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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Bibliographic Details
Published in:Statistics & probability letters Vol. 48; no. 3; pp. 287 - 292
Main Authors: Pal, Nabendu, Shen, Wei-Hsiung, Sinha, Bimal K.
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 01.07.2000
Elsevier
Series:Statistics & Probability Letters
Subjects:
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
Online Access:Get full text
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Description
Summary: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