Lβ = l0 versus H1 : Lβ 6= l0 ,

(45)

with L being an r × k known matrix of rank r and l0 being an known r-vector with r ≤ k. Note that,
this particular hypothesis (45) belongs to the general class of hypothesis in (19) with h (η) = Lβ − l0 and
H (η) = LT . Now for testing (45), we can consider the family of Wald-type test statistics presented in Section
2.2, given by
i−1 


T h
b − l0 .
b τ ) = n Lβ
b − l0
bτ )LT
Lβ
(46)
Wn (θ
LΣτ (θ
τ

τ

Based on our Theorem 6, the null hypothesis given in (45) will be rejected if we have that
bτ ) > χ2 .
Wn (θ
r,α

(47)

Further, following discussions in Section 2.2, this proposed Wald-type test is consistent at any fixed alternatives
and one can obtain an approximation to its power function at any fixed alternatives.

T
Next, suppose the true null parameter value is θ 0 = β T0 , φ0
and consider the sequence of contiguous
alternatives H1,n : β n = β 0 + n−1/2 d with d ∈ Rk − {0}. This is also equivalent to the alternative contiguous
hypothesis H1,n : Lβ n = l0 + n−1/2 d∗ with d∗ = Ld ∈ Rr − {0}. Under these contiguous alternatives, the
proposed Wald-type test statistics have the asymptotic distribution as non-central chi-square with degrees of
freedom r and non-centrality parameter
h
i−1
h
i−1
δ = dT LT LΣτ (θ 0 )LT
Ld = d∗T LΣτ (θ 0 )LT
d∗ .
(48)
Then, the asymptotic power at these contiguous alternatives can easily be obtained through the upper cumulative
distribution functions of the above non-central chi-square distributions. We will examine their behavior for some
special cases of GLM in the next section.
Next, considering the robustness of the proposed Wald-type tests under GLM, the first order influence
function of the test statistics and the level influence functions are always identically zero under contamination
in any fixed direction or in all directions following the general theory developed in Section 3. For both types
of contaminations, the non-zero second order influence function of the proposed Wald-type test statistics (46)
under fixed-design GLM is given by
h
i−1
(2)
LIFi0 (ti0 ; T τ , Fθ0 )
IFi0 (ti0 ; Wτ , Fθ0 ) = 2IFi0 (ti0 ; T τ , Fθ0 )T LT LΣτ (θ 0 )LT
h
i−1
IF (2) (t; Wτ , Fθ0 ) = 2IF (t; T τ , Fθ0 )T LT LΣτ (θ 0 )LT
LIF (t; T τ , Fθ0 ),
bτ under
where IFi0 (ti0 ; T τ , Fθ0 ) and IF (t; T τ , Fθ0 ) are corresponding influence functions of the MDPDE θ
the fixed-design GLM for contamination in the i0 -th direction and all directions respectively. These influence
functions of the MDPDE under fixed-design GLM have been studied by Ghosh and Basu (2016); using the
explicit form of these IFs, the second order IFs of our test statistics become

T


1
1
(2)
S τ ,i0 (ti0 ; β 0 , φ0 ) L∗0,τ
S τ ,i0 (ti0 ; β 0 , φ) ,
IFi0 (ti0 ; Wτ , F(β0 ,φ0 ) ) = 2
n
n
" n
#T
" n
#
X
X
1
1
IF (2) (t; Wτ , F(β0 ,φ0 ) ) = 2
S τ ,i (ti ; β 0 , φ0 ) L∗0
S τ ,i (ti ; β 0 , φ0 ) ,
n i=1
n i=1
13

h
i−1
T
where L∗0,τ = Ψ−1
LΣτ (β 0 , φ0 )LT
LΨ−1
n,τ (β 0 , φ0 )L
n,τ (β 0 , φ0 ) and


K1 (ti , xTi β, φ)f τ (ti , xTi β, φ) − γ 1,τ (xi ) xi
.
S τ ,i (ti ; β, φ) = 

K2 (ti , xTi β, φ)f τ (ti , xTi β, φ) − γ 2,τ (xi )


(49)

