Credible Means are exact Bayesian for Exponential Families

Abstract

The credibility formula used in casualty insurance experience rating is known to be exact for certain prior-likelihood distributions, and is the minimum least-squares unbiased estimator for all others We show that credibility is, in fact, exact for all simple exponential families where the mean is the sufficient statistic, and is also exact in an extended sense for all regular distributions with their natural conjugate priors where there is a fixed-dimensional sufficient statistic ACKNOWLEDGEMENT

Credit Portfolios, Credibility Theory, and Dynamic Empirical Bayes

“…where Q is the risk-neutral measure under which the drift of the GBM is r instead of μ, and 66 . Therefore, the probability of default on the debt at maturity is…”

Exact Credibility and Tweedie Models

“…The function b(q) is twice differentiable with a unique inverse for the first derivative bЈ(q). With = 1 and all w i = 1, (2.1) is the exponential family with canonical parameter considered by Jewell (1974). Following Jørgensen (1997), the models defined by (2.1) are called exponential dispersion models (EDMs).…”

“…Now, the frequency function in (2.1) is defined in terms of the canonic parameter q = h(m), rather than the expectation m, and in our case this parameter becomes q ik = h(m i u k ). We make the corresponding transformation of the random effect and introduce the random variable Q k = h(U k ), which corresponds to the risk parameter in Jewell (1974) and other sources on credibility theory, taking values q k = h(u k ).…”

“…Now, the U k 's are of course non-observable and must be estimated. We will follow Jewell (1974) and assume that the density function of Q = h(U) is the natural conjugate prior (or associate conjugate) to the family in (2.1), which is given by…”

“…Exact credibility, on the other hand, refers to the situation where we can find distributional assumptions under which this linear credibility estimator is optimal (in mean square error) in the unrestricted class of all functions of the observations. A famous theorem by Jewell (1974) states that exact credibility occurs when observations are drawn from a one-parameter exponential distribution with natural conjugate prior for the risk parameter. Another way to put this is that Jewell gives distributional assumptions under which the Bühlmann (1967) linear credibility estimator is the unrestricted best estimator.…”