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Minimax estimation of a lower‐bounded scale parameter of a gamma distribution for scale‐invariant squared‐error loss
Author(s) -
Eeden Constance Van
Publication year - 1995
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 - 10.2307/3315365
Subject(s) - mathematics , minimax , estimator , minimax estimator , scale parameter , pointwise , mean squared error , invariant estimator , efficient estimator , bounded function , combinatorics , statistics , minimum variance unbiased estimator , mathematical optimization , mathematical analysis
Let X have a gamma distribution with known shape parameter θr;aL and unknown scale parameter θ. Suppose it is known that θ ≥ a for some known a > 0. An admissible minimax estimator for scale‐invariant squared‐error loss is presented. This estimator is the pointwise limit of a sequence of Bayes estimators. Further, the class of truncated linear estimators C = {θ ρ |θ ρ (x) = max(a, ρ), ρ > 0} is studied. It is shown that each θ ρ is inadmissible and that exactly one of them is minimax. Finally, it is shown that Katz's [Ann. Math. Statist., 32, 136–142 (1961)] estimator of θ is not minimax for our loss function. Some further properties of and comparisons among these estimators are also presented.