Some measures of robustness for unbiased estimators in one-parameter natural exponential families with quadratic variance function

Abstract

In this paper two measures to highlight the possible effect of an observation on the UMVU estimate are proposed. Our study is based in expansions in terms of orthogonal polynomials for the UMVUE when sampling from a NEF-QVF. We obtain the conditional bias and the asymptotic mean sensitivity curve (AMSC) for the UMVUE. We observe that these measures depend on parametric function under consideration at the true and unknown value of the parameter. We study in detail their properties and relationships as well as to the Hampel's influence function. In fact, we note that the AMSC also verifies for the UMVUE in the NEF-QVF some of most relevant properties of influence function. Also a case-deletion influence diagnostic and some simulations are included to illustrate our results.

Some uses of orthogonal polynomials in statistical inference

“… Lower bounds for the variance of T n ( S n ). The effect of an observation on T n ( S n ).Two robustness measures to assess the effect of a fixed observation, x ∈ support( X 1 ), on T n ( S n ) were proposed in Barranco‐Chamorro and Moreno‐Rebollo 5 . The measures are the conditional bias and the asymptotic mean sensitivity curve (AMSC) for T n ( S n ).…”

“…Two robustness measures to assess the effect of a fixed observation, x ∈ support( X 1 ), on T n ( S n ) were proposed in Barranco‐Chamorro and Moreno‐Rebollo 5 . The measures are the conditional bias and the asymptotic mean sensitivity curve (AMSC) for T n ( S n ).…”

“…Every random variable (rv) X (or random vector) with finite moments generates a set of orthogonal polynomials (OPS), which can be used to obtain properties related to the distribution of X . This technique has been used in statistical inference, mainly connected to the exponential family of distributions, as it can be seen for instance in Abbey and David, 1 Morris, 2,3 López‐Blázquez, 4 and Barranco‐Chamorro and Moreno‐Rebollo 5 …”