Advertisement

Indian Journal of Pure and Applied Mathematics

, Volume 49, Issue 4, pp 633–650 | Cite as

Statistical Inference for the Location and Scale Parameters of the Skew Normal Distribution

  • Wenhao GuiEmail author
  • Lei Guo
Article
  • 55 Downloads

Abstract

In this paper, we consider the problem of estimating the location and scale parameters of the skew normal distribution introduced by Azzalini. For this distribution, the classic maximum likelihood estimators(MLEs) do not take explicit forms. We approximate the likelihood equations and derive explicit estimators of the parameters. The bias and variance of the estimators are investigated and Monte Carlo simulation studies show that the estimators are as efficient as the classic MLEs. We demonstrate that the probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic normality are unsatisfactory, especially when the sample size is small. The use of unconditional simulated percentage points of these quantities is suggested. Finally, a numerical example is used to illustrate the proposed inference methods.

Key words

Skew normal distribution maximum-likelihood estimator Monte Carlo simulation probability coverage pivotal quantity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B. Arnold and N. Balakrishnan, Relations, bounds and approximations for order statistics, 53 (2012), Springer Science and Business Media.Google Scholar
  2. 2.
    B. C. Arnold, R. J. Beaver, R. A. Groeneveld, and W. Q. Meeker, The non-truncated marginal of a truncated bivariate normal distribution, Psychometrika 58(3) (1993), 471–488.MathSciNetCrossRefGoogle Scholar
  3. 3.
    A. Asgharzadeh, L. Esmaily, and S. Nadarajah, Approximate mles for the location and scale parameters of the skew logistic distribution, Statistical Papers, 54(2) (2013), 391–411.MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. Azzalini, A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, (1985), 171–178.Google Scholar
  5. 5.
    A. Azzalini, The skew-normal distribution and related multivariate families, Scandinavian Journal of Statistics, (2005), 159–188.Google Scholar
  6. 6.
    A. Azzalini and A. Capitanio, Statistical applications of the multivariate skew normal distribution, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61(3) (1999), 579–602.MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Azzalini and A. Dalla Valle, The multivariate skew-normal distribution, Biometrika, 83(4) (1996), 715–726.MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Chiogna, Some results on the scalar skew-normal distribution, Journal of the Italian Statistical Society, 7(1) (1998), 1–13.CrossRefGoogle Scholar
  9. 9.
    T. S. Ferguson, A course in large sample theory, Vol. 49 (1996), Chapman and Hall London.Google Scholar
  10. 10.
    S. Ghosh and D. Dey, Estimation of the parameters of skew normal distribution by approximating the ratio of the normal density and distribution functions, Dissertations and Theses - Gradworks, (2010).Google Scholar
  11. 11.
    R. C. Gupta and N. Brown, Reliability studies of the skew-normal distribution and its application to a strength-stress model, Communications in Statistics-Theory and Methods, 30(11) (2001), 2427–2445.MathSciNetCrossRefGoogle Scholar
  12. 12.
    N. Henze, A probabilistic representation of the’ skew-normal’ distribution, Scandinavian Journal of Statistics, (1986), 271–275.Google Scholar
  13. 13.
    A. Hossain and J. Beyene, Application of skew-normal distribution for detecting differential expression to Microrna data, Journal of Applied Statistics, 42(3) (2015), 477–491.MathSciNetCrossRefGoogle Scholar
  14. 14.
    B. Liseo and N. Loperfido, A bayesian interpretation of the multivariate skew-normal distribution, Statistics & probability letters, 61(4) (2003), 395–401.MathSciNetCrossRefGoogle Scholar
  15. 15.
    B. Liseo and N. Loperfido, Default bayesian analysis of the skew-normal distribution (2004).Google Scholar
  16. 16.
    B. Liseo and N. Loperfido, A note on reference priors for the scalar skew-normal distribution, Journal of Statistical Planning and Inference, 136(2) (2006), 373–389.MathSciNetCrossRefGoogle Scholar
  17. 17.
    W. Ning and G. Ngunkeng, An empirical likelihood ratio based goodness-of-fit test for skew normality, Statistical Methods & Applications, 22(2) (2013), 209–226.MathSciNetCrossRefGoogle Scholar
  18. 18.
    H. V. Roberts, Data analysis for managers with Minitab: Harry V. Roberts, Scientific Press, (1991).Google Scholar
  19. 19.
    N. Sartori, Bias prevention of maximum likelihood estimates for scalar skew normal and skew t distributions, Journal of Statistical Planning and Inference, 136(12) (2006), 4259–4275.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Indian National Science Academy 2018

Authors and Affiliations

  1. 1.Department of MathematicsBeijing Jiaotong UniversityBeijingP. R. China

Personalised recommendations