Abstract
Minimum Renyi’s pseudodistance estimators (MRPEs) enjoy good robustness properties without a significant loss of efficiency in general statistical models, and, in particular, for linear regression models (LRMs). In this line, Castilla et al. considered robust Wald-type test statistics in LRMs based on these MRPEs. In this paper, we extend the theory of MRPEs to Generalized Linear Models (GLMs) using independent and nonidentically distributed observations (INIDO). We derive asymptotic properties of the proposed estimators and analyze their influence function to asses their robustness properties. Additionally, we define robust Wald-type test statistics for testing linear hypothesis and theoretically study their asymptotic distribution, as well as their influence function. The performance of the proposed MRPEs and Wald-type test statistics are empirically examined for the Poisson Regression models through a simulation study, focusing on their robustness properties. We finally test the proposed methods in a real dataset related to the treatment of epilepsy, illustrating the superior performance of the robust MRPEs as well as Wald-type tests.
Highlights
Generalized linear models (GLMs) were first introduced by Nelder and Wedderburn [1] and later expanded upon by McCullagh and Nelder [2]
The GLMs represent a natural extension of the standard linear regression models, which enclose a large variety of response variable distributions, including distributions of count, binary, or positive values
We denote by μi the expectation of the random variable Yi and we assume that there exists a monotone differentiable function, so called link function g, verifying g(μi ) = xiT β, with β = ( β 1, . . . , β k ) ∈ Rk (k < n) the regression parameter vector
Summary
Generalized linear models (GLMs) were first introduced by Nelder and Wedderburn [1] and later expanded upon by McCullagh and Nelder [2]. Considered robust Wald-type tests based on the minimum DPD estimator, but assuming random explanatory variables for the GLM. The main purpose of this paper is to introduce new robust Wald-type tests based on the MRPE under fixed (not random) explanatory variables. Broniatowski et al [16] presented robust estimators for the parameters of the linear regression model (LRM) with random explanatory variables and Castilla et al [17] considered Wald-type test statistics, based on MRPE, for the LRM. The robustness of all the previous estimators is based on density power weight, f (y, θ )l , which gives a small weight to outliers observations This idea was developed by Basu et al [15] for the minimum. We empirically examine the performance of the proposed robust estimators and Wald-type test statistics for the Poisson regression model through a simulation study in Section 5, and we illustrate its applicability with real data sets for binomial and Poisson regression
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