Skip to main content
SpringerLink
Log in

A new bivariate exponential distribution for modeling moderately negative dependence

  • Published:
Statistical Methods & Applications Aims and scope Submit manuscript

Abstract

This paper introduces a new bivariate exponential distribution, called the Bivariate Affine-Linear Exponential distribution, to model moderately negative dependent data. The construction and characteristics of the proposed bivariate distribution are presented along with estimation procedures for the model parameters based on maximum likelihood and objective Bayesian analysis. We derive Jeffreys prior and discuss its frequentist properties based on a simulation study and MCMC sampling techniques. A real data set of mercury concentration in largemouth bass from Florida lakes is used to illustrate the methodology.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Arnold BC, Strauss D (1988) Bivariate distributions with exponential conditionals. J Am Stat Assoc 83:522–527

    Article  MATH  MathSciNet  Google Scholar 

  • Barndorff-Nielsen O (1978) Information and exponential families. Wiley, New York

    MATH  Google Scholar 

  • Basu AP (1988) Multivariate exponential distribution and their application in reliability. In: Krishnaiah PR, Sen PK (eds) Handbook of statistics-7. North-Holland, New York, pp 467–477

    Google Scholar 

  • Berger J, De Oliveira V, Sanso B (2001) Objective Bayesian analysis of spatially correlated data. J Am Stat Assoc 96:1361–1374

    Article  MATH  Google Scholar 

  • Berger J, Bernardo J, Sun D (2009) The formal definition of reference priors. Ann Stat 37:905–938

    Article  MATH  MathSciNet  Google Scholar 

  • Blom G (1958) Statistical estimates and transformed beta-variables. Wiley, New York

    MATH  Google Scholar 

  • Chambers J, Cleveland W, Kleiner B, Tukey P (1983) Graphical methods for data analysis. Chapman &Hall/CRC, London

    MATH  Google Scholar 

  • Corless R, Gonnet G, Hare D, Jeffrey D, Knuth D (1996) On the Lambert W function. Adv Comput Math 5:329–359

    Article  MATH  MathSciNet  Google Scholar 

  • Doss DC, Graham RC (1975) Construction of multivariate linear exponential distribution from univariate marginals. Sankhya A 37:257–268

    MATH  MathSciNet  Google Scholar 

  • Downton F (1970) Bivariate exponential distributions in reliability theory. J R Stat Soc B 32:408–417

    MATH  MathSciNet  Google Scholar 

  • Embrechts P, McNeil A, Straumann D (2002) Correlation and dependence in risk management: properties and pitfalls. In: Dempster M (ed) Risk management: value at risk and beyond. Cambridge University Press, Cambridge, pp 176–223

    Chapter  Google Scholar 

  • FAO/WHO (2003) Joint FAO/WHO expert committee on food additives: summary and conclusions. Sixty first meeting, food and agriculture organization of the United Nations and World Health Organization, Rome

  • Franco M, Vivo JM (2010) A multivariate extension of Sarhan and Balakrishnan’s bivariate distribution and its ageing and dependence properties. J Multivar Anal 101:491–499

    Article  MATH  MathSciNet  Google Scholar 

  • Franco M, Kundu D, Vivo JM (2011) Multivariate extension of modified Sarhan–Balakrishnan bivariate distribution. J Stat Plan Inference 141:3400–3412

    Article  MATH  MathSciNet  Google Scholar 

  • Freund JE (1961) A bivariate extension of the exponential distribution. J Am Stat Assoc 56:971–977

    Article  MATH  MathSciNet  Google Scholar 

  • Gelman A, Carlin J, Stern H, Rubin D (2003) Bayesian data analysis. Chapman & Hall/CRC, Boca Raton

    Google Scholar 

  • Genest C, Remillard B (2008) Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models. Ann Inst Henri Poincare B 44:1096–1127

    Article  MATH  MathSciNet  Google Scholar 

  • Genest C, Remilliard B, Beaudoin D (2008) Goodness-of-fit tests for copulas: a review and a power study. Insur Math Econ 44:199–213

    Article  Google Scholar 

  • Gumbel EJ (1960) Bivariate exponential distributions. J Am Stat Assoc 55:698–707

    Article  MATH  MathSciNet  Google Scholar 

  • Gupta RD, Kundu D (1999) Generalized exponential distributions. Aust NZ J Statist 41:173–188

    Article  MATH  MathSciNet  Google Scholar 

  • Gupta RD, Kundu D (2001) Generalized exponential distributions: different methods of estimation. J Stat Comput Simul 69:315–338

    Article  MATH  MathSciNet  Google Scholar 

  • Gupta RD, Kundu D (2002) Generalized exponential distributions: statistical inferences. J Stat Theory Appl 1:101–118

    MathSciNet  Google Scholar 

  • Gupta RD, Kundu D (2007) Generalized exponential distributions: existing methods and recent developments. J Stat Plan Inference 137:3537–3547

    Article  MATH  MathSciNet  Google Scholar 

  • Hougaard P (1986) A class of multivariate failure time distributions. Biometrika 73:671–678

