Minimum Disparity Inference based on Tangent Disparities

International Journal of Information and Management Sciences

22 (2011), 1-25

Minimum Disparity Inference Based on Tangent Disparities

Chanseok Park and Ayanendranath Basu

Clemson University and Indian Statistical Institute

Abstract

This paper introduces the new family of “tangent disparities” based on the tangent func-

tion and considers robust inference based on it. The properties of the resulting inference

procedures are studied. The estimators are asymptotically efficient with asymptotic break-

down points of 1/2 at the model. The corresponding tests are equivalent to the likelihood

ratio test under the null. Numerical studies substantiate the theory developed and compare

the performance of the methods with those based on the Hellinger distance.

Keywords: Tangent disparity, efficiency, breakdown point, robustness, residual adjust-

ment function.

1. Introduction

Let FΘ = {Fθ : θ ∈ Θ ⊂ Rp} be a parametric class of distributions modeling the

true distribution G. Assume that G ∈ G, the class of all distributions having probability

density functions (pdf’s) with respect to a dominating measure. The model FΘ is also

assumed to be a subclass of G. We aim to develop robust inference procedures about

θ, without suffering a corresponding loss in efficiency typically suffered by M -estimators

(Hampel, Ronchetti, Rousseeuw and Stahel [9]).

Among others, Beran [5], Tamura and Boos [22], and Simpson [19, 20] demonstrated

that the simultaneous goals of asymptotic efficiency and robustness can be achieved by

using inference procedures based on the Hellinger distance (HD). Lindsay [12] generalized

the earlier work based on the HD to a general class of disparities generating estimators

that are both robust and first order efficient for discrete models. A disparity is a measure

of discrepancy between a nonparametric density estimator obtained from the data and

the model density. In this paper we develop and investigate the properties of the inference

procedures resulting from a new subclass of disparities based on the tangent function.

This branch of minimum distance estimation actually dates further back. Csiszar [6]

and Ali and Silvey [2] had earlier characterized this class of divergences independently.

Pardo [14] has provided an excellent account of such divergences. However Beran [5]

seems to be the first to point out the robustness applications of such distances. Lindsay

2 CHANSEOK PARK AND AYANENDRANATH BASU

[12] studied the geometry of these distances and identified the general structure which

leads to the robust behavior of the corresponding minimum distance estimators.

The rest of the paper is organized as follows: Section 2 contains a review of minimum

disparity inference. Section 3 develops the class of the tangent disparities, and Section 4

studies asymptotic efficiency of the minimum disparity estimators. Section 5 gives a

breakdown point analysis. Section 6 discusses robust tests of hypotheses. Numerical

illustrations and concluding remarks are given in Sections 7 and 8 respectively.

2. Minimum Disparity Estimation

For the parametric setup of Section 1, let X1,X2, . . . ,Xn be a random sample from

the distribution G (with pdf g), and let

gn(x) =1

nhn

n∑

i=1

w

(

x − Xi

hn

)

(1)

define a nonparametric density estimator of g, where w is a smooth family of kernel

functions with bandwidth hn. For discrete models, we will, without loss of generality, let

the sample space be {0, 1, 2, . . .} and take gn to be the empirical density function, where

gn(x) is the relative frequency of the value x in the sample. Define the Pearson residual

at a point x as

δ(x) =gn(x) − fθ(x)

fθ(x).

Let C(·) be a real-valued, thrice differentiable convex function on [−1,∞) with C(0) = 0.

Lindsay [12] constructed the disparity ρC between gn and fθ defined as

ρC(gn, fθ) =

∫

C(

δ)

fθ, (2)

the integral being with respect to the dominating measure. Under the assumptions on C,

ρC is nonnegative and equals zero if and only if gn ≡ fθ. Examples of disparities include

the likelihood disparity (LD) and the squared HD, defined by LD(g, fθ) =∫

g log (g/fθ)

and HD(g, fθ) =∫

(√

g−√

fθ )2. The LD is a version of the Kullback-Leibler divergence,

and in discrete models is minimized by the maximum likelihood estimator (MLE) of θ.

Let ∇ represent the gradient with respect to θ. For any real valued function u(x) we

will let u′(x) and u′′(x) denote its first and second derivatives with respect to x. Mini-

mization of the disparity ρC(gn, fθ) over θ ∈ Θ gives the minimum disparity estimator

corresponding to the function C. The HD produces the minimum Hellinger distance esti-

mator (MHDE). Under differentiability of the model, the minimum disparity estimating

equation becomes

−∇ρC =

∫

A(δ)∇fθ = 0, (3)

where A(δ) ≡ (δ + 1)C ′(δ) − C(δ). The function A(δ) is increasing on [−1,∞); with-

out affecting the estimating properties of the disparity ρC , it can be redefined to satisfy

MINIMUM DISPARITY INFERENCE 3

A(0) = 0 and A′(0) = 1. This standardized function A(δ) is called the residual adjust-

ment function (RAF) of the disparity. As the estimating equations in (3) are otherwise

equivalent, the form of the RAF determines the specific properties of the estimator like

how strongly the large outlying observations (manifesting themselves as large positive

values of δ) are downweighted. The RAF of the LD and the HD are given respectively

by A(δ) = δ and A(δ) = 2[√

δ + 1− 1]. The curvature A2 = A′′(0) of the RAF measures

how fast the function curves down from the line A(δ) = δ at δ = 0. Large negative values

of A2 provide greater downweighting relative to the MLE, while A2 = 0 indicates a form

of second order efficiency (Rao [16, 17]).

3. The Tangent Disparities

We introduce the new family of tangent disparities (TDα) between two densities g

and f indexed by a single parameter α ∈ [0, 1) as:

TDα(g, f) =

∫[

4

απg(x) tan

(

απ

2

g(x) − f(x)

g(x) + f(x)

)

− g(x) + f(x)

]

. (4)

Let (θ) = TDα(g, fθ) and define the TDα estimation functional Tα : G → Θ as Tα(G)

satisfying

(Tα(G)) = inft∈Θ

(t).

In case Tα(G) is multiple-valued, the notation Tα(G) will represent one of the possible

values chosen arbitrarily. The functional Tα(·) is Fisher consistent since Tα(Fθ) = θ for

all α in our range of interest. By the definition above, the minimum tangent disparity

estimator (MTDE) is Tα(Gn), where Gn is the cumulative distribution function (CDF)

of the kernel density estimator gn based on the data.

The C(·) function and RAF for the TDα disparity are given by

C(δ) =4

απ(δ + 1) tan

(

απ

2

δ

δ + 2

)

− δ, (5)

A(δ) = 4

(

δ + 1

δ + 2

)2

sec2

(

απ

2

δ

δ + 2

)

− 1. (6)

The case for α = 0 corresponds to the continuous limit as α → 0. In the TDα family

A′′(0) = (π2α2 − 4)/8, which equals zero only when α = 2/π. With α = 0, TDα is the

symmetric chi-square (SCS) deviance, which is defined as

SCS(g, f) = 2

∫

(g − f)2

g + f,

see Markatou, Basu and Lindsay [13]. In the following we present a couple of useful

mathematical results about the proposed TDα family.


