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Bayesian Adaptive Bandwidth Kernel Density Estimation of Irregular Multivariate Distributions

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  • Shuowen Hu

    ()

  • D.S. Poskitt

    ()

  • Xibin Zhang

    ()

Abstract

Kernel density estimation is an important technique for understanding the distributional properties of data. Some investigations have found that the estimation of a global bandwidth can be heavily affected by observations in the tail. We propose to categorize data into low- and high-density regions, to which we assign two different bandwidths called the low-density adaptive bandwidths. We derive the posterior of the bandwidth parameters through the Kullback-Leibler information. A Bayesian sampling algorithm is presented to estimate the bandwidths. Monte Carlo simulations are conducted to examine the performance of the proposed Bayesian sampling algorithm in comparison with the performance of the normal reference rule and a Bayesian sampling algorithm for estimating a global bandwidth. According to Kullback-Leibler information, the kernel density estimator with low-density adaptive bandwidths estimated through the proposed Bayesian sampling algorithm outperforms the density estimators with bandwidth estimated through the two competitors. We apply the low-density adaptive kernel density estimator to the estimation of the bivariate density of daily stock-index returns observed from the U.S. and Australian stock markets. The derived conditional distribution of the Australian stock-index return for a given daily return in the U.S. market enables market analysts to understand how the former market is associated with the latter.

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Bibliographic Info

Paper provided by Monash University, Department of Econometrics and Business Statistics in its series Monash Econometrics and Business Statistics Working Papers with number 21/10.

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Length: 35 pages
Date of creation: Dec 2010
Date of revision:
Handle: RePEc:msh:ebswps:2010-21

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Keywords: conditional density; global bandwidth; Kullback-Leibler information; marginal likelihood; Markov chain Monte Carlo; S&P500 index;

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References

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  1. Sain, Stephan R., 2002. "Multivariate locally adaptive density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 39(2), pages 165-186, April.
  2. M. Jácome & I. Gijbels & R. Cao, 2008. "Comparison of presmoothing methods in kernel density estimation under censoring," Computational Statistics, Springer, vol. 23(3), pages 381-406, July.
  3. Sangjoon Kim, Neil Shephard & Siddhartha Chib, . "Stochastic volatility: likelihood inference and comparison with ARCH models," Economics Papers W26, revised version of W, Economics Group, Nuffield College, University of Oxford.
  4. Adelchi Azzalini & Antonella Capitanio, 2003. "Distributions generated by perturbation of symmetry with emphasis on a multivariate skew "t"-distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 367-389.
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  8. Holmes, Michael P. & Gray, Alexander G. & Isbell Jr., Charles Lee, 2010. "Fast kernel conditional density estimation: A dual-tree Monte Carlo approach," Computational Statistics & Data Analysis, Elsevier, vol. 54(7), pages 1707-1718, July.
  9. Stephan R. Sain, 2002. "Zero-Bias Locally Adaptive Density Estimators," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 29(3), pages 441-460.
  10. Marron, J. S. & Nolan, D., 1988. "Canonical kernels for density estimation," Statistics & Probability Letters, Elsevier, vol. 7(3), pages 195-199, December.
  11. Shuowen Hu & D.S. Poskitt & Xibin Zhang, 2010. "Bayesian Adaptive Bandwidth Kernel Density Estimation of Irregular Multivariate Distributions," Monash Econometrics and Business Statistics Working Papers 21/10, Monash University, Department of Econometrics and Business Statistics.
  12. Zhang, Xibin & King, Maxwell L. & Hyndman, Rob J., 2006. "A Bayesian approach to bandwidth selection for multivariate kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 50(11), pages 3009-3031, July.
  13. Tae-Hwy Lee & Yong Bao & Burak Saltoglu, 2006. "Evaluating predictive performance of value-at-risk models in emerging markets: a reality check," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 25(2), pages 101-128.
  14. Tarn Duong & Martin L. Hazelton, 2005. "Cross-validation Bandwidth Matrices for Multivariate Kernel Density Estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 32(3), pages 485-506.
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Cited by:
  1. Zougab, Nabil & Adjabi, Smail & Kokonendji, Célestin C., 2014. "Bayesian estimation of adaptive bandwidth matrices in multivariate kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 75(C), pages 28-38.
  2. Shuowen Hu & D.S. Poskitt & Xibin Zhang, 2010. "Bayesian Adaptive Bandwidth Kernel Density Estimation of Irregular Multivariate Distributions," Monash Econometrics and Business Statistics Working Papers 21/10, Monash University, Department of Econometrics and Business Statistics.

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