Alharbi, A. A. G. On the convergence of the Bhattacharyya bounds in the multiparametric case. (English) Zbl 0812.62056 Appl. Math. 22, No. 3, 339-349 (1994). 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. Cited in 1 Document MSC: 62H10 Multivariate distribution of statistics 62E10 Characterization and structure theory of statistical distributions 60E05 Probability distributions: general theory 62F10 Point estimation Keywords:characterizations; diagonal of covariance matrix; exponential family; diagonality; Bhattacharyya matrix; Meixner hypergeometric distributions; bivariate distribution; normal; Poisson; binomial; negative binomial; gamma; exponential distribution; generalized Bhattacharyya bounds; best unbiased estimator; minimum variance unbiased estimator; geometric distribution Citations:Zbl 0448.60014; Zbl 0252.60008; Zbl 0436.60017; Zbl 0447.62053; Zbl 0285.62011 PDFBibTeX XMLCite \textit{A. A. G. Alharbi}, Appl. Math. 22, No. 3, 339--349 (1994; Zbl 0812.62056) Full Text: DOI EuDML