    MATH  MathSciNet  Google Scholar 

  • Iyer S, Manjunath D, Manivasakan R (2002) Bivariate exponential distributions using linear structures. Sankhya A 64:156–166

    MATH  MathSciNet  Google Scholar 

  • Jeffreys H (1961) Theory of probability. Oxford University Press, Oxford

    MATH  Google Scholar 

  • Johnson NL, Kotz S (1975) A vector multivariate hazard rate. J Multivar Anal 5:53–66

    Article  MATH  MathSciNet  Google Scholar 

  • Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1. Wiley, New York

    MATH  Google Scholar 

  • Kass R, Wasserman L (1996) The selection of prior distributions by formal rules. J Am Stat Assoc 88:1343–1370

    Article  Google Scholar 

  • Klauer KC (1986) Non-exponential families of distributions. Metrika 33:299–305

    Article  MATH  MathSciNet  Google Scholar 

  • Kundu D, Gupta RD (2008) Generalized exponential distribution: Bayesian estimations. Comp Stat Data Anal 52:1873–1883

    Article  MATH  MathSciNet  Google Scholar 

  • Kundu D, Gupta RD (2009) Bivariate generalized exponential distribution. J Multivar Anal 100:581–593

    Article  MATH  MathSciNet  Google Scholar 

  • Lange TL, Royals HE, Connor LL (1993) Influence of water chemistry on mercury concentration in largemouth bass from Florida lakes. Trans Am Fish Soc 122:74–84

    Article  Google Scholar 

  • Lehmann EL (1966) Some concepts of dependence. Ann Math Stat 37:1137–1153

    Article  MATH  Google Scholar 

  • Lehmann EL, Casella G (2003) Theory of point estimation. Springer, New York

    Google Scholar 

  • Marshall AW, Olkin I (1967a) A multivariate exponential distribution. J Am Stat Assoc 62:30–44

    Article  MATH  MathSciNet  Google Scholar 

  • Marshall AW, Olkin I (1967b) A generalized bivariate exponential distribution. J Appl Prob 4:291–302

    Article  MATH  MathSciNet  Google Scholar 

  • Maz’ya V, Shaposhnikova T (1998) Jacques Hadamard, a universal mathematician. AMS/LMS, Providence

    MATH  Google Scholar 

  • Miller RG (1974) The Jackknife—a review. Biometrika 61:1–15

    MATH  MathSciNet  Google Scholar 

  • Nadarajah S, Kotz S (2006) Reliability for some bivariate exponential distributions. Math Probl Eng. doi:10.1155/MPE/2006/41652

  • Nelsen R (2006) An introduction to copulas. Springer, New York

    MATH  Google Scholar 

  • Prudnikov AP, Brychkov YA, Marichev OI (1986) Integrals and series, vol 1. Gordon and Breach, Amsterdam

    MATH  Google Scholar 

  • Regoli G (2009) A class of bivariate exponential distributions. J Multivar Anal 100:1261–1269

    Article  MATH  MathSciNet  Google Scholar 

  • Sarhan AM, Balakrishnan N (2007) A new class of bivariate distributions and its mixture. J Multivar Anal 98:1508–1527

    Article  MATH  MathSciNet  Google Scholar 

  • Spizzichino F (2001) Subjective probability models for lifetimes. Chapman & Hall/CRC, Boca Raton

    Book  MATH  Google Scholar 

  • Waagepetersen R, Ibanez-Escriche N, Sorensen D (2008) A comparison of strategies for Markov chain Monte Carlo computation in quantitative genetics. Genet Sel Evol 40:161–176

    Google Scholar 

  • Ware FJ, Royals H, Lange T (1991) Mercury contamination in Florida largemouth bass. Proc Ann Conf SEAFWA 44:5–12

    Google Scholar 

  • Wiener JG (1987) Metal contamination of fish in low pH-lakes and potential implications for piscivorous wildlife. Trans North Am Wild Nat Resour Conf 52:645–657

    Google Scholar 

Download references

Acknowledgments

The authors are thankful to the associate editor and the two referees for their valuable comments and suggestions which certainly helped to improve the paper. The first author is also thankful to the Higher Education Commission of Pakistan for their financial support of this project.

Author information

Authors and Affiliations

  1. Department of Statistics, Alpen-Adria University of Klagenfurt, 9020 , Klagenfurt, Austria

    Muhammad Mohsin, Hannes Kazianka, Jürgen Pilz & Albrecht Gebhardt

  2. Department of Statistics, COMSATS Institute of Information Technology, Lahore, Pakistan

    Muhammad Mohsin

Authors
  1. Muhammad Mohsin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hannes Kazianka
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jürgen Pilz
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Albrecht Gebhardt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hannes Kazianka.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mohsin, M., Kazianka, H., Pilz, J. et al. A new bivariate exponential distribution for modeling moderately negative dependence. Stat Methods Appl 23, 123–148 (2014). https://doi.org/10.1007/s10260-013-0246-3

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10260-013-0246-3

Keywords

Search

Navigation