Abstract
The present paper deals with the robustness of estimators and tests for ordinal response models. In this context, gross-errors in the response variable, specific deviations due to some respondents’ behavior, and outlying covariates can strongly affect the reliability of the maximum likelihood estimators and that of the related test procedures. The paper highlights that the choice of the link function can affect the robustness of inferential methods, and presents a comparison among the most frequently used links. Subsequently robust $M$-estimators are proposed as an alternative to maximum likelihood estimators. Their asymptotic properties are derived analytically, while their performance in finite samples is investigated through extensive numerical experiments either at the model or when data contaminations occur. Wald and $t$-tests for comparing nested models, derived from $M$-estimators, are also proposed. $M$ based inference is shown to outperform maximum likelihood inference, producing more reliable results when robustness is a concern.
Highlights
In recent years, the interest in ordinal data has been constantly growing since these data convey relevant information on several scientific and applied research areas, such as medicine, psychology, sociology, political sciences, economics, marketing, and so on
The most popular approach to ordered response models initially advocated by McCullagh (1980) is based on the assumption that a latent variable drives the response and the model is embedded within the Generalized Linear Model framework as formalized by Nelder and Wedderburn (1972) and McCullagh and Nelder (1989)
When the model is estimated by M LEs, the tests are based on the Likelihood Ratio (LR) statistic, whereas when M estimation is performed the tests are carried out through the Wald-type statistic introduced in (16)
Summary
The interest in ordinal data has been constantly growing since these data convey relevant information on several scientific and applied research areas, such as medicine, psychology, sociology, political sciences, economics, marketing, and so on. A different perspective, more related to the psychological process of selection, leads to the cub models (Piccolo, 2003; Iannario and Piccolo, 2016), so called because they parameterize the probability of a given response as a mixture of a shifted Binomial and a discrete Uniform random variable. This approach does not require the specification of a model for the latent variable and describes directly the effect of the covariates on the feeling and the uncertainty underlying the respondents’ choices. We investigate the choice of the tuning constant with respect to the efficiency and the robustness of the corresponding estimator
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