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Unimodal density estimation using Bernstein polynomials

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  • Turnbull, Bradley C.
  • Ghosh, Sujit K.

Abstract

The estimation of probability density functions is one of the fundamental aspects of any statistical inference. Many data analyses are based on an assumed family of parametric models, which are known to be unimodal (e.g., exponential family, etc.). Often a histogram suggests the unimodality of the underlying density function. Parametric assumptions, however, may not be adequate for many inferential problems. A flexible class of mixture of Beta densities that are constrained to be unimodal is presented. It is shown that the estimation of the mixing weights, and the number of mixing components, can be accomplished using a weighted least squares criteria subject to a set of linear inequality constraints. The mixing weights of the Beta mixture are efficiently computed using quadratic programming techniques. Three criteria for selecting the number of mixing weights are presented and compared in a small simulation study. More extensive simulation studies are conducted to demonstrate the performance of the density estimates in terms of popular functional norms (e.g., Lp norms). The true underlying densities are allowed to be unimodal symmetric and skewed, with finite, infinite or semi-finite supports. A code for an R function is provided which allows the user to input a data set and returns the estimated density, distribution, quantile, and random sample generating functions.

Suggested Citation

  • Turnbull, Bradley C. & Ghosh, Sujit K., 2014. "Unimodal density estimation using Bernstein polynomials," Computational Statistics & Data Analysis, Elsevier, vol. 72(C), pages 13-29.
    • Handle: RePEc:eee:csdana:v:72:y:2014:i:c:p:13-29
    DOI: 10.1016/j.csda.2013.10.021
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    References listed on IDEAS

    as
    1. Wolters, Mark A., 2012. "A Greedy Algorithm for Unimodal Kernel Density Estimation by Data Sharpening," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 47(i06).
    2. Alexandre Leblanc, 2010. "A bias-reduced approach to density estimation using Bernstein polynomials," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(4), pages 459-475.
    3. Thas, O. & Ottoy, J. P., 2003. "Some generalizations of the Anderson-Darling statistic," Statistics & Probability Letters, Elsevier, vol. 64(3), pages 255-261, September.
    4. Markowitz, Harry M & Usmen, Nilufer, 1996. "The Likelihood of Various Stock Market Return Distributions, Part 1: Principles of Inference," Journal of Risk and Uncertainty, Springer, vol. 13(3), pages 207-219, November.
    5. Kevin Fergusson & Eckhard Platen, 2006. "On the Distributional Characterization of Daily Log-Returns of a World Stock Index," Applied Mathematical Finance, Taylor & Francis Journals, vol. 13(1), pages 19-38.
    6. Sonia Petrone & Larry Wasserman, 2002. "Consistency of Bernstein polynomial posteriors," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(1), pages 79-100, January.
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    Cited by:

    1. Frédéric Ouimet, 2021. "General Formulas for the Central and Non-Central Moments of the Multinomial Distribution," Stats, MDPI, vol. 4(1), pages 1-10, January.
    2. Rasool Roozegar & Saralees Nadarajah & Eisa Mahmoudi, 2022. "The Power Series Exponential Power Series Distributions with Applications to Failure Data Sets," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(1), pages 44-78, May.
    3. Ouimet, Frédéric, 2021. "Asymptotic properties of Bernstein estimators on the simplex," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    4. Ghosh, Sujit K. & Burns, Christopher B. & Prager, Daniel L. & Zhang, Li & Hui, Glenn, 2018. "On nonparametric estimation of the latent distribution for ordinal data," Computational Statistics & Data Analysis, Elsevier, vol. 119(C), pages 86-98.
    5. Zhou, Haiming & Huang, Xianzheng, 2022. "Bayesian beta regression for bounded responses with unknown supports," Computational Statistics & Data Analysis, Elsevier, vol. 167(C).
    6. Burns, Christopher & Prager, Daniel & Ghosh, Sujit & Goodwin, Barry, 2015. "Imputing for Missing Data in the ARMS Household Section: A Multivariate Imputation Approach," 2015 AAEA & WAEA Joint Annual Meeting, July 26-28, San Francisco, California 205291, Agricultural and Applied Economics Association.
    7. Liu, Bowen & Ghosh, Sujit K., 2020. "On empirical estimation of mode based on weakly dependent samples," Computational Statistics & Data Analysis, Elsevier, vol. 152(C).

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