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Nonparametric estimation of multivariate scale mixtures of uniform densities

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  • Pavlides, Marios G.
  • Wellner, Jon A.

Abstract

Suppose that U=(U1,…,Ud) has a Uniform([0,1]d) distribution, that Y=(Y1,…,Yd) has the distribution G on R+d, and let X=(X1,…,Xd)=(U1Y1,…,UdYd). The resulting class of distributions of X (as G varies over all distributions on R+d) is called the Scale Mixture of Uniforms class of distributions, and the corresponding class of densities on R+d is denoted by FSMU(d). We study maximum likelihood estimation in the family FSMU(d). We prove existence of the MLE, establish Fenchel characterizations, and prove strong consistency of the almost surely unique maximum likelihood estimator (MLE) in FSMU(d). We also provide an asymptotic minimax lower bound for estimating the functional f↦f(x) under reasonable differentiability assumptions on f∈FSMU(d) in a neighborhood of x. We conclude the paper with discussion, conjectures and open problems pertaining to global and local rates of convergence of the MLE.

Suggested Citation

  • Pavlides, Marios G. & Wellner, Jon A., 2012. "Nonparametric estimation of multivariate scale mixtures of uniform densities," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 71-89.
    • Handle: RePEc:eee:jmvana:v:107:y:2012:i:c:p:71-89
    DOI: 10.1016/j.jmva.2012.01.001
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    References listed on IDEAS

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    1. Dragi Anevski, 2003. "Estimating the Derivative of a Convex Density," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 57(2), pages 245-257, May.
    2. Polonik, Wolfgang, 1997. "Minimum volume sets and generalized quantile processes," Stochastic Processes and their Applications, Elsevier, vol. 69(1), pages 1-24, July.
    3. Wong, George Y. C. & Yu, Qiqing, 1999. "Generalized MLE of a Joint Distribution Function with Multivariate Interval-Censored Data," Journal of Multivariate Analysis, Elsevier, vol. 69(2), pages 155-166, May.
    4. Polonik, W., 1995. "Density Estimation under Qualitative Assumptions in Higher Dimensions," Journal of Multivariate Analysis, Elsevier, vol. 55(1), pages 61-81, October.
    5. Jongbloed, Geurt, 2000. "Minimax lower bounds and moduli of continuity," Statistics & Probability Letters, Elsevier, vol. 50(3), pages 279-284, November.
    6. Biau, Gérard & Devroye, Luc, 2003. "On the risk of estimates for block decreasing densities," Journal of Multivariate Analysis, Elsevier, vol. 86(1), pages 143-165, July.
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    Cited by:

    1. Chee, Chew-Seng & Wang, Yong, 2014. "Least squares estimation of a k-monotone density function," Computational Statistics & Data Analysis, Elsevier, vol. 74(C), pages 209-216.

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