On some inadmissibility results for the scale parameters of selected gamma populations

• Published in: Journal of statistical planning and inference 2006-07, Vol.136 (7), p.2340-2351
• Main Authors: Misra, Neeraj, van der Meulen, Edward C., Vanden Branden, Karlien
• Format: Article
• Language: English
• Subjects: Distribution theory; Exact sciences and technology; General topics; Inadmissible estimators; Linear inference, regression; Mathematical analysis; Mathematics; Natural estimator; Probability and statistics; Random parameter; Scale and permutation invariant estimator; Scale parameter; Sciences and techniques of general use; Special functions; Squared error loss function; Statistics
• ISSN: 0378-3758; 1873-1171
• DOI: 10.1016/j.jspi.2005.08.020

Let X 1 and X 2 be two independent gamma random variables, having unknown scale parameters λ 1 and λ 2 , respectively, and common known shape parameter p ( > 0 ) . Define, M = 1 , if X 1 < X 2 , M = 2 , if X 1 > X 2 and J = 3 - M . We consider the componentwise estimation of the random parameters λ M and λ J , under the squared error loss functions L 1 ( λ ̲ , δ 1 ) = ( δ 1 - λ M ) 2 and L 2 ( λ ̲ , δ 2 ) = ( δ 2 - λ J ) 2 , respectively. We derive a general result which provides sufficient conditions for a scale and permutation invariant estimator of λ M (or λ J ) to be inadmissible under the squared error loss function. In situations where these sufficient conditions are satisfied, this result also provides dominating estimators. Since, under the squared error loss function, X i / ( p + 1 ) , i = 1 , 2 , is the best scale invariant estimator of λ i for the component problem, estimators δ 1 , c 1 ( X ̲ ) = X M / ( p + 1 ) and δ 2 , c 1 ( X ̲ ) = X J / ( p + 1 ) are the natural analogs of X 1 / ( p + 1 ) and X 2 / ( p + 1 ) for estimating λ M and λ J , respectively. From the general result we derive, it follows that the natural estimators δ 1 , c 1 ( X ̲ ) = X M / ( p + 1 ) and δ 2 , c 1 ( X ̲ ) = X J / ( p + 1 ) are inadmissible for estimating λ M and λ J , respectively, within the class of scale and permutation invariant estimators and the dominating scale and permutation invariant estimators are obtained. For the estimation of λ J , improvements over various estimators derived by Vellaisamy [1992. Inadmissibility results for the selected scale parameter. Ann. Statist. 20, 2183–2191], which are known to dominate the natural estimator δ 2 , c 1 ( · ) , are obtained. It is also established that any estimator which is a constant multiple of X J is inadmissible for estimating λ J . For 0 < p < 1 , an open problem concerning the inadmissibility of the uniformly minimum variance unbiased estimator of λ J is resolved. Finally, we derive another general result which provides relations between the problems of estimation after selection and estimation of ranked parameters. Applications of this result to the problem of estimating the largest scale parameter of k ( ⩾ 2 ) gamma populations are provided.