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Smooth semiparametric and nonparametric Bayesian estimation of bivariate densities from bivariate histogram data


  • Lambert, Philippe


Penalized B-splines combined with the composite link model are used to estimate a bivariate density from a histogram with wide bins. The goals are multiple: they include the visualization of the dependence between the two variates, but also the estimation of derived quantities like Kendall's tau, conditional moments and quantiles. Two strategies are proposed: the first one is semiparametric with flexible margins modeled using B-splines and a parametric copula for the dependence structure; the second one is nonparametric and is based on Kronecker products of the marginal B-spline bases. Frequentist and Bayesian estimations are described. A large simulation study quantifies the performances of the two methods under different dependence structures and for varying strengths of dependence, sample sizes and amounts of grouping. It suggests that Schwarz's BIC is a good tool for classifying the competing models. The density estimates are used to evaluate conditional quantiles in two applications in social and in medical sciences.

Suggested Citation

  • Lambert, Philippe, 2011. "Smooth semiparametric and nonparametric Bayesian estimation of bivariate densities from bivariate histogram data," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 429-445, January.
  • Handle: RePEc:eee:csdana:v:55:y:2011:i:1:p:429-445

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    References listed on IDEAS

    1. Koo, Ja-Yong & Kooperberg, Charles, 2000. "Logspline density estimation for binned data," Statistics & Probability Letters, Elsevier, vol. 46(2), pages 133-147, January.
    2. Jullion, Astrid & Lambert, Philippe, 2007. "Robust specification of the roughness penalty prior distribution in spatially adaptive Bayesian P-splines models," Computational Statistics & Data Analysis, Elsevier, vol. 51(5), pages 2542-2558, February.
    3. Lambert, Philippe & Eilers, Paul H.C., 2009. "Bayesian density estimation from grouped continuous data," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1388-1399, February.
    4. Hanson, Timothy E., 2006. "Inference for Mixtures of Finite Polya Tree Models," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1548-1565, December.
    5. Tommi Harkanen, 2000. "Caries on Permanent Teeth: A Non-parametric Bayesian Analysis," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(4), pages 577-588.
    6. Guadalupe Gómez & M. Calle & Ramon Oller, 2004. "Frequentist and Bayesian approaches for interval-censored data," Statistical Papers, Springer, vol. 45(2), pages 139-173, April.
    7. Yang, Mingan & Hanson, Timothy & Christensen, Ronald, 2008. "Nonparametric Bayesian estimation of a bivariate density with interval censored data," Computational Statistics & Data Analysis, Elsevier, vol. 52(12), pages 5202-5214, August.
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