Clearly these influence functions will be bounded whenever the function S τ ,i (ti ; β 0 , φ0 ) is bounded in ti . However, due to the particular exponential form of the density in GLM, we have Kj (ti , xTi β, φ) is a polynomial
function of ti and the integral γ j,τ (xi ) is bounded for any given finite xi for each j = 1, 2. Hence, for any
τ > 0, the function S τ ,i (ti ; β 0 , φ0 ) will bounded in ti and it will be unbounded at τ = 0. This implies that
the proposed Wald-type tests with τ > 0 under fixed-design GLM will be robust compared to the non-robust
classical Wald-test at τ = 0.
We can similarly also check the power robustness of our proposal at τ > 0 under fixed-design GLM by
deriving the form of PIF from Theorem 12, whose boundedness again depends directly on the boundedness of
the function S τ ,i (ti ; β 0 , φ0 ) with respect to the contamination points ti s. In particular, the form of the PIF
under contiguous alternatives H1,n for the present case of fixed-design GLM simplifies to
" n
#
h
i−1
X
1
S τ ,i (ti ; β 0 , φ0 ) ,
L
P IF (t; Wn , F(β0 ,φ0 ) ) = Kr∗ (δ) dT LT LΣτ (β 0 , φ0 )LT
n i=1
where δ is as given by Equation (48). We will further study the behavior of these influence functions for some
particular examples of GLM in the next section.
It is worthwhile to note that the GLM considered in this paper as an special case of general non-homogeneous
set-up is different from the usual GLM with stochastic covariates (random design); here we are assuming that
the values of covariates (design-points) xi are fixed and known previously. The problem of robust hypothesis
testing under GLM with random design has been considered in Basu et al. (2017a,b).

5
5.1

Examples and Illustrations
Testing Significance of a Normal Linear Regression Model

As our first illustrative example, we will consider the most common and simplest case of GLM, namely the normal
regression model where the model density f (yi , xTi β, φ) is normal with mean xTi β and common variance φ > 0.
This model has a simpler representation given by
yi = xTi β + εi , i = 1, . . . , n,
where εi s are independent normally distributed errors with mean 0 and variance φ. When the design points xi s
are pre-fixed, we can apply the results derived above to construct and study robust Wald-type tests for general
linear hypothesis under this simpler model. In particular, for illustration, let us consider the problem of testing
for the significance of this linear model characterized by the hypothesis
H0 : β = β 0 versus H1 : β 6= β 0 ,

(50)

where β 0 is a known k-vector of hypothesized regression coefficients (usually a zero vector) and we assume φ
to be unknown under both hypotheses. The classical F-test for this problem is a version of the classical Wald
test based on the MLE of the parameters θ = (β T , φ)T and hence known to be highly non-robust. We will now
study the performances of the proposed Wald-type tests for this hypothesis.
Note that the hypothesisin (50) under
the normal linear regression model belongs to the class of general

Ik 0
linear hypothesis with L =
with I k being the identity matrix of order k and l0 = (β T0 0)T . So,
0 0
using the results of the previous section, the robust Wald-type test statistics for this testing problem simplifies
to

T h
i−1 

b ,φ
b )=n β
b −β
b ,φ
b )LT
b −β ,
Wn (β
LΣ
(
β
β
(51)
τ
τ
τ
τ
0
τ
τ
τ
0
b and φ
b are the MDPDE of β and φ respectively with tuning parameter τ and have asymptotic joint
where β
τ
τ
covariance matrix Στ (β 0 , φ0 ) at the true null parameter values (β 0 , φ0 ). Ghosh and Basu (2013) studied the
14

properties of these MDPDEs under fixed design normal linear model in detail. In particular it follows that,
under assumptions (R1)–(R2) of their paper (also listed in Appendix A), asymptotically

√  T
b ,φ
b )T − (β T , φ )T
n (β
τ
0
τ
0
follows a k + 1-variate normal distribution with mean vector 0 and covariance matrix given by
 β −1

υτ C x
0
Στ (β 0 , φ0 ) =
,
0T
υ φτ
1
(X T X) with X T = [x1 · · · xn ] being the design matrix and
n
#
"


3/2
5/2
τ2
4φ2
τ2
β
φ
2
2
2
υτ = φ 1 +
, υτ =
2(1 + 2τ ) 1 +
− τ (1 + τ ) .
1 + 2τ
(2 + τ 2 )2
1 + 2τ

where C x = lim

n→∞

Using these expressions, our proposed Wald-type test statistics (51) for testing (50) further simplifies to
b ,φ
b )= n
Wn ( β
τ
τ
b
φ
τ