Lemma 1. Let

D(g, f) =4

απg tan

(

απ

2

g − f

g + f

)

− g + f

be the integrand in (4). Then

D(g, f) ≤ D(0, f)I(g ≤ f) + D(g, 0)I(f < g) ≤ D(0, f) + D(g, 0),

where I(·) is the indicator function.

Proof. For fixed f and g ∈ (0, f), look at D(g, f) as a function of g.

∂

∂gD(g, f) =

4

απtan

(

απ

2

g − f

g + f

)

+4gf

(g + f)2sec2

(

απ

2

g − f

g + f

)

− 1.

This can be rewritten as a function of δ = g/f − 1 as

Dg(δ) =4

απtan

(

απ

2

δ

δ + 2

)

+4(δ + 1)

(δ + 2)2sec2

(

απ

2

δ

δ + 2

)

− 1.

It is easily shown that Dg(δ) is increasing in δ ∈ (−1, 0), Dg(0) = 0, and Dg(·) is

continuous at δ = 0. Hence ∂∂gD(g, f) < 0 for ∀g ∈ (0, f). Since D(·, f) is strictly

decreasing for g ∈ (0, f) and right-continuous at g = 0, we have D(g, f) ≤ D(0, f) for

g ∈ (0, f) with the equality only when g = 0. Similarly, D(g, f) ≤ D(g, 0) for f ∈ (0, g).

�

The following theorem is a simple consequence of Lemma 1.

Theorem 2. The TDα family for 0 ≤ α < 1 admits the following boundedness result:

0 ≤ TDα(g, f) ≤ 4

απtan

(απ

2

)

,

where the bound for the case α = 0 is the limit of the quantity in the right hand side as

α → 0. The left equality holds when g ≡ f , and the right equality holds when the supports

of the two densities are disjoint almost everywhere.

For better visual understanding of the robustness of the TDα disparity procedures,

we graphically present the RAF as well as the combined weight function wc(δc) (Park,

Basu and Lindsay [15]) for this family for different values of α in Figure 1. Notice that

the RAF has a more dampened response to large values of δ (representing the large

outliers) when α is small. The combined weight function wc(δc) represents the relative

impact, A(δ)/δ, of the observation in the estimating equation compared to maximum

likelihood (ML). Here one defines the combined residual δc

δc(x) =

{

δP (x) : g ≤ fθ

δN (x) : g > fθ


Figure 1: (a) RAF of the TDα family, (b) weight function of the TDα family.

where δN (x) = [g(x) − fθ(x)]/g(x) is the Neyman residual, and δ(x) = δP (x) is theordinary Pearson residual. The combined weight function wc(δc) is defined as

wc(δc) =

A(δc)

δc: −1 ≤ δc < 0

A′(0) : δc = 01 − δc

δcA

( δc

1 − δc

)

: 0 < δc < 1

A′(∞) : δc = 1

. (7)

On the positive side of the δc axis, this amounts to looking at the weights as a functionof the Pearson residuals but in the Neyman scale. For better robustness, it is desirablethat the weight functions converge to 0 as δc → 1. This is seen to be clearly the case forα ≤ 2/π (it is also true but less pronounced for bigger α). The weight functions wc(δc)are better than the RAFs for visual representation as its domain is a bounded intervalwhich illustrates the comparative behavior of the disparity relative to ML as δc → 1 (i.e.,δ = δP → ∞).

Notice that although for large values of α the RAFs curve away from linearity in thewrong direction, their weight functions eventually slide down to zero as δc → 1, indicatingthat while these members of the TDα family may have difficulty in downweighting mildoutliers, they successfully downweight the effect of extreme outliers (see the DrosophilaData I example in Section 7.1).

4. Asymptotic Efficiency of the Estimators

4.1. Efficiency of estimators

Under discrete models, the MTDEα is first order efficient (Lindsay [12]); here weconcentrate on continuous models. The existence, consistency and asymptotic normality


of the MTDEα are given by the following results. For the TDα family, C ′(δ), A(δ),

A′(δ)δ and A′′(δ)δ are bounded on [−1,∞). Many of the results presented here are

slight modifications of the proofs of Tamura and Boos [22], Lindsay [12], and Basu,

Sarkar and Vidyashankar [4]. The next theorem is a simple extension of Proposition 1

of Basu, Sarkar and Vidyashankar [4] and the proof is omitted.

Theorem 3. Assume that (a) the parameter space Θ is compact, (b) for θ1 6= θ2,

fθ1(x) 6= fθ2

(x) on a set of positive dominating measure, and (c) fθ(x) is continuous in θ

for almost every x. Then (i) for any G ∈ G, there exists θ ∈ Θ such that Tα(G) = θ, (ii)

TDα(g, fθ) is a continuous function of θ where g is the pdf of G, (iii) for any Fθ∗ ∈ FΘ,

Tα(Fθ∗) = θ∗ is unique for all α ∈ [0, 1).

Remark 1. As in Beran [5], the above result assumes compactness of Θ, but also applies

when Θ can be embedded within a compact set Θ, and the disparity TDα(g, fθ) can be

extended to a continuous function of θ on Θ. For example, in the location-scale family{

fθ(x) =1

σf

(

x − µ

σ

)

: θ = (µ, σ) ∈ (−∞,∞) × (0,∞)

}

, f continuous,

where the parameter space is not compact, (µ, σ) can be reparameterized as β = (β1, β2),

µ = tan(β1), σ = tan(β2), and (β1, β2) ∈ Θ = (−π/2, π/2) × (0, π/2). The disparity

extends as a continuous function on Θ = [−π/2, π/2] × [0, π/2] which is compact, and

the minimum must occur in Θ. Hence the conclusions of Theorem 3 remain valid.

Theorem 4. Let G ∈ G be the true distribution with pdf g. Given a random sample

X1,X2, . . . ,Xn, let the kernel density estimate gn with CDF Gn be as defined in (1).

If Tα(G) is unique, then under the assumptions of Theorem 3 the functional Tα(Gn) is

continuous at Tα(G) in the sense that Tα(Gn) converges to Tα(G) as gn → g in L1.

Proof. Suppose that gn → g in L1. For convenience, denote (θ) = TDα(g, fθ) and

n(θ) = TDα(gn, fθ). We have

|n(t) − (t)| ≤∫

|C(δn) − C(δ)| ft,

where δn = gn/ft − 1 and δ = g/ft − 1. By the mean value theorem, there exists δ∗

satisfying

C(δn) − C(δ) = C ′(δ∗)(δn − δ),

where δ∗ lies between δn and δ. For the TDα family, it is easily seen that |C ′(δ∗)| is

bounded. Denote K = maxδ |C ′(δ)|. Then we have

|n(t) − (t)| ≤ K

∫

|gn − g| for all t ∈ Θ.

Hence we have

supt

∣

∣n(t) − (t)∣

∣ → 0, (8)


as gn → g in L1. Let θn = arg inft n(t) and θ = arg inft (t). If (θ) ≥ n(θn), then

(θ)− n(θn) ≤ (θn) − n(θn), and if n(θn) ≥ (θ), then n(θn) − (θ) ≤ n(θ)− (θ).

