Skip to main content
SpringerLink
Log in

A view on Bhattacharyya bounds for inverse Gaussian distributions

  • Published:
Metrika Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Alharbi AAG, Shanbhag DN, Thabane L (1997) Some structural properties of the Bhattacharyya matrices. Sankhya Ser A 59: 232–241

    MATH  MathSciNet  Google Scholar 

  • Alzaid AA (1987) A note on the Meixner class. Pakistan J Stat 3: 79–82

    MathSciNet  Google Scholar 

  • Anderson TW (1958) An Introduction to Multivariate statistical analysis. Wiley, New York

    MATH  Google Scholar 

  • Bartosewicz J (1980) The convergence of the Bhattacharyya bounds in the multiparametric case. Appl Math XII(4): 601–608

    Google Scholar 

  • Bhattacharyya A (1946) On some analogues of the amount of information and their use in statistical estimation. Sankhya Ser A 8: 1–14

    Google Scholar 

  • Bhattacharyya A (1947) On some analogues of the amount of information and their use in statistical estimation II. Sankhya Ser A 8: 201–218

    Google Scholar 

  • Bhattacharyya A (1948) On some analogues of the amount of information and their use in statistical estimation (concluded). Sankhya Ser A 8: 315–328

    Google Scholar 

  • Blight BJN, Rao RV (1974) The convergence of Bhattacharyya bounds. Biometrika 61(1): 137–142

    Article  MATH  MathSciNet  Google Scholar 

  • Chhikara RS, Folks JL (1977) The inverse Gaussian distribution as a life time model. Technometrics 19: 461–468

    Article  MATH  Google Scholar 

  • Fend AV (1959) On the attainment of Cramer–Rao and Bhattacharyya bounds for the variance of an estimate. Ann Math Stat 30: 381–388

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Letac G, Mora M (1990) Natural real exponential families with cubic variance functions. Ann Statist 18: 1–37

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Pommeret D (1997) Multidimensional Bhattacharyya matrices and exponential families. J Multivar Anal 63: 105–118

    Article  MATH  MathSciNet  Google Scholar 

  • Seth GR (1949) On the variance of estimates. Ann Math Stat 20: 1–27

    Article  MATH  MathSciNet  Google Scholar 

  • Seshadri V (1988) A U-statistic and estimation for the inverse Gaussian distribution. Stat Probab Lett 7: 47–49

    Article  MATH  MathSciNet  Google Scholar 

  • Shanbhag DN (1972) Some characterizations based on the Bhattacharyya matrix. J Appl Probab 9: 580–587

    Article  MATH  MathSciNet  Google Scholar 

  • Shanbhag DN (1979) Diagonality of the Bhattacharyya Matrix as a characterization. Theory Probab Appl 24: 430–433

    Article  MathSciNet  Google Scholar 

  • Shanbhag DN, Kapoor S (1993) Some questions in characterization theory. Math Scientist 18: 127–133

    MATH  MathSciNet  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Tanaka H, Akahira M (2003) On a family of distributions attaining the Bhattacharyya bound. Ann Inst Stat Math 55: 309–317

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

    G. R. Mohtashami Borzadaran, A. H. Rezaei Roknabadi & M. Khorashadizadeh

Authors
  1. G. R. Mohtashami Borzadaran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. H. Rezaei Roknabadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Khorashadizadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. R. Mohtashami Borzadaran.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00184-009-0245-4

Keywords

Search

Navigation