τ2
1+
1 + 2τ

−3/2 

b −β
β
τ
0

T



b −β ,
Cx β
τ
0

(52)

which coincides with the classical Wald test at τ = 0. Following the theory of Section 4, these Wald-type test
statistics have asymptotic null distribution as χ2k and consistent against any fixed alternatives. To study its
power against contiguous alternatives H1,n : β n = β 0 + n−1/2 d with d ∈ Rk − {0}, note that the asymptotic
distribution of the proposed Wald-type test statistics under H1,n is non-central χ2 with degrees of freedom r
and non-centrality parameter

−3/2 h
i
1
τ2
δ=
dT C x d .
(53)
1+
φ0
1 + 2τ
Clearly the asymptotic contiguous power hof our proposed
test statistics depends on the given fixed values of
i
T
design points through the quantity dx = d C x d along with the tuning parameter τ . Table 1 presents the
empirical values of these contiguous powers over τ for different values of dx , with φ0 = 1 and 5% level of
significance. Note that, as the number (k) of regressors to be tested increases, we need larger values of dx
to achieve any fixed values of the contiguous power; for a given fixed design this corresponds to larger values
of ||d||. Further, for any fixed τ the values of contiguous power increases as dx increases but for any fixed
dx > 0 it decreases as τ increases as expected; the choice dx = 0 leads to the level of the tests for all τ ≥ 0.
However, interestingly, the loss in power compared to the classical Wald test at τ = 0 is not quite significant
at small values of τ > 0. And, against this relatively small price, we will gain substantially robustness against
contamination in data as illustrated below with the specific forms of the influence functions.
Table 1: Contiguous power of the proposed Wald-type test for testing (50) under the normal regression model
k=1
k = 20
τ
τ
dx
0
0.1
0.3
0.5
0.7
1
0
0.1
0.3
0.5
0.7
1
0
0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050
2
0.293 0.290 0.274 0.254 0.234 0.207 0.096 0.096 0.092 0.088 0.083 0.078
5
0.609 0.603 0.574 0.535 0.494 0.437 0.193 0.191 0.179 0.164 0.150 0.133
10 0.885 0.882 0.859 0.825 0.786 0.722 0.402 0.397 0.367 0.331 0.296 0.252
15 0.972 0.971 0.961 0.944 0.921 0.877 0.611 0.604 0.565 0.513 0.461 0.391
20 0.994 0.994 0.990 0.984 0.973 0.950 0.775 0.768 0.730 0.675 0.616 0.531
25 0.999 0.999 0.998 0.996 0.992 0.981 0.883 0.878 0.847 0.800 0.745 0.657
50 1.000 1.000 1.000 1.000 1.000 1.000 0.944 0.941 0.920 0.885 0.840 0.761

Again, based on the general theory developed in Section 4, one can readily check that the second order
influence function of the proposed Wald-type tests at the true null distribution having parameters θ 0 = (β T0 , φ0 )T
15

under the present case simplifies to
(2)

IFi0 (ti0 ; Wτ , Fθ0 )

=

IF (2) (t; Wτ , Fθ0 )

=

2
i
2 τ (ti0 −xTi0 β0 ) h T
2
φ0
(1 + 2τ )3/2 ti0 − xTi0 β 0 e−
xi0 (X T X)−1 C x (X T X)−1 xi0 ,
φ0
2
n
i
X
2 − τ (ti −xTi β0 ) h T
2
3/2
T
φ0
(1 + 2τ )
xi (X T X)−1 C x (X T X)−1 xi .
ti − xi β 0 e
φ0
i=1

Clearly, these influence functions depend on the values of the given fixed design-points in the direction of
contamination. However, for any given finite design-points, they are bounded in contamination points ti s for
each τ > 0 and unbounded at τ = 0. We will explicitly examine their nature for some particular cases of design
matrix; in particular, we consider the following four fixed designs:
Design
Design
Design
Design

