Anna Clara Monti

Inferential Aspects of the Skew Exponential Power Distribution

When the analysis of real data indicates that normality assumptions are untenable, more flexible models that cope with the most prevalent deviations from normality can be adopted. In this context, the skew exponential power (SEP) distribution warrants special attention, because it encompasses distributions having both heavy tails and skewness, it allows likelihood inference, and it includes the normal model as a special case. This article concerns likelihood inference about the parameters of the SEP family. In particular, the information matrix of the maximum likelihood estimators (MLEs) is obtained and finite-sample properties of the estimators are investigated numerically. Special attention is given to the properties of the MLEs and likelihood ratio statistics when the data are drawn from a normal distribution, because this case is relevant for using the SEP distribution to test for normality. Application of the SEP distribution in robust estimation problems is considered for both independent and dependent data. Under moderate deviations from normality, estimators obtained under the SEP distribution are shown to outperform the normal-based estimators and to compete with robust estimators; furthermore, the SEP distribution offers the benefits of having a specified distribution, such as interpretable location and scale parameters under nonnormality.

Robust inference for ordinal response models

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.

Robust portfolio estimation under skew-normal return processes

In this paper, we study issues related to the optimal portfolio estimators and the local asymptotic normality (LAN) of the return process under the assumption that the return process has an infinite moving average (MA) (∞) representation with skew-normal innovations. The paper consists of two parts. In the first part, we discuss the influence of the skewness parameter δ of the skew-normal distribution on the optimal portfolio estimators. Based on the asymptotic distribution of the portfolio estimator ĝ for a non-Gaussian dependent return process, we evaluate the influence of δ on the asymptotic variance V(δ) of ĝ. We also investigate the robustness of the estimators of a standard optimal portfolio via numerical computations. In the second part of the paper, we assume that the MA coefficients and the mean vector of the return process depend on a lower-dimensional set of parameters. Based on this assumption, we discuss the LAN property of the returns distribution when the innovations follow a skew-normal law. The influence of δ on the central sequence of LAN is evaluated both theoretically and numerically.

Testing for sub-models of the skew t-distribution

The skew t-distribution includes both the skew normal and the normal distributions as special cases. Inference for the skew t-model becomes problematic in these cases because the expected information matrix is singular and the parameter corresponding to the degrees of freedom takes a value at the boundary of its parameter space. In particular, the distributions of the likelihood ratio statistics for testing the null hypotheses of skew normality and normality are not asymptotically

Variance stabilization for a scalar parameter

\chi ^2

Approximations to the profile empirical likelihood function for a scalar parameter in the context of M-estimation

χ2. The asymptotic distributions of the likelihood ratio statistics are considered by applying the results of Self and Liang (J Am Stat Assoc 82:605–610, 1987) for boundary-parameter inference in terms of reparameterizations designed to remove the singularity of the information matrix. The Self–Liang asymptotic distributions are mixtures, and it is shown that their accuracy can be improved substantially by correcting the mixing probabilities. Furthermore, although the asymptotic distributions are non-standard, versions of Bartlett correction are developed that afford additional accuracy. Bootstrap procedures for estimating the mixing probabilities and the Bartlett adjustment factors are shown to produce excellent approximations, even for small sample sizes.

Robustness issues for cub models

This note explores the robustness properties of a general class of ineqyality measures which includes the Bonferroni and the Gini indexes as special cases and proposes some modifications in order to make them outlier resistant.