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

Published online by Cambridge University Press:  18 April 2011

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

Keywords

Bandwidth ParameterKernel Density EstimatorMarkov Chain Monte CarloMetropolis-Hastings AlgorithmPower TransformationTransformation Parameter
Type
Papers
Information
Annals of Actuarial Science , Volume 5 , Issue 2 , September 2011 , pp. 181 - 193
Copyright
Copyright © Institute and Faculty of Actuaries 2011

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References

Bolancé, C., Guillén, M., Pelican, E., Vernic, R. (2008). Skewed bivariate models and nonparametric estimation for the CTE risk measure. Insurance: Mathematics and Economics, 43, 386–393.Google Scholar
Bowman, A.W., Azzalini, A. (1997). Applied Smoothing Techniques for Data Analysis. Oxford University Press, London.CrossRefGoogle Scholar
Buch-Larsen, T., Nielsen, J.P., Guillén, M., Bolancé, C. (2005). Kernel density estimation for heavy-tailed distributions using the Champernowne transformation. Statistics, 39(6), 503–518.CrossRefGoogle Scholar
Clements, A.E., Hurn, A.S., Lindsay, K.A. (2003). Mobius-like mappings and their use in kernel density estimation. Journal of the American Statistical Association, 98, 993–1000.CrossRefGoogle Scholar
Härdle, W. (1991). Smoothing Techniques with Implementation in S. Springer-Verlag, New York.CrossRefGoogle Scholar
Hjort, N.L., Glad, I.K. (1995). Nonparametric density estimation with a parametric start. The Annals of Statistics, 23, 882–904.CrossRefGoogle Scholar
Izenman, A.J. (1991). Recent developments in nonparametric density estimation. Journal of the American Statistical Association, 86, 205–224.Google Scholar
Kim, S., Shephard, N., Chib, S. (1998). Stochastic volatility: Likelihood inference and comparison with ARCH models. Review of Economic Studies, 65, 361–393.CrossRefGoogle Scholar
Marron, J.S. (1988). Automatic smoothing parameter selection: A survey. Empirical Economics, 13, 187–208.CrossRefGoogle Scholar
Meyer, R., Yu, J. (2000). BUGS for a Bayesian analysis of stochastic volatility models. Econometrics Journal, 3, 198–215.CrossRefGoogle Scholar
Roberts, G.O. (1996). Markov chain concepts related to sampling algorithms. In Gilks, W.R. Richardson, S., Spiegelhalter, D.J. (Eds.) Markov Chain Monte Carlo in Practice. Chapman & Hall, London, 45–57.Google Scholar
Scott, D.W. (1992). Multivariate Density Estimation: Theory, Practice and Visualisation. John Wiley & Sons, New York.CrossRefGoogle Scholar
Sheather, S.J., Jones, M.C. (1991). A reliable data-based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society, Series B, 53, 683–690.Google Scholar
Simonoff, J.S. (1996). Smoothing Methods in Statistics. Springer, New York.CrossRefGoogle Scholar
Tse, Y.K., Zhang, X., Yu, J. (2004). Estimation of Hyperbolic Diffusion with Markov Chain Monte Carlo Simulation. Quantitative Finance, 4, 158–169.CrossRefGoogle Scholar
Wand, M.P., Jones, M.C. (1995). Kernel Smoothing. Chapman & Hall, London.CrossRefGoogle Scholar
Wand, M.P., Marron, J.S., Ruppert, D. (1991). Transformations in density estimation. Journal of the American Statistical Association, 86, 414, 343–353.Google Scholar
Zhang, X., Brooks, R.D., King, M.L. (2009). A Bayesian approach to bandwidth selection for multivariate kernel regression with an application to state-price density estimation. Journal of Econometrics, 153, 21–32.CrossRefGoogle Scholar
Zhang, X., King, M.L., Hyndman, R.J. (2006). A Bayesian approach to bandwidth selection for multivariate kernel density estimation. Computational Statistics & Data Analysis, 50, 3009–3031.CrossRefGoogle Scholar