1:
2:
3:
4:

xi
xi
xi
xi

= (1, xi1 )T ; xi1 = a, i = 1, . . . , n/2; xi1 = b, i = n/2 + 1, . . . , n.
(two-point design)
= (1, xi1 )T ; xi1 , i = 1, . . . , n, are pre-fixed iid observations from N (µx , σ 2x ). (Fixed-Normal design)
= (1, xi1 )T ; xi1 = i for i = 1, . . . , n.
(Divergent design)
= (1, xi1 , xi2 )T ; xi1 = 1i , xi2 = i12 for i = 1, . . . , n.
(Convergent design)

Note that, the C x matrix is finitely defined and is positive definite for the first two designs with values




1
1
(a + b)
1
µx
2
Cx =
and C x =
1
1 2
2
µx σ 2x + µ2x
2 (a + b)
2 (a + b )
respectively. In our illustrations, we have taken a = 1, b = 2 in design 1 and µx = 0, σ x = 1 in Design 2
so that the first one is asymmetric and non-orthogonal but the second one is symmetric and orthogonal. The
design matrix for Designs 3 and 4 are positive definite for any finite sample sizes, but the corresponding C x
matrices have all elements except their (1, 1)-th element as ∞ and 0 respectively; however, for the computation
of the above fixed sample IFs in these cases we can use the finite sample approximation of C x by n1 (X T X).
Figure 1 presents the second order IF of our test statistics for different contamination direction under these four
designs at the finite sample size n = 50 with β 0 = 1, the vector of ones, φ0 = 1 and different values of τ . The
boundedness of these IFs at τ > 0 clearly indicates the robustness of our proposed Wald-type tests. Further,
the (absolute) supremum of these IFs also decreases as τ increases which implies the increasing robustness
of the proposal with increasing τ . The extent of this increase in robustness for τ > 0 over τ = 0 becomes
more prominent when contamination is in all directions and/or the size of the fixed design matrix increases (an
extreme increment is in the case of Figure 3i).
Noting that the LIF is always zero, we can next study the power influence function also. In the present case,
the PIF can be seen to have the form
P IF (t; Wn , Fθ0 )

=

Kk∗ (δ)

2
n
i
X
 τ (ti −xTi β0 ) h T
2
2φ0
(1 + 2τ )3/2 (1 + τ )−3/2
ti − xTi β 0 e−
d C x (X T X)−1 xi ,
φ0
i=1

where δ is as given by Equation (53). Figure 2 presents these PIFs for the above four designs with different τ
at the finite sample size n = 50 with β 0 = 1, φ0 = 1, d = 10−2 β and 5% level of significance. Again, the power
of the proposed Wald-type tests under the normal model seems to be robust for all τ > 0 and for all the fixed
designs over the classical non-robust choice of τ = 0. Further, the extent of robustness increases as τ increases
or the size of the design matrix decreases.

5.2

Testing for individual regression coefficients in Poisson model for Count Data

Let us consider another popular special class of GLM applicable to the analysis of count responses, namely
T
the Poisson regression model. Here the response yi is assumed to have a Poisson distribution with mean exi β
depending on the given predictor values xi . In terms of the GLM notations of Section 4, the density in (40)
is then a Poisson density with η i = xTi β, known φ = 1 and the logarithmic link function. So, we can obtain
bτ = β
b of the regression parameter θ = β in this case following the general theory of
the robust MDPDE θ
τ
b under the fixed-design Poisson regression model
Section 4. Ghosh and Basu (2016) studied these MDPDEs β
τ
and their properties in detail with examples. In particular, in the notations of Section 4, we have estimating
16

(a) Design 1, i0 = 10

(b) Design 1, i0 = 40

(c) Design 1, all directions

(d) Design 2, i0 = 10

(e) Design 2, i0 = 40

(f) Design 2, all directions

(g) Design 3, i0 = 10

(h) Design 3, i0 = 40

(j) Design 4, i0 = 10

(k) Design 4, i0 = 40

(i)

∗

Design 3, all directions

(l) Design 4, all directions

Figure 1: Second order influence function of the proposed Wald-type test statistics for testing (50) under the
normal regression model with fixed designs 1 – 4 and contamination in the direction i0 = 10, 40 or in all
directions at t = t1 [solid line: τ = 0; dash-dotted line: τ = 0.1; dotted line: τ = 0.3; dashed line: τ = 0.5].
∗
indicates that the values for τ = 0 (solid line) has been shown in multiple of 10−2 for this graph only.

