Academic journal article Journal of Risk and Insurance

# Semi-Parametric Specification Tests for Discrete Probability Models

Academic journal article Journal of Risk and Insurance

# Semi-Parametric Specification Tests for Discrete Probability Models

## Article excerpt

ABSTRACT

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 semi-parametric 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 [chi square] 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 frequency distributions. Initially, we develop the arguments in a general 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.

The structure of the article is as follows. "Current Perspectives" outlines current perspectives of frequency distributions and commonly used specification testing methods. The next section presents the GMM-based testing procedure and briefly summarizes some of the properties of the proposed tests. "Two Illustrative Examples" demonstrates the method. "An Empirical Example" illustrates the empirical relevance of the proposed testing procedure through an application to the United Kingdom comprehensive motor policy data of Johnson and Hey (1971). The conclusion follows.

CURRENT PERSPECTIVES

Frequency Distributions

There are various possible choices of claim frequency distributions. A natural starting place is the Poisson distribution with a constant parameter [lambda], which measures the expected number of accidents. Early work using the Poisson distribution in the context of insurance includes Bohlmann (1909), Lundberg (1909), and Cramer (1930). At times when the homogeneous Poisson does not adequately describe the characteristics of data, one can obtain generalized distributions in two common ways. First, one may consider mixed Poisson distributions by treating [lambda] as the outcome of a random variable. With more flexibility in shape than the homogeneous Poisson, mixed Poisson distributions have been extensively used when the heterogeneity of risks arises. For example, the negative binomial, which is of central importance within the family of mixed Poisson distributions because of its convenient mathematical properties, has been viewed as one of the most important alternatives to the homogeneous Poisson in the cla ssical collective risk theory (Dickson et al., 1998). The Poisson-inverse Gaussian distribution is another widely used mixed Poisson distribution. …

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