Abstract
The minimum divergence estimators have proved to be useful tools in the area of robust inference. The robustness of such estimators are measured using the classical Influence functions. However, in many complex situations like testing a composite hypothesis using divergence require the estimators to be restricted into some subspace of the parameter space. The robustness of these restricted minimum divergence estimators are very important in order to have overall robust inference. In this paper we provide a comprehensive description of the robustness of such restricted estimators in terms of their Influence Function for a general class of density based divergences along with their unrestricted versions. In particular, the robustness of some popular minimum divergence estimators are also demonstrated under certain usual restrictions. Thus this paper provides a general framework for the influence function analysis of a large class of minimum divergence estimators with or without restrictions on the parameters.
Highlights
The minimum divergence approach has proved to be a very useful one in the context of parametric statistical inference
In many practical situations we need to consider restrictions for which the rank is strictly less than r and we can not apply the above Theorem 3.1 directly to obtain the influence function of the corresponding Restricted Minimum Divergence Estimator (RMDE)
Based on the general results obtained in the previous sections, one can describe the influence function analysis and the asympto√tic distributions of any minimum divergence estimators (MDEs) or RMDE provided one can prove only their n-consistency
Summary
The minimum divergence approach has proved to be a very useful one in the context of parametric statistical inference. 2. Density-based minimum divergence estimators (MDEs) and their influence functions: A general form. The following theorem provides a general form of the influence function of the MDEs corresponding to the particular divergence given in Equation (2.1); for brevity in presentation the proof is given in Appendix A.1. Under standard regularity conditions on model densities, the influence function of the minimum divergence functional Tρ corresponding the particular divergences given by Equation (2.1) has the form. From the general theory developed above, the minimum density power divergence estimators with α > 0 are robust with respect to the outliers and that corresponding to α = 0 is non-robust This fact exactly coincides with the corresponding results derived independently in Basu et al [2].
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