Thus

|n(θn) − (θ)| ≤ |n(θn) − (θn)| + |n(θ) − (θ)| ≤ 2 supt

|n(t) − (t)| , (9)

which implies n(θn) → (θ) as gn → g in L1. Using (8) and (9), we obtain

limn→∞

(θn) = (θ). (10)

Then we only have to show that θn → θ as n → ∞. If θn 6→ θ, compactness of Θ ensures

existence of a subsequence {θm} ⊂ {θn} such that θm → θ∗ 6= θ, implying (θm) → (θ∗)

by continuity of (·). By (10), (θ∗) = (θ), which contradicts the uniqueness assumption

of θ = Tα(G). �

Thus under the conditions of Theorem 4, θn = Tα(Gn) is a consistent estimator of

the best fitting best fitting parameter Tα(G) whenever gn → g in L1. When G = Fθ

belongs to the specified model, θn is consistent for θ.

In practice, one may use the bandwidth hn = cnsn where sn is a robust scale es-

timator based on the data and cn is a sequence of positive constants. This gives the

automatic kernel density estimator (Devroye and Gyorfi [8]) of the form

gn(x) =1

ncnsn

n∑

i=1

w

(

x − Xi

cnsn

)

. (11)

In this case, sufficient conditions under which gn → g in L1 are that cn → 0, ncn → ∞and

√n(sn − s) is bounded almost surely, where s is a finite, positive constant.

Remark 2. Suppose that it is difficult to find a compact embedding Θ such that the

disparity extends continuously to all the points of Θ. In such cases, a result of Simpson

([19], Theorem 1) may be applied to show that the existence and continuity results above

still remain valid for the class G∗ of distributions, where any G ∈ G∗ satisfies

infθ∈Θ\Θ∗

TDα(g, fθ) > TDα(g, fθ∗)

for some compact Θ∗ ⊂ Θ and θ∗ ∈ Θ∗. Formally, suppose that fθ(x) is continuous in

θ for each x. Then, for each G ∈ G∗, (i) Tα(G) exists, and (ii) if Tα(G) is unique, then

the condition gn → g in L1 implies that Tα(Gn) → Tα(G) as n → ∞. In particular, if

conditions (b) and (c) of Theorem 3 hold and the parametric family satisfies

inft∈Θ\Θ∗

TDα(fθ, ft) > 0

for some compact Θ∗ ⊂ Θ, then gn → fθ in L1 implies that Tα(Gn) → θ as n → ∞.

Next we introduce some notation. Let uθ(x) = ∇ log fθ(x) be the p-dimensional

vector of ML score function, and let ∇2fθ(x) denote the p × p matrix of second partial


derivatives of fθ. We will denote by uiθ(x) and ∇ijfθ(x) the i-th element of uθ(x) and

(i, j)-th element of ∇2fθ(x) respectively. Then

I(θ) ≡∫

uθuTθ fθ

is the Fisher information matrix, where the superscript T denotes transpose. For deriving

asymptotic normality of the MTDEα we will assume that fθ(x) is twice continuously

differentiable with respect to θ, and TDα(·, fθ) is twice differentiable with respect to θ

under the integral sign. For the latter one, sufficient conditions are: for any θ ∈ Θ, there

exists ǫ > 0, and functions Kiθ(x), Lij

θ (x) and M ijθ (x), i, j = 1, . . . , p, such that for θ†

satisfying ‖θ − θ†‖ < ǫ (‖ · ‖ being the Euclidean norm),

(I) |uiθ†

fθ†(x)| < Kiθ(x),

∫

Kiθ < ∞, i = 1, 2, . . . , p;

(II) |∇ijfθ†(x)| < Lijθ (x),

∫

Lijθ < ∞, i, j = 1, . . . , p;

(III) |uiθ†

(x)ujθ†

(x)fθ†(x)| < M ijθ (x),

∫

M ijθ < ∞, i, j = 1, . . . , p.

Theorem 5. Suppose G = Fθ0, g = fθ0

, Gn and gn are as defined in Theorem 4. Let

{φn} denote any sequence of estimators such that φn = θ0 + op(1), {αn} any sequence

of positive real numbers going to ∞, and IB the indicator function for the set B. In

addition to the conditions (I) – (III) above, assume the following:

(a)∫

|∇ijfφn−∇ijfθ0

| = op(1) and∫

|uiφn

ujφn

fφn− ui

θ0uj

θ0fθ0

| = op(1), i, j = 1, . . . , p,

for all {φn} defined above.

(b)∫

uiθ0

(x+a)ujθ0

(x+a)fθ0(x)dx →

∫

uiθ0

(x)ujθ0

(x)fθ0(x)dx as |a| → 0, i, j = 1, . . . , p,

and the matrix I(θ0) is finite (element-wise).

(c) The kernel w(·) is a density which is symmetric about 0, square integrable, and

twice continuously differentiable with compact support S. The bandwidth hn satisfies

hn → 0, n1/2hn → ∞, and n1/2h4n → 0 as n → ∞.

(d) lim supn→∞

supy∈An

∫

|∇ijfθ0(x+y)uθ0

(x)|dx < ∞ for i, j = 1, . . . , p, where An = {y : y =

hnz, z ∈ S}.

(e) n supt∈S P (|X1 − hnt| > αn) → 0 as n → ∞, for all {αn} defined above.

(f) (n1/2hn)−1∫

∣

∣uθ0I|x|≤αn

∣

∣ → 0, for all {αn} defined above.

(g) sup|x|≤αnsupt∈S{fθ0

(x + hnt)/fθ0(x)} = O(1), for all {αn} defined above.

Then,√

n(

Tα(Gn) − Tα(G))

converges in distribution to N(0, I−1(θ0)).


The proof is given in the Appendix. Condition (a) is a simple continuity condition

at the model stating that the second derivative of fθ with respect to θ is L1-continuous

and Eθ[uiθu

jθ] is continuous in θ. This condition is satisfied by the distributions of the

exponential family. Condition (b) is satisfied if uiθ0

(x) is uniformly continuous in x on

compact sets. Condition (c) represents the key restriction on the bandwidth. Conditions

(e)–(g) have been used and discussed by Tamura and Boos ([22], Theorem 4.1)

4.2. Influence function

Consider the contaminated version of the true distribution G defined by

Gǫ(x) = (1 − ǫ)G(x) + ǫ∆ξ(x),

where ǫ is the contamination proportion and ∆ξ is the degenerate distribution at ξ. For

the functional Tα(·) on the space of distributions, its influence function at G is defined

by

IF(ξ;Tα, G) =∂Tα(Gǫ)

∂ǫ

∣

∣

∣

∣

ǫ=0

.

Differentiation of the estimating equation (3) shows the influence function of Tα to be

IF(ξ;Tα, G) =[

∫

A′(δ)guθuTθ −

∫

A(δ)∇2fθ

]−1[

A′(δ(ξ))uθ(ξ) −∫

A′(δ)guθ

]

, (12)

where θ = Tα(G), δ = g/fθ − 1 and A(·) is as in (6). We note that if G is a model point

Fθ, then the influence function of the Tα functional reduces to I−1(θ)uθ(ξ) suggesting

that the MTDEα is asymptotically fully efficient at the model. On the other hand, the

influence function of the Tα estimator is potentially unbounded. Several authors have

shown that the MHDE and some other density-based minimum divergence estimators

have strong robustness properties in spite of this, exhibiting the limitation of the influence

function approach in this case. See, for example, Beran [5] and Lindsay [12] for useful

discussions.

