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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| Veröffentlicht in: | Statistics & probability letters Jg. 48; H. 3; S. 287 - 292 |
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| Hauptverfasser: | , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Amsterdam
Elsevier B.V
01.07.2000
Elsevier |
| Schriftenreihe: | Statistics & Probability Letters |
| Schlagworte: | |
| ISSN: | 0167-7152, 1879-2103 |
| Online-Zugang: | Volltext |
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| Abstract | 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). |
|---|---|
| AbstractList | Let Y be a positive integer-valued random variable with the probability mass function P[theta](Y=y)=f(y;r)/a([theta]), y=r,r+1,...,[theta], where r is a known positive integer, and [theta][set membership, variant][Theta]={r,r+1,...} is an unknown parameter. We show that, for estimating [theta], cY is inadmissible under both 0-1 and a general loss whenever 01 in the case of sampling with replacement and discuss their bias and mean-squared error ([cY] denotes the integer nearest to cY). 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). |
| Author | Shen, Wei-Hsiung Pal, Nabendu Sinha, Bimal K. |
| Author_xml | – sequence: 1 givenname: Nabendu surname: Pal fullname: Pal, Nabendu organization: Department of Mathematics, University of Southwestern Louisiana, Lafayette, LA 70504, USA – sequence: 2 givenname: Wei-Hsiung surname: Shen fullname: Shen, Wei-Hsiung organization: Department of Statistics, Tunghai University, Taichung, Taiwan – sequence: 3 givenname: Bimal K. surname: Sinha fullname: Sinha, Bimal K. email: sinha@math.umbc.edu organization: Department of Mathematics and Statistics, University of Maryland, Baltimore County Campus, Baltimore, MD 21228, USA |
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| Keywords | 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 |
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| References | Seber (BIB6) 1982 Feller (BIB2) 1968 Hossain (BIB3) 1995; 15 Lehmann (BIB4) 1983 Rohatgi (BIB5) 1976 Boswell, Burnham, Patil (BIB1) 1988 Sengupta, De (BIB7) 1997; 59 Hossain (10.1016/S0167-7152(00)00009-2_BIB3) 1995; 15 Feller (10.1016/S0167-7152(00)00009-2_BIB2) 1968 Rohatgi (10.1016/S0167-7152(00)00009-2_BIB5) 1976 Seber (10.1016/S0167-7152(00)00009-2_BIB6) 1982 Sengupta (10.1016/S0167-7152(00)00009-2_BIB7) 1997; 59 Boswell (10.1016/S0167-7152(00)00009-2_BIB1) 1988 Lehmann (10.1016/S0167-7152(00)00009-2_BIB4) 1983 |
| References_xml | – volume: 59 start-page: 66 year: 1997 end-page: 75 ident: BIB7 article-title: On the estimation of a finite population. publication-title: Sankhyā B – year: 1968 ident: BIB2 publication-title: An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd Edition. – year: 1983 ident: BIB4 publication-title: Theory of Point Estimation. – volume: 15 start-page: 89 year: 1995 end-page: 94 ident: BIB3 article-title: Unknown population size estimation: an urn model approach publication-title: J. Statist. Studies – start-page: 469 year: 1988 end-page: 488 ident: BIB1 article-title: Role and use of composite sampling and capture–recapture sampling in ecological studies. publication-title: Handbook of Statistics, Vol. 6 – year: 1982 ident: BIB6 publication-title: The Estimation of Animal Abundance and Related Parameters, 2nd Edition. – year: 1976 ident: BIB5 publication-title: An Introduction to Probability Theory and Mathematical Statistics. – year: 1968 ident: 10.1016/S0167-7152(00)00009-2_BIB2 – start-page: 469 year: 1988 ident: 10.1016/S0167-7152(00)00009-2_BIB1 article-title: Role and use of composite sampling and capture–recapture sampling in ecological studies. doi: 10.1016/S0169-7161(88)06021-3 – year: 1983 ident: 10.1016/S0167-7152(00)00009-2_BIB4 – volume: 59 start-page: 66 year: 1997 ident: 10.1016/S0167-7152(00)00009-2_BIB7 article-title: On the estimation of a finite population. publication-title: Sankhyā B – volume: 15 start-page: 89 year: 1995 ident: 10.1016/S0167-7152(00)00009-2_BIB3 article-title: Unknown population size estimation: an urn model approach publication-title: J. Statist. Studies – year: 1976 ident: 10.1016/S0167-7152(00)00009-2_BIB5 – year: 1982 ident: 10.1016/S0167-7152(00)00009-2_BIB6 |
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| Snippet | 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,... Let Y be a positive integer-valued random variable with the probability mass function P[theta](Y=y)=f(y;r)/a([theta]), y=r,r+1,...,[theta], where r is a known... |
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| SubjectTerms | 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 |
| Title | Estimation of the support of a discrete distribution |
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