Loss Models: From Data to Decisions, 4th Edition

B.1 Introduction

The 16 models presented in this appendix fall into three classes. The divisions are based on the algorithm by which the probabilities are computed. For some of the more familiar distributions these formulas will look different from the ones you may have learned, but they produce the same probabilities. After each name, the parameters are given. All parameters are positive unless otherwise indicated. In all cases, p_k is the probability of observing k losses.

For finding moments, the most convenient form is to give the factorial moments. The jth factorial moment is μ_(j) = E[N(N − 1) ··· (N − j + 1)]. We have E[N] = μ₍₁₎ and Var(N) = μ₍₂₎ + μ₍₁₎ − μ²₍₁₎.

The estimators presented are not intended to be useful estimators but. rather, provide starting values for maximizing the likelihood (or other) function. For determining starting values, the following quantities are used (where n_k is the observed frequency at k [if, for the last entry, n_k represents the number of observations at k or more, assume it was at exactly k] and n is the sample size):

When the method of moments is used to determine the starting value, a circumflex (e.g., ) is used. For any other method, a tilde (e.g., ) is used. When the starting value formulas do not provide admissible parameter values, a truly ...