5. Breakdown Point Analysis

The breakdown point of a statistical functional is roughly the smallest fraction of

contamination in the data that may cause an arbitrarily extreme value in the estimate.

Here we establish the breakdown point of the functional Tα(G) under general conditions.

Consider the contamination model,

Hǫ,n = (1 − ǫ)G + ǫKn,

where {Kn} is a sequence of contaminating distributions. Let hǫ,n, g and kn be the

corresponding densities. Following Simpson [19], we say there is breakdown in Tα for ǫ

level contamination if there exists a sequence Kn such that |Tα(Hǫ,n) − Tα(G)| → ∞ as

n → ∞. We write below θn = Tα(Hǫ,n), suppressing the α and ǫ subscripts for brevity.


We assume the following conditions for the breakdown point analysis. The conditions

reflect the intuitively worst possible choice of the contamination, and the expected be-

havior or the functional when breakdown does and does not occur respectively. The

proof of the theorem is in the appendix.

Asumption 1. The contaminating sequence {kn}, the truth g(x) and the model fθ(x)

satisfy the following:

A1∫

min{g(x), kn(x)} → 0 as n → ∞. That is, the contamination distribution becomes

asymptotically singular to the true distribution.

A2∫

min{fθ(x), kn(x)} → 0 as n → ∞ uniformly for |θ| ≤ c, for any fixed c. That is

the contamination distribution is asymptotically singular to the specified models.

A3∫

min{g(x), fθn(x)} → 0 as n → ∞ if |θn| → ∞ as n → ∞. That is, large values of

the parameter θ give distributions which become singular to the true distribution.

Theorem 6. Under the assumptions A1 – A3, the asymptotic breakdown point ε∗ of

the TDα estimator is at least 12 at the model for α ∈ [0, 1).

6. The Tangent Disparity Tests

Because of the lack of robustness of the likelihood ratio tests (LRTs), alternative

robust tests have received a lot of attention in the literature. Breakdown robustness

of test functionals have been studied in detail by He, Simpson and Portnoy [10]. From

the disparity point of view, Simpson’s [20] Hellinger deviance test is robust under data

contamination and asymptotically equivalent to the LRT under the model. Agostinelli

and Markatou [1] considered testing of hypotheses based on the weighted likelihood

methodology. Here, we study analogs of the LRT based on the TDα. These tests are

also shown to be robust with an asymptotic behavior similar to that of the LRT at the

model. Under the parametric setup given in Section 1, suppose that the hypotheses of

interest are H0 : θ ∈ Θ0 against H1 : θ ∈ Θ \ Θ0, where Θ0 is a proper subset of Θ. Let

(Tα(G)) = inft∈Θ (t) and (T ∗α(G)) = inft∈Θ0

(t), where (θ) = TDα(g, fθ).

For a random sample of size n with kernel density estimate gn and CDF Gn, denote

the estimators T ∗α(Gn) and Tα(Gn) under the null and under no restriction by θ∗n and

θn, respectively. Define the TDα test statistic as

2n{

TDα(gn, θ∗n) − TDα(gn, θn)}

.

Suppose that Θ0 is given by a set of r ≤ p restrictions on the parameter space defined

by Ri(θ) = 0, 1 ≤ i ≤ r. We assume that the parameter space under H0 can be described

through a parameter ν with p−r independent components such that ν = (ν1, . . . , νp−r)T ,

i.e., θ = b(ν) where b : Rp−r → R

p. Thus θ∗n = b(νn), where νn is the MTDEα in the

ν-formulation in the model based on a random sample of size n. Let G = Fθ0, where θ0

is the true value of the parameter. Under the null hypothesis let ν0 be the true value of ν


parameter. From the arguments of Section 4 it follows that, under H0, νn is a consistent

estimator of ν0 and θ∗n is a consistent estimator of θ0. Define Dν = [∂bi/∂νj ]p×(p−r). Let

J(ν0) denote the information matrix under the ν-formulation, while I(θ0) denotes that

under no restrictions. The LRT has the form

LRT = 2{

n∑

i=1

log fθML(Xi) −

n∑

i=1

log fθ∗ML

(Xi)}

,

where θML and θ∗ML represent the MLEs of θ under no restrictions and under H0, respec-

tively. Asymptotically the statistic LRT has a χ2(r) distribution under H0. The following

theorem is proved in the Appendix which establishes the asymptotic distribution of the

TDα statistic under the null hypothesis.

Theorem 7. Under the conditions of Theorem 5, 2n{

TDα(gn, fθ∗n

) − TDα(gn, fθn)}

−LRT = op(1).

7. Examples and Simulations

7.1. Examples

Drosophila assay: We consider a part of an experiment originally reported by Woodruff

et al. [24], and analyzed by Simpson [19]. The frequencies of frequencies of daughter flies

carrying a recessive lethal mutation on the X-chromosome are considered where the male

parents have been exposed to a certain degree of a chemical. Roughly hundred daughter

flies were sampled for each male. This particular experiment resulted in (xi, fi) = (0, 23),

(1, 7), (2, 3), (91, 1), where xi is the number of daughters carrying the recessive lethal

mutation and fi is the number of male parents having xi such daughters. We will refer to

this as Drosophila Data I . The estimators of θ under a Poisson (θ) model corresponding

to α = 0, 0.1, 0.3, 0.5, 2/π, 0.7, 0.9 for Drosophila Data I are presented in Table 1. All

the TDα estimators (as well as the MHDE) successfully downweight the big outlier, 91,

unlike the MLE.

Table 1: The estimated parameters under the Poisson model for Drosophila Data I. The MLEsare θ = 3.059 and θ = 0.394 with and without the outlier respectively.

α 0 0.1 0.3 0.5 2/π 0.7 0.9 MHDETDα 0.381 0.381 0.384 0.389 0.394 0.396 0.405 0.364

The second example also involves data from Woodruff et al. [24]. The responses now

are the frequencies of frequencies of daughter flies having a recessive lethal mutation on

the X-chromosome where the male parent was either exposed to a dose of chemical or

to control conditions. This data set, also analyzed by Simpson ([20], Table 5) will be

referred to as Drosophila Data II . We model the responses as Poissons with mean θ1

(control), and θ2 (exposed) respectively. For testing H0 : θ1 ≥ θ2 against H1 : θ1 < θ2,

a two sample signed divergence is appropriate. Suppose that random samples of size ni


are available from the population with density fθiand let di be the empirical density of

i-th sample, i = 1, 2. For a divergence ρ(·, ·) between two densities, define the overall

divergence in the two sample case as

D = D(θ1, θ2) =1

n1 + n2{n1ρ(d1, fθ1

) + n2ρ(d2, fθ2)} .

