Credibility using semiparametric models and a loss function with a constancy penalty

Abstract

In credibility ratemaking, one seeks to estimate the conditional mean of a given risk. The most accurate estimator (as measured by squared error loss) is the predictive mean. To calculate the predictive mean one needs the conditional distribution of losses given the parameter of interest (often the conditional mean) and the prior distribution of the parameter of interest. Young (1997. ASTIN Bulletin 27, 273–285) uses kernel density estimation to estimate the prior distribution of the conditional mean. She illustrates her method with simulated data from a mixture of a lognormal conditional over a lognormal prior and finds that the estimated predictive mean is more accurate than the linear Bühlmann credibility estimator. However, generally, in her example, the estimated predictive mean was more accurate only up to the 95th percentile of the marginal distribution of claims. Beyond that point, the credibility estimator occasionally diverged widely from the true predictive mean.To reduce this divergence, we propose using the loss function of Young and De Vylder (2000. North American Actuarial Journal, 4(1), 107–113). Their loss function is a linear combination of a squared-error term and a term that encourages the estimator to be close to constant, especially in the tails of the distribution of claims, where Young (1997) noted the difficulty with her semiparametric approach. We show that by using this loss function, the problem of upward divergence noted in Young (1997) is reduced. We also provide a simple routine for minimizing the loss function, based on the discussion of De Vylder in Young (1998a. North American Actuarial Journal 2, 101–117).