equations given only by (41) with K1 (yi , xTi β, φ) = yi − xTi β , and the required asymptotic variance matrix
Στ can be obtained in terms of Ωn,τ and Ψn,τ as defined in (44).
Here, as our second illustration of the proposed Wald-type testing procedures, we will consider the problem
of testing for the significance of any predictor (say, h-th predictor) in the model. For a fixed integer h between

17

(a) Design 1

(b) Design 2

(c) Design 3

(d) Design 4

Figure 2: Power influence function of the proposed Wald-type test statistics for testing (50) under the normal
regression model with fixed designs 1 – 4 at contamination point t = t1 [solid line: τ = 0; dash-dotted line:
τ = 0.1; dotted line: τ = 0.3; dashed line: τ = 0.5].
1 to k, the corresponding hypothesis is given by
H0 : β h = 0 versus H1 : β h 6= 0,

(54)

where β h denotes the h-th component of the regression vector β. Clearly this important hypothesis in (54) is
a special case of the general linear hypotheses in (45) with r = 1, LT being an k-vector with all entries zero
except the h-th entry as 1 and l0 = 0. So, our proposed Wald-type test statistics for this problem, following the
general theory of Section 4, can be simplified as
2

b )=
Wn ( β
τ

b
nβ
h,τ
,
2
b )
σ hh,τ (β
τ

(55)

2
b
b
where β
h,τ is the h-th entry of β τ denoting the MDPDE of β h and σ hh,τ denote the h-th diagonal element of the
√ b
asymptotic covariance matrix Στ at the null parameter values denoting the null asymptotic variance of nβ
h,τ .
Following Section 4, this test statistics in (55) is asymptotically distributed as χ21 distribution and consistent
at any fixed alternatives. Further, denoting the null parameter value as β 0 having h-th entry β h,0 = 0, the
asymptotic distribution of the proposed Wald-type test statistics under the contiguous alternatives

H1,n : β n with β h,n = n−1/2 d, β l,n = β l,0 for l 6= h, d ∈ R − {0}
is a non-central χ2 distribution with one degree of freedom and non-centrality parameter
δ=

d2
.
σ 2hh,τ (β 0 )

(56)

Note that σ 2hh,τ (β 0 ) has no closed form expression in this case but can be estimated numerically for any fixed
sample size and any given design-matrix by σ
b2hh,τ (β 0 ), the h-th diagonal entry of the matrix
−1
Ψ−1
n,τ (β 0 )Ωn,τ (β 0 )Ψn,τ (β 0 )

18

estimating Στ (β 0 ). Therefore, the effect of given design points can not be separated out explicitly from the
form of asymptotic contiguous power based on this non-central distribution as was the case for previous normal
model. We again consider the four given designs 1–4 from Section 5.1 and numerically compute the asymptotic
contiguous power of the proposed Wald-type tests for testing (54) with different values of h, d and τ assuming
n = 50, β l,0 = 1 for all l 6= h and 5% level of significance; the results are shown in Table 2. Once again, the
power loss is not quite significant for any small positive values of τ . Also, we need larger values of d to attain
any fixed power by the proposed Wald-type test statistics with fixed tuning parameter whenever the values of
the fixed design variables increases.
Table 2: Contiguous power of the
Design
dx
0
0.1
Design 1
0
0.050 0.050
2
0.200 0.199
3
0.388 0.383
5
0.796 0.792
7
0.974 0.973
10
1.000 1.000
Design 2
0
0.050 0.050
1
0.140 0.139
2
0.414 0.408
3
0.743 0.735
5
0.992 0.991
7
1.000 1.000
Design 3
0
0.050 0.050
0.01 1.000 1.000
0.05 1.000 1.000
0.1
1.000 1.000
0.2
1.000 1.000
0.5
1.000 1.000
Design 4
0
0.050 0.050
10
0.153 0.152
20
0.459 0.455
30
0.795 0.792
50
0.996 0.996
70
1.000 1.000