For the divergence test statistic tn = 2n(D0 − D), where D0 and D are the minimizers

of D(·, ·) under the null and without any restrictions respectively, the signed divergence

statistic is given by sn = t1/2n sign(θ2 − θ1) where θ1 and θ2 are the unrestricted minimum

divergence estimators of the parameters; the signed divergence test is asymptotically

equivalent to the signed two sample LRT. For the full data and the cleaned data (after

removing the two large observations from the treated group) the signed divergences and

the associated p-values using the standard normal approximation are given in Table 2.

The significant value in case of the LRT is caused by the two large outliers, but their

presence does not reverse the conclusion for the robust tests. Results of the Hellinger

deviance test (Simpson [20]) are also provided for comparison.

Table 2: The signed divergence statistics (sn) and their p-values for Drosophila Data II.

All observations Outliers Deletedα sn p-value sn p-value0 0.454 0.325 0.509 0.305

0.1 0.465 0.321 0.519 0.3020.3 0.561 0.288 0.608 0.2710.5 0.780 0.218 0.819 0.2062/π 1.005 0.157 1.041 0.1490.7 1.124 0.130 1.160 0.1230.9 1.518 0.065 1.554 0.060HD 0.698 0.243 0.743 0.229ML 2.595 0.002 1.099 0.136

The number of cases of peritonitis: The next example involves the incidence of

peritonitis on n = 390 kidney patients (Table 3). A glimpse of the data, provided by

Prof. P. W. M. John, suggests that a geometric model with θ around 12 may fit the data

well. The observed frequency (Ok) of the number of cases of peritonitis (k) is modeled by

the geometric distribution with success probability θ. For an estimate θn, the expected

frequencies are obtained as Ek = n θn (1−θn)k. The largest number of cases of peritonitis

is k = 12, so we merged all the expected frequencies for k ≥ 12.

To assess the goodness-of-fit of the model, we use the log likelihood ratio statistic which

is given by

G2 = 2

12∑

k=0

Ok log(Ok/Ek).


Table 3: The observed frequencies (Ok) of the number of cases (k) of peritonitis for each of 390kidney patients and the expected frequencies under different methods with the goodness-of-fitlikelihood ratio statistics (G2).

k 0 1 2 3 4 5 6 7 8 9 10 11 12+ G2

Ok 199 94 46 23 17 4 4 1 0 0 1 0 1 —α TDα

0 197.8 97.5 48.0 23.7 11.7 5.7 2.8 1.4 0.7 0.3 0.2 0.1 0.1 10.90.1 197.8 97.5 48.1 23.7 11.7 5.8 2.8 1.4 0.7 0.3 0.2 0.1 0.1 10.90.3 197.4 97.5 48.2 23.8 11.7 5.8 2.9 1.4 0.7 0.3 0.2 0.1 0.1 10.80.5 196.4 97.5 48.4 24.0 11.9 5.9 2.9 1.5 0.7 0.4 0.2 0.1 0.1 10.62/π 195.3 97.5 48.7 24.3 12.1 6.1 3.0 1.5 0.8 0.4 0.2 0.1 0.1 10.50.7 194.4 97.5 48.9 24.5 12.3 6.2 3.1 1.5 0.8 0.4 0.2 0.1 0.1 10.50.9 188.8 97.4 50.2 25.9 13.4 6.9 3.6 1.8 0.9 0.5 0.3 0.1 0.1 10.8HD 199.1 97.5 47.7 23.4 11.4 5.6 2.7 1.3 0.7 0.3 0.2 0.1 0.1 11.1ML 193.5 97.5 49.1 24.7 12.5 6.3 3.2 1.6 0.8 0.4 0.2 0.1 0.1 10.4

In this example the fit provided by the MLE is excellent. The two marginally large

observations at 10 and 12 have little impact since the sample size is so large. Notice that

the fit provided by most of the robust estimates is almost equally good (last column,

Table 3) and clearly better than the HD. The example demonstrates that the proposed

methods can work almost as well as the ML method when the model provides good fit

to the data.

Determinations of the parallax of the sun: We consider Short’s data (Stigler [21],

Data Set 2) for the determination of the parallax of the sun, the angle subtended by the

earth’s radius, as if viewed and measured from the surface of the sun. From this angle

and available knowledge of the physical dimensions of the earth, the mean distance from

earth to sun could be easily determined. For our estimation we have used the kernel

density defined in (11), with the Epanechnikov kernel (w(x) = (3/4)(1 − x2), if |x| < 1,

and w(x) = 0, otherwise). Following Devroye and Gyorfi ([8], pp. 107-108), the mean L1

criterion with the Epanechnikov kernel and Gaussian pdf f leads to a bandwidth of the

form

hn = (15e)1/5(π/32)1/10σ n−1/5 = 1.66σ n−1/5,

where σ is the standard deviation. As the standard deviation is not specified, one can

use hn = 1.66 σ n−1/5, where σ = MAD = median(|Xi − median(Xi)|)/0.674.

Table 4: Fits of a normal N(µ, σ2) to the Short’s data using the maximum likelihood (ML),maximum likelihood without outlier (ML-O), and minimum TDα estimation.

α 0 0.1 0.3 0.5 2/π 0.7 0.9 ML-O ML HDµ 8.378 8.378 8.376 8.377 8.525 8.541 8.280 8.541 8.378 8.386

TDσ 0.335 0.335 0.339 0.350 0.555 0.571 0.971 0.552 0.846 0.352


•• ••• ••• • ••• ••• ••5 6 7 8 9 10

0.0

0.2

0.4

0.6

0.8

1.0

1.2

TDα=0.5

MLKernel density

Figure 2: Density estimates for Short’s 1763 determinations of the parallax of the sun in Dataset 2 (Stigler [21]).’

For the Short’s data, Table 4 gives the values of the minimum TDα estimates of µ and σ

for various values of α under the normal model, the MHDEs, as well as the MLEs for all

the observations and that after deleting the outlier, 5.76. Figure 2 shows the fit for the

kernel density estimate and the normal densities by ML and TDα=0.5 estimators. Notice

that the estimates are highly robust for small values of α. Most of them downweight the

other remaining moderate outliers also, and hence provide a smaller scale than ML-O.

Measurements of the passages of the light: We next consider Newcomb’s light

speed data (Stigler [21]). For the data, Table 5 gives the values of the TDα estimate of µ

and σ for various values of α under the normal model, the MHDEs, as well as the MLEs

for the full data (ML), and that after deleting −44, and −2, the two obvious outliers

(ML-O) – see Figure 3.

Table 5: Fits of a normal N(µ, σ2) to the Newcomb data using the maximum likelihood (ML),maximum likelihood without outlier (ML-O), and minimum TDα estimation.

α 0 0.1 0.3 0.5 2/π 0.7 0.9 ML-O ML HDµ 27.73 27.73 27.74 27.74 27.75 27.75 27.76 27.75 26.21 27.74

TDσ 5.22 5.22 5.22 5.24 5.25 5.26 5.28 5.04 10.66 5.14

We have used the Epanechnikov kernel with bandwidth hn = 1.66MAD n−1/5. Notice

that these estimators exhibit strong outlier resistance properties even for quite large

values of α. A graphic representation is provided in Figure 3, where the normal density

N(µ, σ2), for α = 0.5 is superimposed on a histogram of the Newcomb data, together


•• •••••• •• • •• •• • •• •• • • ••• • •• •• ••• • •••• ••• ••• ••• • •• •••• ••• • ••• •0-40

0.0

2

-20

0.0

4

20

0.0

6

40

0.0

80.0

TDα=0.5

MLKernel density

Figure 3: Density estimates for the Newcomb data

with the N(µML, σ2ML) density and the kernel density estimate. For the robust method,

the normal density fits the main body of the histogram quite well.

