Maximum Weighted Likelihood Estimator in Logistic Regression

The least weighted squares estimator is a well known technique in robust regression. Its likelihood analogy in logistic regression is the maximum weighted likelihood estimator, proposed in Vandev and Neykov (1998) and Mueller and Neykov (2003). This article mentions already proved properties, shows its inconsistency and compare it to the other estimators by an extensive simulation. Introduction For estimation of parameters in regression models with categorical outcome, maximum likelihood estimator (MLE) is commonly used. A disadvantage of this method is high sensitivity to deviation from assumptions, e.g. to outliers in the dataset (see figure 1). To solve this problem, many robust alternatives of MLE were developed. Carroll and Pederson (1993) andWang and Carroll (1995) discussed different types of Mand GM-estimators for binary regression, depending on the type of contamination. However, the breakdown point of these estimators drops down for higher number of covariates (cannot exceed 1/(p+1), where p is the number of covariates). Christmann (1994) proposed a high breakdown point estimator based on least quantile squares estimation (LQS, Rousseeuw and Leroy (1987)). In this paper maximum weighted likelihood estimator will be discussed. Throughout the paper it will be assumed that observations Y1, . . . , Yn are independent and Yi has density fi(yi;β). A vector y = (y1, . . . , yn) ′ denotes a vector of realizations of (Y1, . . . , Yn) and w = (w1, . . . , wn) is a vector of weights. Let li(β) = log f(yi;β). −4 −2 0 2 4 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0