proposed Wald-type test for testing (54) under the Poisson regression model
τ
τ
0.3
0.5
0.7
1
0
0.1
0.3
0.5
0.7
1
h=1
h=2
0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050
0.189 0.178 0.167 0.152 0.378 0.375 0.355 0.331 0.308 0.276
0.364 0.339 0.315 0.282 0.696 0.691 0.663 0.627 0.589 0.534
0.766 0.730 0.692 0.634 0.985 0.984 0.978 0.968 0.954 0.926
0.964 0.950 0.932 0.897 1.000 1.000 1.000 1.000 0.999 0.998
1.000 0.999 0.999 0.996 1.000 1.000 1.000 1.000 1.000 1.000
h=1
h=2
0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050
0.132 0.124 0.117 0.110 0.320 0.316 0.300 0.280 0.261 0.234
0.381 0.351 0.326 0.299 0.846 0.842 0.819 0.786 0.749 0.693
0.700 0.658 0.619 0.574 0.994 0.994 0.991 0.985 0.977 0.959
0.985 0.976 0.965 0.947 1.000 1.000 1.000 1.000 1.000 1.000
1.000 1.000 1.000 0.999 1.000 1.000 1.000 1.000 1.000 1.000
h=1
h=2
0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050
1.000 1.000 1.000 1.000 0.057 0.056 0.056 0.056 0.055 0.055
1.000 1.000 1.000 1.000 0.221 0.219 0.209 0.196 0.183 0.166
1.000 1.000 1.000 1.000 0.662 0.657 0.630 0.593 0.557 0.503
1.000 1.000 1.000 1.000 0.997 0.997 0.996 0.993 0.988 0.976
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
h=2
h=3
0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050
0.145 0.137 0.129 0.119 0.168 0.167 0.159 0.149 0.140 0.128
0.431 0.402 0.373 0.333 0.510 0.506 0.479 0.445 0.412 0.366
0.765 0.728 0.689 0.630 0.845 0.841 0.816 0.781 0.740 0.679
0.994 0.990 0.983 0.968 0.999 0.999 0.998 0.995 0.992 0.981
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Further, the robustness of our proposed Wald-type tests can also be verified by examining the second order
influence function of the test statistics and the power influence functions from their general expressions derived
in Section 4. However, in the present case of Poisson regression model, we cannot have more simplified explicit
expressions for them except for the particular form of K1 (yi , xTi β, φ) as defined above and no K2 (yi , xTi β, φ);
but they can be easily computed numerically for any given fixed design. We again present the numerical values of
the second order influence functions of the proposed Wald-type test statistics for testing significance of the first
slope parameter β 2 (h = 2 in (54)) at different values of τ under the four designs considered in Section 5.1 with
n = 50 and β l,0 = 1 for all l 6= 2; these are presented in Figures 3. The redescending nature of all the influence
functions with increasing τ is again quite clear from the figures, which indicates the increasing robustness of
our proposed Wald-types tests as τ > 0 increases over the non-robust choice τ = 0 (having unbounded IFs).
The nature of the power influence functions in this case are also seen to be very similar implying the robustness
of our proposal at τ > 0; so we skip them for brevity.

19

(a) Design 1, i0 = 10

(b) Design 1, i0 = 40

(c) Design 1, all directions

(d) Design 2, i0 = 10

(e) Design 2, i0 = 40

(f) Design 2, all directions

(g) Design 3, i0 = 10

(h) Design 3, i0 = 40

(i) Design 3, all directions

(j) Design 4, i0 = 10

(k) Design 4, i0 = 40

(l) Design 4, all directions

Figure 3: Second order influence function of the proposed Wald-type test statistics for testing (54) under Poisson
regression model with fixed designs 1 – 4 and contamination in the direction i0 = 10, 40 or in all directions at
t = t1 [solid line: τ = 0; dash-dotted line: τ = 0.1; dotted line: τ = 0.3; dashed line: τ = 0.5].

6

Concluding remarks and the Choice of τ

We have proposed a robust parametric hypothesis testing approach for general non-homogeneous observations
involving a common model parameter. The test statistics have been constructed by generalizing the Wald
test statistics using the robust minimum density power divergence estimator (with parameter τ ≥ 0) of the
underlying common parameter in place of its non-robust maximum likelihood estimator. The properties of the