Telephone-line faults: Welch [23] presented a data set from an experiment to test a

method of reducing faults on telephone lines. This data set was previously analyzed by

using the HD (Simpson [20]). For the Telephone-line faults data, the ordered differences

between the inverse test rates and the inverse control rates are given in Table 6. In

Table 7 are given the values of the TDα estimate of µ and σ for various values of α under

the normal model, the MHDEs, as well as the MLEs for the full data (ML), and those

after deleting −988, the obvious outlier (ML-O) – see Figure 4.

Table 6: Inverse Fault-Rate Differences (D) for telephone lines in 14 pairs of areas.

Pair 1 2 3 4 5 6 7 8 9 10 11 12 13 14D × 105 -988 -135 -78 3 59 83 93 110 189 197 204 229 269 310

Table 7: Fits of a normal N(µ, σ2) to the Telephone line faults data using the maximum likelihood(ML), maximum likelihood without outlier (ML-O), and minimum TDα estimation.

α 0 0.1 0.3 0.5 2/π 0.7 0.9 ML-O ML HDµ 118.6 118.5 118.4 118.1 117.9 117.6 117.5 117.9 38.9 116.77

TDσ 143.3 143.4 143.5 143.8 144.0 144.4 144.7 127.6 310.2 138.43

The corresponding fitted normal density for α = 0.5 is shown in Figure 4 along with the

kernel density obtained by the Epanechnikov kernel and the normal density fit by ML.The


• • • • •••• •••• • •0

0.0

-1000 -500 500

0.0

005

0.0

010

0.0

015

0.0

020

0.0

025

TDα=0.5

MLKernel density

Figure 4: Density estimates for Telephone-line faults data.

bandwidth was determined as in the previous examples. Notice that these estimators

exhibit strong outlier resistance properties even for quite large values of α. We also use

the data to test that the mean of the population µ is zero with variance unspecified,

against µ > 0. The presence of the large negative outlier makes the result insignificant

for the t-test, but the signed TDα=0.5 test statistic leads to rejection using the critical

value of either N(0, 1) or t(13) distribution. In Figure 5 we have plotted the p-values

of these two tests and that of the one-sided matched-pair t-test as the first difference

is replaced by values between −1000 and 4000. Except for a very small region of the

contaminating value, the robust tests retain their significance while the t-test fails to

give significant results for most.

7.2. Simulations

For the sake of brevity and to keep a clear focus we restrict the simulation results to

the discrete Poisson model. Most of the features carry over to continuous models such

as the normal. In the first study, the data are generated from the Poisson distribution

with mean 5, and modeled as the Poisson (θ) distribution. Next data are generated from

the 0.9Poisson (5)+ 0.1Poisson (15) mixture, and the assumed model is still Poisson (θ).

In Table 8, we have presented the estimated biases and the mean square errors (MSEs)

of the TDα estimators of θ against the target value of 5 for sample sizes of n = 20, 50.

The estimators for small values of α perform very well under contamination. Notice the

extremely strong performance of the proposed estimators in comparison to the MHDE.

The poor performance of the HD at the model is certainly due in part to its improper

controlling of inliers (cells with less data than expected under the model), see, e.g., Basu,


0

0.0

20.0

40.0

60.0

80.0

-1000 1000 2000 3000 4000

0.1

0

TDα=0.5 Z testTDα=0.5 t-test

t-test

p-v

alu

e

1st difference.

Figure 5: p-value as the value of the 1st difference changes.

Harris, and Basu [3], a property which is not shared at all by the members of the proposed

family as evidenced by Figure 1.

Next we looked at the powers of the tests for pure data and under contamination.

The results for the nominal level γ = 0.05 are presented in Figure 6 and are based

on sample size 50 with 1000 replications. First we generated data from Poisson (θ)

with θ in the range (3.5, 6.5), and determined the power of each of the tests for the

hypothesis H0 : θ = 5 against the two sided alternative based on both the χ2 critical

values and empirically determined critical values. The results are presented in Figures 6

(a), (b). The thick solid line represents the LRT for each case. Notice that the difference

between the power curves for the chi-square critical values versus the empirical critical

values is really minimal – indicating that the chi-square critical value provides a decent

approximation to the true critical value. This is in stark contrast to the performance of

the Hellinger deviance test which requires a really large sample size for the true level to

be even moderately close to the nominal level (e.g., Simpson [20]; Basu, Harris and Basu

[3]).

Next data are generated from 0.9Poisson (θ) + 0.1Poisson (15) mixture, where θ

ranges from 3.5 to 6.5. The results are presented in Figures 6 (c), (d). For comparison

purposes the power curve of the LRT for the no contamination case is presented with

the other graphs as the thick dotted line. While the power curve of the LRT under

contamination shows a dramatic shift with substantial loss of power at several cases, the

other curves (except for α = 0.9) are largely unchanged in comparison, demonstrating

the relative stability of these test statistics under contamination.


Table 8: Estimated biases and MSEs of TDα and ML estimators. 5000 random samples weredrawn from Poisson (5) and 0.1 Poisson(5) + 0.9 Poisson(15) with sample size n = 20, 50.

Poisson (5) 0.1 Poisson(5) + 0.9 Poisson(15)n = 20 n = 50 n = 20 n = 50

TDα Bias MSE Bias MSE Bias MSE Bias MSE0 -0.0788 0.3016 -0.0414 0.1102 0.0665 0.4077 0.1198 0.1665

0.1 -0.0776 0.3006 -0.0407 0.1100 0.0688 0.4080 0.1219 0.16700.2 -0.0739 0.2978 -0.0387 0.1093 0.0758 0.4090 0.1283 0.16880.3 -0.0676 0.2933 -0.0352 0.1082 0.0879 0.4104 0.1397 0.17250.4 -0.0586 0.2874 -0.0302 0.1069 0.1071 0.4161 0.1577 0.17920.5 -0.0464 0.2808 -0.0235 0.1053 0.1354 0.4284 0.1851 0.19170.6 -0.0306 0.2742 -0.0146 0.1038 0.1782 0.4550 0.2279 0.21632/π -0.0238 0.2720 -0.0107 0.1034 0.1994 0.4717 0.2498 0.23110.7 -0.0103 0.2687 -0.0029 0.1028 0.2471 0.5166 0.3004 0.27060.8 0.0164 0.2673 0.0134 0.1032 0.3747 0.6799 0.4432 0.42000.9 0.0545 0.2790 0.0387 0.1083 0.7039 1.3278 0.8228 1.0543HD -0.1579 0.3316 -0.0986 0.1206 0.0367 0.4387 0.1470 0.1956ML 0.0057 0.2643 0.0037 0.1023 1.0185 1.8215 1.0094 1.3138

8. Concluding Remarks

This paper looks at statistical inference based on the TDα disparity for discrete and

continuous models. This method has nice features such as boundedness of disparities,

consistency and the asymptotic efficiency of the estimators, and asymptotic equivalence

of the TDα test to the LRT. The simulations and real data examples show that this

new method provides excellent robust alternatives to the MLE and the LRT without

sacrificing efficiency.

