Semi-Parametric Specification Tests for Discrete Probability Models

Loss functions play an important role in analyzing insurance portfolios. A fundamental issue in the study of loss functions involves the selection of probability models for claim frequencies. In this article, we propose a semiparametric approach based on the generalized method of moments (GMM) to solve the specification problems concerning claim frequency distributions. The GMM-based testing procedure provides a general framework that encompasses many specification problems of interest in actuarial applications. As an alternative approach to the Pearson 2 and other goodness-of-fit tests, it is easy to implement and should be of practical use in applications involving selecting and validating probability models with complex characteristics. Introduction Virtually all insurance problems are about the building of a mathematical model that can be used to quantify the loss function and to predict future insurance costs. The usual starting place of such practice is the search of a model for the claim frequency distribution. In many cases, when confronting a large collection of distributions from which to choose, one has to narrow the selection to a single model. The chosen model should provide a balance between simplicity and conformity to the available data. In this article, we provide a semi-parametric approach based on the generalized method of moments (GMM) to solve the specification problems concerning claim frequencydistributions. Initially,wedevelop the arguments in ageneral setting. Then, for illustrative purposes, we focus on a few important special cases. As a general framework, the GMM-based testing procedure provides much of the flexibility needed to encompass a variety of specification problems.An appealing feature of this semi-parametric approach is that it does not require complete knowledge of the distribution but only demands the specification of a set of moment conditions that the model should satisfy. Since it depends only upon moment restrictions of the model of interest, it is easy to implement even when the problem involves complex distributional forms. Yue Fang is in the Decision Sciences department, Lundquist College of Business, University of Oregon. The author is grateful to the editor and two anonymous referees for their helpful comments on an earlier draft of the article.