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A law of the iterated logarithm for Grenander’s estimator

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  • Dümbgen, Lutz
  • Wellner, Jon A.
  • Wolff, Malcolm

Abstract

In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If f(t0)>0, f′(t0)<0, and f′ is continuous in a neighborhood of t0, then blalim supn→∞(n2loglogn)1/3(f̂n(t0)−f(t0))=|f(t0)f′(t0)/2|1/32M almost surely where M≡supg∈GTg=(3/4)1/3andTg≡argmaxu{g(u)−u2}; here G is the two-sided Strassen limit set on R. The proof relies on laws of the iterated logarithm for local empirical processes, Groeneboom’s switching relation, and properties of Strassen’s limit set analogous to distributional properties of Brownian motion; see Strassen [26].

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:12:p:3854-3864
    DOI: 10.1016/j.spa.2016.04.012
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    References listed on IDEAS

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    1. Groeneboom,Piet & Jongbloed,Geurt, 2014. "Nonparametric Estimation under Shape Constraints," Cambridge Books, Cambridge University Press, number 9780521864015, November.
    2. Einmahl, John H. J., 1997. "Poisson and Gaussian approximation of weighted local empirical processes," Stochastic Processes and their Applications, Elsevier, vol. 70(1), pages 31-58, October.
    3. 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.
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