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A Bayesian Approach to Parameter Estimation for Kernel Density Estimation via Transformations

  • Qing Liu, David Pitt, Xibin Zhang and Xueyuan Wu

Abstract

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 exhibits 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. In the current literature, there have been some developments in the area of estimating densities based on transformed data, where bandwidth selection usually depends on pre-determined transformation parameters. Moreover, in the bivariate situation, the transformation parameters were estimated for each dimension individually. We use a Bayesian sampling algorithm 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 simultaneously within a Metropolis-Hastings sampling procedure. Moreover, we demonstrate that the correlation between the two dimensions is better captured through the bivariate density estimator based on transformed data.

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Corresponding author

Contact address Qing Liu, Centre for Actuarial Studies, Faculty of Business and Economics, The University of Melbourne, VIC 3010, Australia. E-mail: q.liu5@pgrad.unimelb.edu.au
David Pitt, Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, NSW 2109, Australia. E-mail: david.pitt@mq.edu.au.
Xibin Zhang, Department of Econometrics and Business Statistics, Monash University, VIC 3145, Australia. E-mail: xibin.zhang@monash.edu
Xueyuan Wu, Centre for Actuarial Studies, Faculty of Business and Economics, The University of Melbourne, VIC 3010, Australia. E-mail: xueyuanw@unimelb.edu.au.

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A Bayesian Approach to Parameter Estimation for Kernel Density Estimation via Transformations

  • Qing Liu, David Pitt, Xibin Zhang and Xueyuan Wu

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