Skip to main content
SpringerLink
Log in

LBI tests of independence in bivariate exponential distributions

  • Tests
  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

The locally best invariant test for the hypothesis of independence in bivariate distributions with exponentially distributed marginals is derived. The model consists of a family of bivariate exponential distributions with probability density function

$$f_\theta (x_1 ,x_2 ;\lambda _1 ,\lambda _2 ) = \lambda _1 \lambda _2 \exp [ - (\lambda _1 x_1 + \lambda _2 x_2 )]g(\lambda _1 x_1 ,\lambda _2 x_2 ;\theta )$$

with unknown scale parameter γ j (j=1, 2) and association parameter ϑ which includes the independence situation. The locally best invariant (LBI) test is derived and the asymptotic null and nonnull distributions are also derived under some regularity conditions. The results are applied to the Gumbel (1960,J. Amer. Statist. Assoc.,55, 698–707), Frank (1979,Aequationes Math.,19, 194–226, and Cook and Johnson (1981),J. Roy. Statist. Soc. Ser. B,43, 210–218) families.

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

  • Cook, R. D. and Johnson, M. E. (1981). A family of distributions for modelling nonelliptically symmetric multivariate data.J. Roy. Statist. Soc. Ser. B,43, 210–218.

    Google Scholar 

  • Frank, M. J. (1979). On the simultaneous associativity ofF(x, y) andx+y − F(x, y), Aequationes Math.,19, 194–226.

    Google Scholar 

  • Genest, C. (1987). Frank's family of bivariate distributions,Biometrika,74, 549–555.

    Google Scholar 

  • Genest, C. and MacKay, R. J. (1986). Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données,Canad. J. Statist.,14, 145–159.

    Google Scholar 

  • Gumbel, E. J. (1960). Bivariate exponential distributions,J. Amer. Statist. Assoc.,55, 698–707.

    Google Scholar 

  • Marshall, B. W. and Olkin, I. (1988). Families of multivariate distributions,J. Amer. Statist. Assoc.,83, 834–841.

    Google Scholar 

  • Oakes, D. (1982). A model for association in bivariate survival data,J. Roy. Statist. Soc. Ser. B,44, 414–442.

    Google Scholar 

  • Wijsman, R. A. (1967) Cross-sections of orbits and their applications to densities of maximal invariants,Proc. Fifth Berkeley Symp. on Math. Statist. Prob., Vol. 1, 389–400, Univ. of California Press, Berkeley.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Départment de mathématiques et de statistique, Université de Montréal, succursale A, C.P. 6128, H3C 3J7, Montréal, Canada

    Martin Bilodeau

  2. The Institute of Economic Research, Hitotsubashi University, 186, Kunitachi, Tokyo, Japan

    Takeaki Kariya

  3. Graduate School of Business, University of Chicago, USA

    Takeaki Kariya

Authors
  1. Martin Bilodeau
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takeaki Kariya
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The first author would like to thank the National Sciences and Engineering Research Council of Canada and the Fonds pour la formation de chercheurs et l'aide `a la recherche of Québec for their financial support.

About this article

Cite this article

Bilodeau, M., Kariya, T. LBI tests of independence in bivariate exponential distributions. Ann Inst Stat Math 46, 127–136 (1994). https://doi.org/10.1007/BF00773598

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00773598

Key words and phrases

Search

Navigation