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The Logarithmic Super Divergence and its use in Statistical Inference

This paper introduces a new superfamily of divergences that is similar in spirit to the S-divergence family introduced by Ghosh et al. (2013). This new family serves as an umbrella that contains the logarithmic power divergence family (Renyi, 1961; Maji, Chakraborty and Basu 2014) and the logarithmic density power divergence family (Jones et al., 2001) as special cases. Various properties of this new family and the corresponding minimum distance procedures are discussed with particular emphasis on the robustness issue; these properties are demonstrated through simulation studies. In particular the method demonstrates the limitation of the first order influence function in assessing the robustness of the corresponding minimum distance procedures.


Introduction
The density-based minimum divergence approach, which includes both χ 2 type (Csisźar, 1963) and Bregman (Bregman, 1967) divergences, has long history. A prominent member of the class of density-based divergences is the Pearson's χ 2 (Pearson, 1900) which started its journey from the very early days of formal research in statistics. From the robustness perspective, however, Beran's 1977 work is the first useful reference in the literature of density-based minimum divergence inference. In the present paper we focus on a new subclass of density based divergences which encompasses some variants of the power divergence measure of Cressie and Read (1984) and the density power divergence of Basu et al. (1998) and discuss possible applications in statistical inference. Among many other things, our analysis highlights the limitation of the first order influence function analysis as an indicator of the robustness of these procedures.
In this article our primary aim is to describe some statistical uses of the proposed superfamily of divergences. To keep this focus clear, we will push most of the technical details including the proofs of the asymptotic distribution to a separate article, and will simply state the relevant theoretical results appropriately in the present context. The asymptotic results will be presented in Maji, Ghosh and Basu (2014).
The rest of the paper is organized as follows.

The Logarithmic Super Divergence and Parametric Estimation
We first define the generalized S-divergence (GSD) family. Given two probability density functions g and f with respect to the same measure, the GSD family is defined, as a function of two real parameters β and γ, as where A = 1 + γ(1 − β) and B = β − γ(1 − β), and ψ(x) is a function with suitable properties. Note that ψ(x) = x in (1) recovers the S-divergence family considered by Ghosh et al. (2013); the function ψ(x) = log x generates another family of divergences which we will refer to as the logarithmic super divergence (logarithmic S-divergence or LSD for short). The generalizaion given in (1) is in the spirit of the general form considered by Kumar and Basu (2014) in relation to the density power divergence measure. However we will defer the exploration of the properties of this generalized divergence (including the properties that ψ must possess to be statistically useful) to a sequel paper, and concentrate on the properties of the LSD family in the present paper. The Logarithmic S-Divergence (LSD) has the form where A and B are as defined earlier. It has to be noted that, A + B = 1 + β. For β = 0 (A = 1 + γ, B = −γ), this family coincides with the logarithmic power divergence (LPD) family with parameter γ where LPD has the form while γ = 0 gives the logarithmic density power divergence (LDPD) family with parameter β where LDPD has the form LDPD β (g, f ) = log f 1+β − 1 + 1 β log f β g + 1 β log g 1+β , β ≥ 0. (4) Clearly, for β = 0 and γ = 0, this family coincides with the likelihood disparity (LD) where LD has the form LD(g, f ) = g log g f .
This is a version of the Kullback-Leibler divergence. On the other hand, the value β = 1 generates the divergence irrespective of the value of γ. Jones et al. (2001) have presented a comparison of the method based on DPD and LDPD, where a (weak) preference for DPD was indicated. Later on Fujisawa and Eguchi (2008) and Eguchi (2013) have reported some advantages for LDPD for parameter estimation under heavy contamination. Similar comparison between the S-divergence and the logarithmic S-divergence remain among our agenda for future work.
Theorem 1. Given two densities g and f , the measure LSD β,γ (g, f ) represents a genuine statistical divergence for all β ≥ 0 and γ ∈ R.
Proof. A simple application of Holder's inequality establishes the above result.

