Estimation of the support of a discrete distribution

Let Y be a positive integer-valued random variable with the probability mass function P θ(Y=y)=f(y;r)/a(θ), y=r,r+1,…,θ , where r is a known positive integer, and θ∈Θ={r,r+1,…} is an unknown parameter. We show that, for estimating θ, cY is inadmissible under both 0–1 and a general loss whenever 0<...

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Vydané v:	Statistics & probability letters Ročník 48; číslo 3; s. 287 - 292
Hlavní autori:	Pal, Nabendu, Shen, Wei-Hsiung, Sinha, Bimal K.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Amsterdam Elsevier B.V 01.07.2000 Elsevier
Edícia:	Statistics & Probability Letters
Predmet:	0–1 loss Admissible Admissible Hammersley-Chapman-Robbins inequality Mean-squared error Minimax Population size Squared error loss 0-1 loss Exact sciences and technology Hammersley–Chapman–Robbins inequality Mathematics Mean-squared error Minimax Nonparametric inference Population size Probability and statistics Sciences and techniques of general use Squared error loss Statistics 62G05 Minimax Admissible Hammersley–Chapman–Robbins inequality Squared error loss Population size Mean-squared error 0–1 loss Bayes estimation Admissibility Sample size Hammersley Chapman Robins inequality Discrete distribution Minimaxity Minimax method Sampling without replacement Support estimation Mean square error
ISSN:	0167-7152, 1879-2103
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Shrnutí:	Let Y be a positive integer-valued random variable with the probability mass function P θ(Y=y)=f(y;r)/a(θ), y=r,r+1,…,θ , where r is a known positive integer, and θ∈Θ={r,r+1,…} is an unknown parameter. We show that, for estimating θ, cY is inadmissible under both 0–1 and a general loss whenever 0<c<1. Under some mild conditions on f(y;r), we prove that Y is admissible and minimax under both 0–1 and squared error loss. As an application, we consider the problem of estimating the size θ of a finite population whose elements are labeled from 1 to θ, based on a simple random sample of size n under both with and without replacement. Admissibility and minimaxity of Y, the largest number observed in the sample, under 0–1 and squared error loss hold under both sampling situations. We propose two integer-valued estimators of θ of the form [cY] for c>1 in the case of sampling with replacement and discuss their bias and mean-squared error ( [cY] denotes the integer nearest to cY).
ISSN:	0167-7152 1879-2103
DOI:	10.1016/S0167-7152(00)00009-2