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Grenander functionals and Cauchy's formula

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  • Piet Groeneboom

Abstract

Let f^n be the nonparametric maximum likelihood estimator of a decreasing density. Grenander characterized this as the left‐continuous slope of the least concave majorant of the empirical distribution function. For a sample from the uniform distribution, the asymptotic distribution of the L2‐distance of the Grenander estimator to the uniform density was derived in an article by Groeneboom and Pyke by using a representation of the Grenander estimator in terms of conditioned Poisson and gamma random variables. This representation was also used in an article by Groeneboom and Lopuhaä to prove a central limit result of Sparre Andersen on the number of jumps of the Grenander estimator. Here we extend this to the proof of the main result on the L2‐distance of the Grenander estimator to the uniform density and also prove a similar asymptotic normality results for the entropy functional. Cauchy's formula and saddle point methods are the main tools in our development.

Suggested Citation

  • Piet Groeneboom, 2021. "Grenander functionals and Cauchy's formula," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(1), pages 275-294, March.
    • Handle: RePEc:bla:scjsta:v:48:y:2021:i:1:p:275-294
    DOI: 10.1111/sjos.12449
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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.
    2. P. Groeneboom & H. P. Lopuhaa, 1993. "Isotonic estimators of monotone densities and distribution functions: basic facts," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 47(3), pages 175-183, September.
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