Improving on the minimum risk equivariant estimator of a location parameter which is constrained to an interval or a half-interval

Éric Marchand, William Strawderman

Research output: Contribution to journalArticle

20 Citations (Scopus)

Abstract

For location families with densities f0(x - θ), we study the problem of estimating θ for location invariant loss L(θ, d) = ρ(d - θ), and under a lower-bound constraint of the form θ ≥ a. We show, that for quite general (f0, ρ), the Bayes estimator δU with respect to a uniform prior on (a, ∞) is a minimax estimator which dominates the benchmark minimum risk equivariant (MRE) estimator. In extending some previous dominance results due to Katz and Farrell, we make use of Kubokawa's IERD (Integral Expression of Risk Difference) method, and actually obtain classes of dominating estimators which include, and are characterized in terms of δU. Implications are also given and, finally, the above dominance phenomenon is studied and extended to an interval constraint of the form θ ∈ [a, b].

Original languageEnglish (US)
Pages (from-to)129-143
Number of pages15
JournalAnnals of the Institute of Statistical Mathematics
Volume57
Issue number1
DOIs
StatePublished - Jan 1 2005

Fingerprint

Equivariant Estimator
Location Parameter
Risk Difference
Minimax Estimator
Bound Constraints
Bayes Estimator
Interval
Difference Method
Benchmark
Lower bound
Estimator
Invariant
Form
Family
Class

All Science Journal Classification (ASJC) codes

  • Statistics and Probability

Keywords

  • Constrained parameter space
  • Dominating estimators
  • Location family
  • Lower-bounded parameter
  • Minimax estimation
  • Minimum risk equivariant estimator

Cite this

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abstract = "For location families with densities f0(x - θ), we study the problem of estimating θ for location invariant loss L(θ, d) = ρ(d - θ), and under a lower-bound constraint of the form θ ≥ a. We show, that for quite general (f0, ρ), the Bayes estimator δU with respect to a uniform prior on (a, ∞) is a minimax estimator which dominates the benchmark minimum risk equivariant (MRE) estimator. In extending some previous dominance results due to Katz and Farrell, we make use of Kubokawa's IERD (Integral Expression of Risk Difference) method, and actually obtain classes of dominating estimators which include, and are characterized in terms of δU. Implications are also given and, finally, the above dominance phenomenon is studied and extended to an interval constraint of the form θ ∈ [a, b].",
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N2 - For location families with densities f0(x - θ), we study the problem of estimating θ for location invariant loss L(θ, d) = ρ(d - θ), and under a lower-bound constraint of the form θ ≥ a. We show, that for quite general (f0, ρ), the Bayes estimator δU with respect to a uniform prior on (a, ∞) is a minimax estimator which dominates the benchmark minimum risk equivariant (MRE) estimator. In extending some previous dominance results due to Katz and Farrell, we make use of Kubokawa's IERD (Integral Expression of Risk Difference) method, and actually obtain classes of dominating estimators which include, and are characterized in terms of δU. Implications are also given and, finally, the above dominance phenomenon is studied and extended to an interval constraint of the form θ ∈ [a, b].

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