A Bayesian approach to parameter estimation for kernel density estimation via transformations

In this paper, we present a Markov chain Monte Carlo (MCMC) simulation algorithm for estimating parameters in the kernel density estimation of bivariate insurance claim data via transformations. Our data set consists of two types of auto insurance claim costs and exhibit a high-level of skewness in the marginal empirical distributions. Therefore, the kernel density estimator based on original data does not perform well. However, the density of the original data can be estimated through estimating the density of the transformed data using kernels. It is well known that the performance of a kernel density estimator is mainly determined by the bandwidth, and only in a minor way by the kernel choice. In the current literature, there have been some developments in the area of estimating densities based on transformed data, but bandwidth selection depends on pre-determined transformation parameters. Moreover, in the bivariate situation, each dimension is considered separately and the correlation between the two dimensions is largely ignored. We extend the Bayesian sampling algorithm proposed by Zhang, King and Hyndman (2006) and present a Metropolis-Hastings sampling procedure to sample the bandwidth and transformation parameters from their posterior density. Our contribution is to estimate the bandwidths and transformation parameters within a Metropolis-Hastings sampling procedure. Moreover, we demonstrate that the correlation between the two dimensions is well captured through the bivariate density estimator based on transformed data.