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Transformation-based nonparametric estimation of multivariate densities

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  • Chang, Meng-Shiuh
  • Wu, Ximing

Abstract

We present a probability-integral-transformation-based estimator of multivariate densities. Given a sample of random vectors, we first transform the data into their corresponding marginal distributions. The marginal densities and the joint density of the transformed data are estimated nonparametrically. The joint density of the original data is constructed as the product of the density of the transformed data and marginal densities, which coincides with the copula representation of multivariate densities. We show that the Kullback–Leibler Information Criterion (KLIC) between the true density and its estimate can be decomposed into the KLIC of the marginal densities and that of the copula density. We derive the convergence rate of the proposed estimator in terms of the KLIC and propose a supervised hierarchical procedure of model selection. Monte Carlo simulations demonstrate the good performance of the estimator. An empirical example on the US and UK stock markets is presented. The estimated conditional copula density provides useful insight into the joint movements of the US and UK markets under extreme Asian markets.

Suggested Citation

  • Chang, Meng-Shiuh & Wu, Ximing, 2015. "Transformation-based nonparametric estimation of multivariate densities," Journal of Multivariate Analysis, Elsevier, vol. 135(C), pages 71-88.
    • Handle: RePEc:eee:jmvana:v:135:y:2015:i:c:p:71-88
    DOI: 10.1016/j.jmva.2014.11.010
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    1. Aurore Delaigle & Peter Hall & Jiashun Jin, 2011. "Robustness and accuracy of methods for high dimensional data analysis based on Student's t‐statistic," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(3), pages 283-301, June.
    2. Kooperberg, Charles & Stone, Charles J., 1991. "A study of logspline density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 12(3), pages 327-347, November.
    3. Cadima, Jorge & Cerdeira, J. Orestes & Minhoto, Manuel, 2004. "Computational aspects of algorithms for variable selection in the context of principal components," Computational Statistics & Data Analysis, Elsevier, vol. 47(2), pages 225-236, September.
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    1. Nan Yang & Yu Huang & Dengxu Hou & Songkai Liu & Di Ye & Bangtian Dong & Youping Fan, 2019. "Adaptive Nonparametric Kernel Density Estimation Approach for Joint Probability Density Function Modeling of Multiple Wind Farms," Energies, MDPI, vol. 12(7), pages 1-15, April.
    2. Perloff, Jeffrey M. & Schlenker, Wolfram & Sears, Molly & Wu, Ximing, 2020. "Crop Failures from Temperature and Precipitation Shocks: Implications for U.S. Crop Insurance," 2020 Annual Meeting, July 26-28, Kansas City, Missouri 304540, Agricultural and Applied Economics Association.

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