20

proposed test have been studied for both simple and composite hypotheses under general non-homogeneous setup and applied to the cases of several fixed design GLMs. In particular, it has been observed that the proposed
tests have simple chi-square asymptotic limit under null hypothesis in contrast to the linear combination of chisquare limit for the robust tests of Ghosh and Basu (2017) making its application much easier under complex
models. Also, the tests are always consistent at any fixed alternatives and have bounded second order influence
functions of the test statistics and bounded power influence functions for τ > 0 implying robustness of our
proposal.
Further, in each of the examples considered, we have seen that the asymptotic power of the proposed
Wald-type tests under any contiguous alternatives as well as the extent of robustness depends on the tuning
parameter τ . In particular, as τ increases, the contiguous power decreases slightly from its highest value at τ = 0
corresponding to the non-robust classical MLE based Wald test but the robustness increases significantly. Thus,
the tuning parameter τ yields a trade-off between asymptotic contiguous power and robustness of these Waldtype tests; note the similarity with the trade-off between asymptotic efficiency and robustness of the underlying
MDPDE as studied in Ghosh and Basu (2013). In fact one can explicitly examine, from the theoretical results
derived here, that the dependence of the power and robustness of the proposed Wald-type tests comes directly
through the similar dependence of the efficiency and robustness of the MDPDE used in constructing the test
statistics. Hence a proper choice of the tuning parameter τ balancing the asymptotic power and robustness
can be equivalently obtained by balancing the corresponding trade-off for the underlying MDPDE. This latter
problem under the non-homogeneous set-up has been proposed and studied by Ghosh and Basu (2013, 2015,
2016), where it is proposed that a data-driven estimate of the mean square error of the MDPDE be minimized
to obtain the optimum tuning parameter for any given practical dataset. The same optimum τ can also be used
as well for applying our proposed Wald-type tests for any practical hypothesis testing problems. However, more
detailed investigation on this issue could ba an interesting future research work.
Another possible future extension of the present paper will be to construct similar robust testing procedures
for two independent samples of non-homogeneous data. This problem is of high practical relevance as one
can then use the construction to test between two regression lines from fixed design clinical trials occurring
frequently in medical sciences and epidemiology. The corresponding problem with homogeneous sample has
been recently tackled by Ghosh et al. (2017) which should be extended to the cases with non-homogeneous data
and fixed-design regressions as in the present paper. We hope to pursue some such extensions in future.

A

Assumptions

Assumptions required for Asymptotic distributions of the MDPDE under non-homogeneous data
(Ghosh and Basu, 2013):
(A1) For all i = 1, . . . , n, the support of model distribution given by χ = {y|fi,θ (y) > 0} is independent of i
and θ and is the same as the support of true distribution Gi for each i.
(A2) There exists an open subset ω ⊆ Θ that contains the best fitting parameter θ g = T τ (G) and, at each
θ ∈ ω, the model density fi,θ (y) is thrice continuously differentiable with respect to θ for almost all y ∈ χ
and all i = 1, . . . , n.
R
R
(A3) fi,θ (y)1+α dy and fi,θ (y)α gi (y)dy can be differentiated three times with respect to θ, and the derivatives
can be taken under the integral sign for any i = 1, . . . , n.
(A4) The matrix J i,τ is positive definite for any i = 1, . . . , n, and, inf [min eigenvalue of Ψn,τ ] > 0.
n

1
α


fi,θ (y) . For all θ ∈ Θ, i = 1, . . . , n, and all j, h, l = 1, . . . , k,
(A5) Define Vi,θ (y) =
fi,θ (y)
dy − 1 +
the (j, h, l)-th (third order) partial derivatives of Vi,θ (y) is bounded in absolute value by some functions
n
h
i
1X
(i)
(i)
Mjhl (y) satisfying
Egi Mjhl (Y ) = O(1).
n i=1
R

1+α



α

(1)

(2)

(A6) For all j, h = 1, . . . , k, define N ijh (y) = ∇j Vi,θ (y) and N ijh (y) = ∇jh Vi,θ (y) − Egi (∇jh Vi,θ (y)). Then,
we have
( n
)
1X
(l)
(l)
lim sup
Egi |N ijh (y)|I(|N ijh (y))| > N ) = 0, l = 1, 2.
N →∞ n>1 n
i=1
21

(A7) For any  > 0,
(
lim

n→∞

)
n
h
√ i
1X
−1/2
−1/2
2
Eg ||Ωn ∇Vi,θ (y)|| I(||Ωn ∇Vi,θ (y)|| >  n)
= 0.
n i=1 i

(57)

Assumptions required for Asymptotic distributions of the MDPDE under Normal Fixed-Design
Linear Model (Ghosh and Basu, 2013):
The values of given design point xi = (x1i , . . . , xki )T are such that
n

