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On the convergence of the Bhattacharyya bounds in the multiparametric case. (English) Zbl 0812.62056

Summary: D. N. Shanbhag [J. Appl. Probab. 9, 580-587 (1972; Zbl 0252.60008); Theory Probab. Appl. 24, 430-433 (1979); translation from Teor. Veroyatn. Primen. 24, 424-427 (1979; Zbl 0436.60017)] showed that the diagonality of the Bhattacharyya matrix characterizes the set of normal, Poisson, binomial, negative binomial, gamma or Meixner hypergeometric distributions. In this note, using Shanbhag’s techniques, we show that if a certain generalized version of the Bhattacharyya matrix is diagonal, then the bivariate distribution is either normal, Poisson, binomial, negative binomial, gamma, or Meixner hypergeometric.
J. Bartoszewicz [Zastosow. Mat. 16, 601-608 (1980; Zbl 0447.62053)] extended the result of B. J. Blight and P. V. Rao [Biometrika 61, 137-143 (1974; Zbl 0285.62011)] to the multiparameter case. He gave an application of this result when independent samples come from the exponential distribution, and also evaluated the generalized Bhattacharyya bounds for the best unbiased estimator of \(P(Y<X)\). We show that there are misprints in these results, give corrections and obtain the generalized Bhattacharyya bounds for the variance of the minimum variance unbiased estimator of \(P(Y<X)\) when independent observations are taken from a normal or geometric distribution.

MSC:

62H10 Multivariate distribution of statistics
62E10 Characterization and structure theory of statistical distributions
60E05 Probability distributions: general theory
62F10 Point estimation
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