The current standard in minimum disparity estimation is still provided by the HD.

It is imperative that any new research work based on disparities should establish its

position and demonstrate its usefulness vis-a-vis the existing standard. In our examples

and simulations we have demonstrated that our methods are extremely competitive with

those based on the HD, and often outperform the latter, exhibiting the potential practical

value of the new methods.

Appendix. Proofs

Proof of Theorem 5. By the condition (c) and Theorem 1 in Chapter 1 of Devroye

[7], we have as n → ∞∫

|gn(x) − fθ0(x)|dx

a.s.−→ 0.

For convenience, denote n(θ) = TDα(gn, fθ) and θn = Tα(Gn). Since θn minimizes n(·)over Θ, the Taylor series expansion of ∇n(θn) at θ0 yields

0 =√

n∇n(θn) =√

n∇n(θ0) +√

n∇2n(θ†)(θn − θ0),


0.0

0.0

0.0

0.0

0.2

0.2

0.2

0.2

0.4

0.4

0.4

0.4

0.5

0.6

0.6

0.6

0.6

0.8

0.8

0.8

0.8

1.0

1.0

1.0

1.0

3.53.5

3.53.5

4.04.0

4.04.0

4.54.5

4.54.5

5.05.0

5.05.0

5.55.5

5.55.5

6.06.0

6.06.0

6.56.5

6.56.5

θθ

θθ

Pow

er

Pow

er

Pow

er

Pow

er

LRTLRT

LRT(pure)LRT(pure)

α = 0α = 0

α = 0.3α = 0.3

α = 2/πα = 2/π

α = 0.9α = 0.9

(a) (b)

(c) (d)

Figure 6: Estimated powers for the tests under consideration testing H0 : θ = 5 versus H1 : θ 6= 5with level γ = 0.05. 1000 random samples with sample size n = 50 were drawn from Poisson (θ)for (a) and (b); and from 0.9 Poisson(θ) + 0.1 Poisson(15) for (c) and (d). For (a) and (c), χ2

critical value used. For (b) and (d), empirical critical value used.

where θ† is a point on the line segment L(θ0, θn). It follows that

√n(θn − θ0) = −∇2n(θ†)−1√n∇n(θ0).

Now it suffices to prove that

∇2n(θ†)P−→ I(θ0) (13)


and

−√

n∇n(θ0)D−→ N(0, I(θ0)). (14)

First we prove (13). Let δn(θ) = gn/fθ − 1. Differentiating with respect to θ, we

have

∇2n(θ) = −∫

A(δn)∇2fθ +

∫

A′(δn)(δn + 1)uθuTθ fθ

It is easily seen that |A(δ)| and |A′(δ)(δ + 1)| are bounded for all δ. Let us denote

B1 = supδ |A(δ)| and B2 = supδ |A′(δ)(δ + 1)|. It follows from assumption (a) that as

n → ∞∣

∣

∣

∫

A(δn(θ†))(∇2fθ† −∇2fθ0)∣

∣

∣≤ B1

∫

|∇2fθ† −∇2fθ0| P−→ 0, (15)

and∣

∣

∣

∫

A′(δn(θ†)){

δn(θ†) + 1}

(uθ†uTθ†fθ† − uθ0

uTθ0

fθ0)∣

∣

∣≤ B2

∫

|uθ†uTθ†fθ† − uθ0

uTθ0

fθ0| P−→ 0.

(16)

Since A(0) = 0 and δn(θ†) → 0 as n → ∞, using the dominated convergence theorem we

have∫

A(δn(θ†))∇2fθ0

P−→ 0,

and hence by (15)∫

A(δn(θ†))∇2fθ†P−→ 0.

Using assumption (b), A′(0) = 1 and δn(θ†) → 0 as n → ∞, we have∫

A′(δn(θ†)){

δn(θ†) + 1}

uθ0uT

θ0fθ0

P−→∫

uθ0uT

θ0fθ0

,

by the dominated convergence theorem and hence by (16)∫

A′(δn(θ†)){

δn(θ†) + 1}

uθ†uTθ†fθ†

P−→∫

uθ0uT

θ0fθ0

.

Therefore we have (13).

Next we prove (14). Note that

−√

n∇n(θ0) =√

n

∫

A(δn(θ0))∇fθ0,

and from the equations (3.12) and (3.13) of Beran [5],

√n

∫

δn(θ0)∇fθ0=

√n

∫

(gn − fθ0)uθ0

D−→ N(0, I(θ0)).

Therefore, it is enough to prove that

∣

∣

∣

√n

∫

{

A(δn(θ0)) − δn(θ0)}

∇fθ0

∣

∣

∣

P−→ 0. (17)


It can be shown that A′(δ) and A′′(δ)(δ + 1) are bounded. Thus using Lemma 25 of

Lindsay [12], we have a finite B such that∣

∣A(r2 − 1) − (r2 − 1)∣

∣ ≤ B × (r − 1)2 for all

r ≥ 0. Evidently we have∣

∣

∣A(δn(θ0)) − δn(θ0)

∣

∣

∣=

∣

∣

∣A

(

(√

gn/fθ0)2 − 1

)

−[

(√

gn/fθ0)2 − 1

]

∣

∣

∣

≤ B × (√

gn/fθ0− 1

)2.

Using this and (17), we have

∣

∣

∣

√n

∫

{

A(δn(θ0)) − δn(θ0)}

∇fθ0

∣

∣

∣≤ B

√n

∫

(√gn −

√

fθ0

)2|uθ0|.

Using the inequality (a − b)2 ≤ (a − c)2 + (b − c)2, we have

√n

∫

(√gn −

√

fθ0

)2|uθ0| ≤ T1 + T2,

where T1 =√

n∫ (√

gn −√

Egn(X))2|uθ0

| and T2 =√

n∫ (√

fθ0−

√

Egn(X))2|uθ0

|.The term T1 represents the Hellinger deviation of the estimator Gn from its mean and its

convergence to zero in probability has been established by Tamura and Boos [22] using

conditions (e), (f) and (g). The term T2 represents the bias in the Hellinger metric and

its convergence to zero in probability follows from conditions (c) and (d). This completes

the proof. �

Proof of Theorem 6. Let θn be the minimizer of TDα(hǫ,n, fθ). Given a level of

contamination ǫ, if possible, breakdown occurs, that is there exists a sequence {Kn}such that |θn| → ∞ where θn = Tα(Hǫ,n). Then

TDα(hǫ,n, fθn) =

∫

An

D(hǫ,n(x), fθn(x)) +

∫

Acn

D(hǫ,n(x), fθn(x)),

where An = {x : g(x) > max(kn(x), fθn(x))} and D(·, ·) is as in Lemma 1. From A1,

∫

Ankn(x) → 0, and from A3,

∫

Anfθn

(x) → 0 as n → ∞. Similarly from A1 and

A3,∫

Acn

g(x) → 0 as n → ∞. Thus under g(·), the set Acn converges to a set of zero

probability, while under kn(·) and fθn(·), the set An converges to a set of zero probability.

