Abstract
Shanbhag (J Appl Probab 9:580–587, 1972; Theory Probab Appl 24:430–433, 1979) 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 (J Appl Probab 9:580–587, 1972; Theory Probab Appl 24:430–433, 1979) and Pommeret (J Multivar Anal 63:105–118, 1997) techniques, we evaluated the general form of the 5 × 5 Bhattacharyya matrix in the natural exponential family satisfying \({f(x|\theta)=\frac{\exp\{xg(\theta)\}}{\beta(g(\theta))}\psi(x)}\) with cubic variance function (NEF-CVF) of θ. We see that the matrix is not diagonal like distribution with quadratic variance function and has off-diagonal elements. In addition, we calculate the 5 × 5 Bhattacharyya matrix for inverse Gaussian distribution and evaluated different Bhattacharyya bounds for the variance of estimator of the failure rate, coefficient of variation, mode and moment generating function due to inverse Gaussian distribution.
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References
Alharbi AAG, Shanbhag DN, Thabane L (1997) Some structural properties of the Bhattacharyya matrices. Sankhya Ser A 59: 232–241
Alzaid AA (1987) A note on the Meixner class. Pakistan J Stat 3: 79–82
Anderson TW (1958) An Introduction to Multivariate statistical analysis. Wiley, New York
Bartosewicz J (1980) The convergence of the Bhattacharyya bounds in the multiparametric case. Appl Math XII(4): 601–608
Bhattacharyya A (1946) On some analogues of the amount of information and their use in statistical estimation. Sankhya Ser A 8: 1–14
Bhattacharyya A (1947) On some analogues of the amount of information and their use in statistical estimation II. Sankhya Ser A 8: 201–218
Bhattacharyya A (1948) On some analogues of the amount of information and their use in statistical estimation (concluded). Sankhya Ser A 8: 315–328
Blight BJN, Rao RV (1974) The convergence of Bhattacharyya bounds. Biometrika 61(1): 137–142
Chhikara RS, Folks JL (1977) The inverse Gaussian distribution as a life time model. Technometrics 19: 461–468
Fend AV (1959) On the attainment of Cramer–Rao and Bhattacharyya bounds for the variance of an estimate. Ann Math Stat 30: 381–388
Fosam EB (1993) Characterizations and structural aspect of probability distributions. Ph.D. Thesis, Sheffield University
Khan RA (1984) On UMVU estimator and Bhattacharyya bounds in exponential distributions. J Stat Plan Inference 9: 199–206
Letac G, Mora M (1990) Natural real exponential families with cubic variance functions. Ann Statist 18: 1–37
Mohtashami Borzadaran GR (2001) Results related to the Bhattacharyya matrices. Sankhya, vol 63, Series A, Pt. 1, pp 113–117
Mohtashami Borzadaran GR (2006) A note via diagonality of the 2 × 2 Bhattacharyya matrices. J Math Sci Inf 1(2): 73–78
Noriaki S, Kosei I (1985) UMUV estimators of the mode and limits of an interval for the inverse Gaussian distribution. Comm Stat Theory Method 14(5): 1151–1161
Pommeret D (1997) Multidimensional Bhattacharyya matrices and exponential families. J Multivar Anal 63: 105–118
Seth GR (1949) On the variance of estimates. Ann Math Stat 20: 1–27
Seshadri V (1988) A U-statistic and estimation for the inverse Gaussian distribution. Stat Probab Lett 7: 47–49
Shanbhag DN (1972) Some characterizations based on the Bhattacharyya matrix. J Appl Probab 9: 580–587
Shanbhag DN (1979) Diagonality of the Bhattacharyya Matrix as a characterization. Theory Probab Appl 24: 430–433
Shanbhag DN, Kapoor S (1993) Some questions in characterization theory. Math Scientist 18: 127–133
Tanaka H (2003) On a relation between a family of distributions attaining the Bhattacharyya bound and that of linear combinations of the distributions from an exponential family. Comm Stat Theory Methods 32(10): 1885–1896
Tanaka H, Akahira M (2003) On a family of distributions attaining the Bhattacharyya bound. Ann Inst Stat Math 55: 309–317
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Mohtashami Borzadaran, G.R., Rezaei Roknabadi, A.H. & Khorashadizadeh, M. A view on Bhattacharyya bounds for inverse Gaussian distributions. Metrika 72, 151–161 (2010). https://doi.org/10.1007/s00184-009-0245-4
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DOI: https://doi.org/10.1007/s00184-009-0245-4