Estimating Equation of the LSD
Consider a parametric class of model densities {f θ : θ ∈ Θ ⊆ R p } and suppose that our interest is in estimating θ. Let G denote the distribution function corresponding to the true density g. The minimum LSD functional T β,γ (G) at G is defined through the relation A simple differentiation gives us the estimating equation for θ, which is For β = 0 (A = 1 + γ, B = −γ), the equation becomes the same as the estimating equation of the logarithmic power divergence family with parameter γ. For γ = 0 (A = 1, B = β), on the other hand, it is the estimating equation for the LDPD measure. It takes the value θ when the true density g = f θ is in the model; when it does not, θ g β,γ = T β,γ (G) represents the best fitting parameter, and f θ g is the model element closest to g in terms of logarithmic super divergence. For simplicity in the notation, we suppress the scripts and refer to θ g β,γ as simply θ when there is no scope for confusion.

Influence Function
The influence function is one of the most important heuristic tools in robust inference. Consider the minimum LSD functional T β,γ (G). The value θ = T β,γ (G) solves the equation (8). Consider the estimating equation at the mixture contamination density g (x) = (1− ) g(x)+ I y (x) where I y (x) is the indicator function at y. Let θ be the corresponding functional which solves the estimating equation in this case. Taking a derivative of both sides of this estimating equation and evaluating at = 0, the influence function is found Under the parametric set-up of Section 2.1, consider a discrete family of distributions. We will use the term "density function" generally for the sake of a unified notation, irrespective of whether the distribution is discrete or continuous. Let X 1 , . . . , X n be a random sample from the true distribution having density function g and let the distribution have support χ = {0, 1, 2, · · · }. Denote the relative frequency at x from the data by r n (x) = 1 n n i=1 I x (X i ). Representing the logarithmic S-divergence in terms of the parameter β and γ (as given in Section 2), letθ β,γ be the estimator obtained by minimizing LSD β,γ (ĝ, f θ ) over θ ∈ Θ, whereĝ is a suitable nonparametric density estimate of g; in the discrete case the vector of relative frequencies r n based on the sample data is the canonical choice forĝ.
In this paper we will primarily describe the statistical applications of the minimum distance procedures that are generated by the logarithmic S-divergence. However, for the sake of completeness, we also present the asymptotic distri-bution of the estimators which has been separately established in Maji, Ghosh and Basu (2014).
When g is replaced by r n , the estimating equation (8) may be expressed as 4 Testing Parametric Hypothesis using the LSD Measures

One Sample problem
We consider a parametric family of densities F = {f θ : θ ∈ Θ ⊆ R p } as introduced earlier. Suppose we are given a random sample X 1 , . . . , X n of size n from the population. Based on this sample, we want to test the hypothesis When the model is correctly specified and the null hypothesis is correct, f θ0 is the data generating density. We consider the test statistics based on the LSD with parameter β and γ defined by where LSD β,γ (fθ β,γ , f θ0 ) has the form given in (2). Then the following theorem becomes useful in obtaining the critical values of the test statistics in (21).
To see the robustness properties of the LSD based test, we study the influence function analysis of the test statistics as in Hampel et al. (1986), Ghosh and Basu (2014) etc. We define the corresponding LSD based test functional (LSDT) for one sample simple hypothesis problem as described above as (ignoring the sample size dependent multiplier) where T β,γ (G) is the minimum LSD functional defined in Section 2.2. Then, considering the contaminated distribution G associated with g , Hampel's first-order influence function of the LSDT functional turns out to be zero at the null distribution G = F θ0 . However, corresponding second order influence Therefore the robustness of the LSDT functional depends directly on the robustness of the minimum LSD estimator used in constructing the test statistics. So, following the arguments of Section 2.2 it follows that, the proposed test will have bounded influence function whenever β > 0 implying its robustness and has unbounded influence function at β = 0 implying the lack of robustness. Figure 2 shows the second order influence function of the N (θ, 1) model at the simple null θ = 0; the equivalence with the corresponding influence function of the minimum LSD estimator presented in Figure 1 is quite clear.

Two Sample Problem
Again consider a parametric family of densities {f θ : θ ∈ Θ ⊆ R p } as above in one sample problem, but here we are given two random samples X 1 , . . . , X n of size n and Y 1 , . . . , Y m of size m from two populations having parameters θ 1 and θ 2 respectively and based on these two samples, we want to test for the homogeneity of the two samples, i.e. to test the hypothesis We will consider the estimator (1)θ β,γ and (2)θ β,γ of θ 1 and θ 2 respectively, obtained by minimizing the LSD having parameter β, γ and then as before, we consider the test statistic based on the LSD with parameter β and γ as follows We present the asymptotic distribution of the test statistics S β,γ and A β (·) as defined in previous section and

