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Computing confidence intervals for log-concave densities

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  • Azadbakhsh, Mahdis
  • Jankowski, Hanna
  • Gao, Xin

Abstract

In Balabdaoui, Rufibach, and Wellner (2009), pointwise asymptotic theory was developed for the nonparametric maximum likelihood estimator of a log-concave density. Here, the practical aspects of their results are explored. Namely, the theory is used to develop pointwise confidence intervals for the true log-concave density. To do this, the quantiles of the limiting process are estimated and various ways of estimating the nuisance parameter appearing in the limit are studied. The finite sample size behavior of these estimated confidence intervals is then studied via a simulation study of the empirical coverage probabilities.

Suggested Citation

  • Azadbakhsh, Mahdis & Jankowski, Hanna & Gao, Xin, 2014. "Computing confidence intervals for log-concave densities," Computational Statistics & Data Analysis, Elsevier, vol. 75(C), pages 248-264.
    • Handle: RePEc:eee:csdana:v:75:y:2014:i:c:p:248-264
    DOI: 10.1016/j.csda.2014.01.020
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    References listed on IDEAS

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    1. Moulinath Banerjee & Jon A. Wellner, 2005. "Confidence Intervals for Current Status Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(3), pages 405-424, September.
    2. Madeleine Cule & Richard Samworth & Michael Stewart, 2010. "Maximum likelihood estimation of a multi‐dimensional log‐concave density," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(5), pages 545-607, November.
    3. Piet Groeneboom & Geurt Jongbloed & Jon A. Wellner, 2008. "The Support Reduction Algorithm for Computing Non‐Parametric Function Estimates in Mixture Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(3), pages 385-399, September.
    4. Walther G., 2002. "Detecting the Presence of Mixing with Multiscale Maximum Likelihood," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 508-513, June.
    5. Dümbgen, Lutz & Rufibach, Kaspar, 2011. "logcondens: Computations Related to Univariate Log-Concave Density Estimation," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 39(i06).
    6. Chang, George T. & Walther, Guenther, 2007. "Clustering with mixtures of log-concave distributions," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6242-6251, August.
    7. Duong, Tarn, 2007. "ks: Kernel Density Estimation and Kernel Discriminant Analysis for Multivariate Data in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 21(i07).
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    Cited by:

    1. Dümbgen, Lutz & Wellner, Jon A. & Wolff, Malcolm, 2016. "A law of the iterated logarithm for Grenander’s estimator," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3854-3864.

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