Actuarial statistics I: likelihood characterizations.

Abstract

The present work establishes some likelihood properties of Bernoulli mixed discretecontinuous models with special emphasis on actuarial models. Potential statistical applications include one-parametric deformations from discrete to continuous families of distributions, perturbed models for Robust Statistics, models with unobservable events of use in the (re)insurance of large claims, insurance claims models with outliers, distribution-free reinsurance models via stop-loss extremal distributions (Section 1). Based on a mean equivalence relation for parametric random functions, a mean equivalence class of log-likelihoods is considered. The corresponding systems of sample mean equations lead to a class of maximum likelihood mean equivalent estimators, or shortly likelihood estimators (Section 2). For Bernoulli mixed discrete-continuous models a likelihood estimator, which is asymptotically equal to the maximum likelihood estimator, is constructed. Sometimes it is equal to the maximum likelihood estimator. This is the case for a specific insurance claims model with outliers. Furthermore this estimator appears to be useful as estimation method for the unknown parameters of a stop-loss ordered extremal distribution associated to the class of positive risks with fixed mean and variance (Section 3). The mainly theoretical Section 4 studies the relationship between pseudo-estimators and orthogonal parameters. After some motivation for the use of pseudo-estimators has been given, the well-known information inequality for pseudo-estimators is recalled. It extends the classical Cramer-Rao lower bound for unbiased estimators. Several examples, which illustrate the usefulness of the generalized inequality, are presented. As a main result, a substantial generalization of a characterization obtained previously by the author is derived. It characterizes unbiased pseudo-estimators related to the class of maximum likelihood mean equivalent estimators (introduced in Section 2) under an orthogonal condition of the parameter of interest. A thorough discussion and several examples follow. In particular a complete likelihood characterization of one-parametric maximum likelihood estimators is formulated (Corollary 4.2). The latter result is applied in Section 5 to provide rational justifications for the use of some categories of statistical models with location or scale parameters encountered in the fields of Economy and Social Sciences. First two simpler variants of classical results are formulated. Then practically quite useful characterizations of the compound Poisson, binomial and negative binomial gamma models in collective risk theory are derived. In Section 6 sufficient conditions for the sample mean to be a somewhat robust likelihood estimator in a Bernoulli mixed discrete-continuous model are given. These conditions characterize in Section 7 a mean scaled gamma outlier model encountered in the recent actuarial literature.