TY - JOUR
T1 - A Bayesian approach to parameter estimation for kernel density estimation via transformations
AU - Liu, Qing
AU - Pitt, David
AU - Zhang, Xibin
AU - Wu, Xueyuan
N1 - Funding Information:
We have also computed the conditional tail expectation as in Bolancé et al. (). However, our results tend to underestimate the empirical conditional tail expectations. This is not surprising because our sampling algorithms were developed based on the Kullback-Leibler information, under which our results are optimal in terms of the entire density rather than the tails of the density. Further research could focus on finding the optimal bandwidth and transformation parameters for bivariate kernel density estimation via transformations, which give a more accurate estimate of the tail of the joint density. The third author acknowledges financial support from the Australian Research Council under the discovery project DP1095838.
Publisher Copyright:
Copyright © Institute and Faculty of Actuaries 2011.
PY - 2011
Y1 - 2011
N2 - 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.
AB - 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.
KW - Bandwidth Parameter
KW - Kernel Density Estimator
KW - Markov Chain Monte Carlo
KW - Metropolis-Hastings Algorithm
KW - Power Transformation
KW - Transformation Parameter
UR - http://www.scopus.com/inward/record.url?scp=84977866674&partnerID=8YFLogxK
U2 - 10.1017/S1748499511000030
DO - 10.1017/S1748499511000030
M3 - Article
SN - 1748-4995
VL - 5
SP - 181
EP - 193
JO - Annals of Actuarial Science
JF - Annals of Actuarial Science
IS - 2
ER -