Advertisement

Metrika

, Volume 68, Issue 1, pp 31–49 | Cite as

A new class of inverse Gaussian type distributions

  • Antonio Sanhueza
  • Víctor Leiva
  • N. Balakrishnan
Article

Abstract

The elliptical laws are a class of symmetrical probability models that include both lighter and heavier tailed distributions. These models may adapt well to the data, even when outliers exist and have other good theoretical properties and application perspectives. In this article, we present a new class of models, which is generated from symmetrical distributions in \({\mathbb{R}}\) and generalize the well known inverse Gaussian distribution. Specifically, the density, distribution function, properties, transformations and moments of this new model are obtained. Also, a graphical analysis of the density is provided. Furthermore, we estimate parameters, propose asymptotic inference and discuss influence diagnostics by using likelihood methods for the new distribution. In particular, we show that the maximum likelihood estimates parameters of the new model under the t kernel are down-weighted for the outliers. Thus, smaller weights are attributed to outlying observations, which produce robust parameter estimates. Finally, an illustrative example with real data shows that the new distribution fits better to the data than some other well known probabilistic models.

Keywords

Elliptical distributions Diagnostics Kurtosis Likelihood methods Moments 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Balakrishnan N (ed) (1992) Handbook of the logistic distribution. Marcel Dekker, New YorkMATHGoogle Scholar
  2. Chhikara RS, Folks JL (1989) The inverse Gaussian distribution. Marcel Dekker, New YorkMATHGoogle Scholar
  3. Cook RD (1986) Assessment of local influence (with discussion). J R Stat Soc B 48:133–169MATHGoogle Scholar
  4. Cook RD, Weisberg S (1982) Residuals and influence in regression. Chapman & Hall, LondonMATHGoogle Scholar
  5. Fang KT, Kotz S, Ng KW (1990) Symmetric multivariate and related distribution. Chapman, LondonGoogle Scholar
  6. Fernandez C, Steel M (1999) Multivariate t regression models: pitfalls and inference. Biometrika 86:153–167MATHCrossRefMathSciNetGoogle Scholar
  7. Galea M, Riquelme M, Paula GA (2000) Diagnostics methods in elliptical linear regression models. Braz J Prob Stat 14:167–184MATHMathSciNetGoogle Scholar
  8. Galea M, Leiva V, Paula GA (2004) Influence diagnostics in log-Birnbaum-Saunders regression models. J Appl Stat 31:1049–1064MATHCrossRefMathSciNetGoogle Scholar
  9. Gupta AK, Varga T (1993) Elliptically contoured models in statistics. Kluwer, DordrechtMATHGoogle Scholar
  10. Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1. Wiley, New YorkMATHGoogle Scholar
  11. Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, vol 2. Wiley, New YorkMATHGoogle Scholar
  12. Kleiber C, Kotz S (2003) Statistical size distribution in economics and actuarial sciences. Wiley, New YorkGoogle Scholar
  13. Lange KL, Little JA, Taylor MG (1989) Robust statistical modelling using the t distribution. J Am Stat Soc 84:881–896MathSciNetGoogle Scholar
  14. Leiva V, Barros M, Paula GA, Galea M (2007) Influence diagnostics in log-Birnbaum–Saunders regression models with censored data. Comp Stat Data Anal 51:5694–5707CrossRefGoogle Scholar
  15. Mudholkar GS, Natarajan R (2002) The inverse Gaussian analogues of symmetry, skewness and kurtosis. Ann Inst Stat Math 54:138–54MATHCrossRefMathSciNetGoogle Scholar
  16. Panjer HH (2006) Operational risk: modeling analytics. Wiley, New YorkMATHGoogle Scholar
  17. Saunders SC (1974) A family of random variables closed under reciprocation. J Am Stat Soc 69:533–539MATHGoogle Scholar
  18. Seshadri V (1993) The inverse Gaussian distribution: a case study in exponential families. Claredon, New YorkGoogle Scholar
  19. Seshadri V (1999) The inverse Gaussian distribution: statistical theory and applications. Springer, New YorkMATHGoogle Scholar
  20. Taylor J, Verbyla A (2004) Joint modelling of location and scale parameters of t distribution. Stat Model 4:91–112MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Antonio Sanhueza
    • 1
  • Víctor Leiva
    • 2
  • N. Balakrishnan
    • 3
  1. 1.Department of Mathematics and StatisticsUniversidad de La FronteraTemucoChile
  2. 2.Department of StatisticsUniversidad de ValparaísoValparaísoChile
  3. 3.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

Personalised recommendations