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Abstract
In this paper a robust version of the classical Wald test statistics for linear hypothesis in the logistic regression model is introduced and its properties are explored. We study the problem under the assumption of random covariates although some ideas with non random covariates are also considered. The family of tests considered is based on the minimum density power divergence estimator instead of the maximum likelihood estimator and it is referred to as the Wald-type test statistic in the paper. We obtain the asymptotic distribution and also study the robustness properties of the Wald type test statistic. The robustness of the tests is investigated theoretically through the influence function analysis as well as suitable practical examples. It is theoretically established that the level as well as the power of the Wald-type tests are stable against contamination, while the classical Wald type test breaks down in this scenario. Some classical examples are presented which numerically substantiate the theory developed. Finally a simulation study is included to provide further confirmation of the validity of the theoretical results established in the paper.
Highlights
Experimental settings often include dichotomous response data, wherein a Bernoulli model may be assumed for independent response variables Y1, ..., Yn, withPr(Yi = 1) = πi and Pr(Yi = 0) = 1 − πi, i = 1, ..., n.In many cases, a series of explanatory variables xi0, ..., xik may be associated with each Yi
Our interest in this paper is to present a family of Wald-type test statistics based on the robust minimum density power divergence estimator for testing the general linear hypothesis given in (1.3)
Combining above results and simplifying, we get the required expression of PIF as presented in the theorem. It is clear from the above theorem that, the asymptotic level of the proposed Wald-type test statistic will be unaffected by a contiguous contamination for any values of the tuning parameter λ, whereas the power influence function will be bounded whenever the influence function of the MDPDE is bounded
Summary
Experimental settings often include dichotomous response data, wherein a Bernoulli model may be assumed for independent response variables Y1, ..., Yn, with. Notice that if we consider M = Ik+1 and m = β0, we get the Wald-type test statistic presented by Bianco and Martinez (2009) based on a weighted Bianco and Yohai (1996) estimator. In the recent years several authors have attempted to derive robust estimates of the parameters in the logistic regression model; see for instance Pregibon (1982), Morgenthaler (1992), Carroll and Pederson (1993), Christmann (1994), Bianco and Yohai (1996), Croux and Haesbroeck (2003), Bondell (2005; 2008) and Hobza et al (2008; 2017). Our interest in this paper is to present a family of Wald-type test statistics based on the robust minimum density power divergence estimator for testing the general linear hypothesis given in (1.3). The Wald-type test statistics, based on the minimum density power divergence estimator, are presented, together with their asymptotic properties.
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