Thus on An, D(hǫ,n(x), fθn(x)) → D((1 − ǫ)g(x), 0) as n → ∞ and

∣

∣

∣

∫

An

D(hǫ,n(x), fθn(x)) −

∫

D((1 − ǫ)g(x), 0)∣

∣

∣→ 0

by dominated convergence theorem. Notice that∫

D((1 − ǫ)g(x), 0) =

∫

D((1 − ǫ)g(x), 0) = (1 − ǫ)

[

4

απtan

(απ

2

)

− 1

]

.

Similarly we have∣

∣

∣

∫

Acn

D(hǫ,n(x), fθn(x)) −

∫

D(ǫkn(x), fθn(x))

∣

∣

∣→ 0.


Notice that∫

D(ǫkn(x), fθn(x)) ≥ C(ǫ − 1) by Jensen’s inequality. It follows that

lim infn→∞

TDα(hǫ,n, fθn) ≥ C(ǫ − 1) + (1 − ǫ)

[

4

απtan

(απ

2

)

− 1

]

= a1(ǫ),

(say). We will have a contradiction to our assumption that {kn} is a sequence for which

breakdown occurs if we can show that there exists a constant value θ∗ in the parameter

space such that for the same sequence {kn},

lim supn→∞

TDα(hǫ,n, fθ∗) < a1(ǫ) (18)

as then the {θn} sequence above could not minimize TDα for every n. We will show

that this is true for all ǫ < 1/2 under the model where θ∗ is the minimizer of∫

D((1 −ǫ)g(x), fθ(x)). Using analogous techniques, assumptions A1, A2, and Lemma 1 we

obtain, for any fixed θ,

limn→∞

TDα(hǫ,n, fθ) = ǫ

[

4

απtan

(απ

2

)

− 1

]

+

∫

D((1 − ǫ)g(x), fθ(x))

≥ ǫ

[

4

απtan

(απ

2

)

− 1

]

+ infθ

∫

D((1 − ǫ)g(x), fθ(x)). (19)

with equality for θ = θ∗. Let a2(ǫ) = ǫ[

4απ tan

(

απ2

)

− 1]

+∫

D((1 − ǫ)g(x), fθ∗(x)).

Notice from (19) that among all fixed θ the divergence TDα(hǫ,n, fθ) is minimized in the

limit by θ∗.

If g(·) = fθt(·), that is the true distribution belongs to the model,

∫

D((1− ǫ)fθt(x),

fθt(x)) = C(−ǫ) which is also the lower bound (over θ ∈ Θ) for

∫

D((1− ǫ)fθt(x), fθ(x)).

Thus in this case θ∗ = θt, and from (19),

limn→∞

TDα(hǫ,n, fθ) = C(−ǫ) + ǫ

[

4

απtan

(απ

2

)

− 1

]

(20)

As a result asymptotically there is no breakdown for ǫ level contamination when a3(ǫ) <

a1(ǫ), where a3(ǫ) is the right hand sides of equation (20). Note that a1(ǫ) and a3(ǫ)

are strictly decreasing and increasing respectively in ǫ, and a1(1/2) = a3(1/2), so that

asymptotically there is no breakdown and lim supn→∞ |Tα(Hǫ,n)| < ∞ for ǫ < 1/2. �

Proof of Theorem 7. Denote n(θ) = TDα(gn, fθ) and θn = Tα(Gn). By a Taylor

series expansion at θn, we have

n(θ∗n) = n(θn) + ∇n(θn)T (θ∗n − θn) +1

2(θ∗n − θn)T∇2n(θ†)(θ∗n − θn),

where θ† is a point on the line segment L(θn, θ∗n). Using ∇n(θn) = 0, it follows that

2n{

n(θ∗n)−n(θn)}

= n(θ∗n−θn)T I(θ0)(θ∗n−θn)+n(θ∗n−θn)T {∇2n(θ†)−I(θ0)}(θ∗n−θn).

(21)


Now we show that, under H0

√n(θML − θ∗ML) =

√n(θ − θ∗) + op(1), (22)

√n(θML − θ∗ML) = Op(1),

√n(θn − θ∗n) = Op(1), (23)

∇2n(θ†) − I(θ0) = op(1). (24)

From the theory of maximum likelihood inference we know that

√n(θML − θ0) = I−1(θ0)

√n

[

1

n

n∑

i=1

uθ0(Xi)

]

+ op(1). (25)

Using ∇n(θn) = 0 and a Taylor expansion of ∇n(θ) at θ0, we have

√n(θn − θ0) =

{

∇2n(θ‡)}−1{ −

√n∇n(θ0)

}

where θ‡ is a point on the line segment L(θ0, θn). By (13), we get ∇2n(θ‡) = I(θ0)+op(1)

and from (17)

−√

n∇n(θ0) =√

n

∫

gnuθ0+ op(1) =

√n

[

1

n

n∑

i=1

uθ0(Xi)

]

+ op(1),

where the second equality above follows from the proof of Theorem 4 in Beran [5].

Therefore,√

n(θn − θ0) = I−1(θ0)√

n

[

1

n

n∑

i=1

uθ0(Xi)

]

+ op(1). (26)

Now by (25) and (26) we have

√n(θML − θn) = op(1) (27)

and √n(θML − θ0)

D−→ N(0, I−1(θ0)), (28)

√n(θn − θ0)

D−→ N(0, I−1(θ0)). (29)

Likewise, under H0, it can be shown that

√n(θ∗ML − θ∗n) = op(1) (30)

and √n(θ∗ML − θ0)

D−→ N(0,Dν0J−1(ν0)D

Tν0

), (31)

√n(θ∗n − θ0)

D−→ N(0,Dν0J−1(ν0)D

Tν0

). (32)

By (27) and (30) it follows that

√n(θML − θ∗ML) =

√n(θn − θ∗n) + op(1),


which is equation (22). Note that equation (23) follows from the use of (28) together

with (31), and (29) together with (32). Equation (24) follows from (13). Using (22), we

have

n(θ∗n − θn)T I(θ0)(θ∗n − θn) = n(θ∗ML − θML)T I(θ0)(θ

∗ML − θML) + op(1). (33)

Therefore, by (21), (23), (24) and (33) we have

2n{

n(θ∗n) − n(θn)}

= n(θ∗ML − θML)T I(θ0)(θ∗ML − θML) + op(1).

Thus the result follows that from the proof of Theorem 4.4.4 of Serfling [18] where it is

shown that, under the null hypothesis

n(θ∗ML − θML)T I(θ0)(θ∗ML − θML)

D−→ χ2(r). �

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Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA.

E-mail: [email protected]

Major area(s): Applied statistics, Quality and reliability engineering, Robust inference.

Bayesian and Interdisciplinary Research Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700

108, India.

E-mail: [email protected]

Major area(s): Minimum distance inference, Robust statistical methods, Discrete data.

(Received March 2011; accepted March 2011)

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Minimum Disparity Inference based on Tangent Disparities