(R1) sup max |xji | = O(1), sup max |xji xli | = O(1), and
n>1 1≤i≤n

n>1 1≤i≤n

1X
|xji xli xhi | = O(1), for all j, l, h = 1, . . . , k.
n i=1



1
T
(R2) inf min eigenvalue of (X X) > 0.
n
n

References
Aerts, S. and Haesbroeck, G. (2016). Robust asymptotic tests for the equality of multivariate coefficients of
variation Test, doi: 10.1007/s11749-016-0504-4.
Basu, A., Harris, I. R., Hjort, N. L., and Jones, M. C. (1998). Robust and efficient estimation by minimising a
density power divergence. Biometrika, 85, 549–559.
Basu, A., Shioya, H. and Park, C. (2011). Statistical Inference: The Minimum Distance Approach. Chapman &
Hall/CRC, Boca de Raton.
Basu, A., Mandal, A., Martin, N. and Pardo, L. (2016) Generalized Wald-type tests based on minimum density
power divergence estimators. Statistics, 50(1), 1–26.
Basu, A., Ghosh, A., Mandal, A., Martin, N. and Pardo, L. (2017a) A Wald-type test statistic for testing linear
Hypothesis in logistic regression models based on minimum density power divergence estimator. Electronic
Journal of Statistics, 11, 2741–2772.
Basu, A., Ghosh, A., Mandal, A., Martin, N. and Pardo, L. (2017b) Robust Wald-type tests in GLM with
random design based on minimum density power divergence estimators. Pre-print.
Beran, R. (1982). Robust estimation in models for independent non-identically distributed data. The Annals of
Statistics, 10, 2, 418-428.
Cochran, W. G. (1952). The χ2 test of goodness of fit. Annals of Mathematical Statistics, 23, 15-28.
Ghosh, A., and Basu, A. (2013). Robust estimation for independent non-homogeneous observations using density
power divergence with applications to linear regression. Electronic Journal of Statistics, 7, 2420–2456.
Ghosh, A., and Basu, A. (2015). Robust Estimation for Non-Homogeneous Data and the Selection of the Optimal
Tuning Parameter: The DPD Approach Journal of Applied Statistics, 42(9), 2056–2072.
Ghosh, A., and Basu, A. (2016). Robust Estimation in Generalized Linear Models : The Density Power Divergence Approach. Test, 25(2), 269–290.
Ghosh, A., and Basu, A. (2017). Robust Bounded Influence Tests for Independent but Non-Homogeneous
Observations. Statistica Sinica, doi:10.5705/ss.202015.0320.
Ghosh, A., Basu, A., and Pardo, L. (2015). On the robustness of a divergence based test of simple statistical
hypotheses. Journal of Statistical Planning and Inference, 116, 91–108.
Ghosh, A., Mandal, A., Martin, N. and Pardo, L. (2016). Influence analysis of robust Wald-type tests. Journal
of Multivariate Analysis, 147, 102–126.

22

Ghosh, A., Martin, N., Basu, A., and Pardo, L. (2017). A New Class of Robust Two-Sample Wald-Type Tests.
ArXiv Pre-print, arXiv:1702.04552 [stat.ME].
Hampel, F. R., Ronchetti, E., Rousseeuw, P. J., and Stahel W.(1986). Robust Statistics: The Approach Based
on Influence Functions. New York, USA: John Wiley & Sons.
Heritier, S. and Ronchetti, E. (1994). Robust bounded-influence tests in general parametric models. Journal of
the American Statistical Association, 89, 897–904.
Huber, P. J. (1983). Minimax aspects of bounded-influence regression (with discussion). Journal of the American
Statistical Association, 69, 383-393.
Muller, C. (1998). Optimum robust testing in linear models. The Annals of Statistics, 26, 3, 1126-1146.
Rousseeuw, P. J. and Ronchetti, E. (1979). The influence curve for tests. Research Report 21, Fachgruppe für
Statistik, ETH, Zurich.
Rousseeuw, P. J. and Ronchetti, E. (1981). Influence curves for general statistics. J. Comput. Appl. Math., 7,
161–166.
Toma, A. and Broniatowski, M. (2010). Dual divergence estimators and tests: robustness results. Journal of
Multivariate Analysis, 102, 20–36.

23

