MINIMUM DISTANCE ESTIMATION OF LOSS DISTRIBUTIONS

Loss distributions have a number of uses in the pricing and reserving of casualty insurance. Many authors have recommended maximum likelihood for the estimation of the parameters. It has the advantages of asymptotic opti-mality (in the sense of mean square error) and applicabii-ity (the likelihood function can always be written). Also, it is possible to estimate the variance of the estimate, a use-ful rool in assessing the accuracy of any results. The only disadvantage of maximum likelihood is thar the objective function does not relate to the actuarial problem being investigated. Minimum distance estimates can be tailored fo reflect the goals of the analysis and, as such, should give more appropriate answers. The purpose of this paper is to demonstrate that these estimates hare rhe second and third desirable qualities bt,ith maximum likelihood. 1, DEFINITIONS, NOTATION, AND AGENDA We start with a definition of a minimum distance estimate. Let G(c; fl) be any function of c that is uniquely related to AC; 6), the probability density function (pdf) of the population. By uniquely re-lated we mean that if you know A you can obtain G and vice versa. Call G the model functional. Let f,(c) be the empirical density. It assigns probability l/n to each of the n observations in the sample. Let G,(c) be found from f, in the same way that G is from f. Call G,, the empirical functional. The objective function i