Performance of the Minimum LSD Estimator : Simulation in the Poisson Model
To explore the performance of the proposed minimum LSD estimators, we have done several simulation studies under the Poisson model with sample size of n = 50. We simulate data from a Poisson distribution with parameter θ = 4 and compute the empirical bias and the MSE of the minimum LSD estimators of θ based on 1000 replications. The results obtained are reported in Tables  1 and 2 respectively. Clearly both the bias and MSE are quite small for any (β, γ) combination; however the MSE increases slightly with β. Next to study the robustness properties of the minimum LSD estimators we repeat the above study, but introduce a contamination in the simulated samples by replacing 10% of it by P oisson(θ = 12) observations. The corresponding values of the empirical bias and MSE, against the target value of θ = 3, are presented in Tables 3 and 4 respectively. Note that, the minimum LSD estimators are seen to be robust for all β ∈ [0, 1] if γ < 0 and for suitably large values of β if γ ≥ 0. However, the estimators corresponding to small β close to zero and γ ≥ 0.    The numeral examples and simulation results presented in the previous section clearly shows that the robustness of minimum LSD estimators in terms of its bias and MSE under data contamination depends on the parameter γ for smaller values of β. However, according to the classical literature, its first order influence function suggests that (see Section 2.2) its robustness will be independent of the parameter γ for all values of β. Thus, the classical approach of robustness measure through the first order influence fails in the case of minimum divergence estimation with the logarithmic super divergence family. Similar limitations of the first order influence functions was also observed by   Lindsay (1994) and Ghosh et al. (2013) for the case of power divergence family and the S-divergence family; accordingly they have proposed some alternative measure of robustness. In this section, we use some of those alternative measures to explain the robustness of the proposed minimum LSD estimators.

Higher Order Influence Analysis
The higher (second) order influence function analysis for studying the robustness of a minimum divergence estimators was used by Lindsay (1994)  Theorem 5. Under the above mentioned set-up with a scalar parameter θ, if true distribution belonging to the model family then the second order influence function of the minimum LSD estimator defined by the estimating equation (8) is where u θ (y) = ∂ ∂θ u θ (y). Example (Poisson Mean): Let us now consider a numerical simulation to study the performance of the above second order influence analysis through its application in case of the Poisson model with mean θ. Using the special structure of one parameter exponential family, of which Poisson distribution is a special case, we compute the first and second order bias approximation using their respective expressions as given above and in Section 2.2. However, for brevity, we will only present some particular simulation result with θ = 4, the contamination point y = 12 and specific (β, γ) combinations and the corresponding bias plots are shown in Figures 3, 4 and 5 respectively for γ = 0, γ > 0 and γ < 0.
Comments on Figure 3 (γ = 0): As expected both first order and second order influence function for β = 0 gives a straight line. The bias approximation decreases as β increases for both first and second order influence function. The difference of approximation between first and second order decreases as β increases.
Comments on Figure 4 (γ > 0): Keeping γ fixed the difference between bias approximation among first and second order decreases as β increases.
Comments on Figure 5 (γ < 0): As expected for this case the bias approximation is more for the first order influence function compared to the second order but the difference among two types of influence function shows same behavior compared to the case γ > 0.

A Breakdown Point Result : Location Model
Another popular alternative to the influence function analysis is the breakdown point theory; following Simpson (1987) we will say that the estimator T The particular property of this family, that helps to make the calculations simpler, is which is independent of the parameter θ. Using this and the increasing nature of the logarithmic function, the minimum LSD estimator for a location model is seen to be the maximizer of only the one integral term f B g A whenever A > 0 and B > 0. However, under the same location model the minimum for the minimum LSD estimator for γ < 0 the power usually does not go to 1 but for low β and high negative value of γ it goes very close. For γ close to 0 and low β, the power becomes less than 0.5 but this is not much common throughout the table. For sample size 100, the power is usually becomes 1 except for very few combinations of (β, γ) and this fact is maintained for sample size 50 also. As shown earlier, for β = 1, the divergence is independent of γ, that fact is also evident from the result that both level and power for all values of γ is same for β = 1.

Conclusion
Logarithmic super divergence family acts as a super family of both LPD and LDPD family. Its usage in both statistical estimation and testing of hypotheses have been studied. Along with the limitation of the first order influence function and the breakdown point under location model have also extensively studied. Computational exercises have shown that there exist a region of the  parameter which usually performs better where outliers are present in the observations.