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Asymptotic behavior of the grenander estimator at density flat regions
Author(s) -
Carolan Chris,
Dykstra Richard
Publication year - 1999
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3316111
Subject(s) - estimator , mathematics , convolution (computer science) , asymptotic distribution , monotone polygon , distribution (mathematics) , constant (computer programming) , mathematical analysis , limiting , point (geometry) , fixed point , statistics , geometry , computer science , mechanical engineering , machine learning , artificial neural network , engineering , programming language
Over forty years ago, Grenander derived the MLE of a monotone decreasing density f with known mode. Prakasa Rao obtained the asymptotic distribution of this estimator at a fixed point x where f ' ( x ) < 0. Here, we obtain the asymptotic distribution of this estimator at a fixed point x when f is constant and nonzero in some open neighborhood of x . This limiting distribution is expressible as the convolution of a closed‐form density and a rescaled standard normal density. Groeneboom (1983) derived the aforementioned closed‐form density and we provide an alternative, more direct derivation.