Bayesian prediction with an asymmetric criterion in a nonparametric model of insurance risk
Wojciech Niemiro · Statistics · 2006
We consider a nonparametric Bayesian insurance risk model. The claims are seen as a marked point process (T i , Y i ), where T i is the time of occurrence of the ith claim and Y i is its size. We assume that this is a nonhomogeneous Poisson process on ℝ+ 2 with intensity measure P×Θ. Here P describes the exposure to risk and it is known, whereas Θ is regarded as an unknown risk characteristic. According to the Bayesian paradigm, we assume that the measure Θ is random. Processes with independent increments are used as prior distributions. In particular, Gamma processes are conjugate priors. The problem is to predict the sum of future claims in a given period, given the past of the process. We consider the asymmetric criterion LINEX (linear-exponential) that penalizes underestimation of claims more severely than overestimation. For the conjugate Gamma prior, we construct the best predictor. Under a relaxed assumption on the prior distribution, we construct the best linear predictor.