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Optimal reparametrization and large sample likelihood inference for the location-scale skew-normal model

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Abstract

Motivated by results in Rotnitzky et al. (2000), a family of parametrizations of the location-scale skew-normal model is introduced, and it is shown that, under each member of this class, the hypothesis H 0: λ = 0 is invariant, where λ is the asymmetry parameter. Using the trace of the inverse variance matrix associated to a generalized gradient as a selection index, a subclass of optimal parametrizations is identified, and it is proved that a slight variant of Azzalini’s centred parametrization is optimal. Next, via an arbitrary optimal parametrization, a simple derivation of the limit behavior of maximum likelihood estimators is given under H 0, and the asymptotic distribution of the corresponding likelihood ratio statistic for this composite hypothesis is determined.

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References

  1. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.

    MATH  Google Scholar 

  2. A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Stat. Theory Appl., 12 (1985), 171–178.

    MathSciNet  MATH  Google Scholar 

  3. A. Azzalini, Further results on a class of distributions which includes the normal ones, Statistica, 46 (1986), 199–208.

    MathSciNet  MATH  Google Scholar 

  4. A. Azzalini and A. Capitanio, Statistical applications of the multivariate skew normal distribution, J. R. Stat. Soc., Ser. B, Stat. Methodol., 61 (1999), 579–602.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Azzalini and A. Dalla Valle, The skew normal distribution, Biometrika, 83 (1996), 715–726.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Bilodeau and D. Brenner, Theory of Multivariate Statistics, Springer, New York, 1999.

    MATH  Google Scholar 

  7. A. A. Borobkov, Mathematical Statistics, Gordon & Breach, Amsterdam, The Netherlands, 1998.

    Google Scholar 

  8. R. Cavazos-Cadena and G. González-Farías, Existence and consistency of maximum likelihood estimators for the scalar location-scale skew-normal model, Adv. Appl. Math. Sci., 4 (2010), 83–111.

    MathSciNet  MATH  Google Scholar 

  9. A. Dalla Valle, The skew-normal distribution, Skew Elliptical Distributions and Their Applications, Genton M. G. (ed.), Chapman & Hall, London, 2004, 3–24.

    Google Scholar 

  10. M. Chiogna, A note on the asymptotic distribution of the maximum likelihood estimator for the scalar skew-normal distribution, Stat. Methods Appl., 14 (2005), 331–341.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. G. Genton, Skew Elliptical Distributions and Their Applications, Chapman & Hall, London, 2004.

    Book  MATH  Google Scholar 

  12. E. I. Lehmann and G. Casella, Theory of Point Estimation, Second Edition, Springer, New York, 1998.

    MATH  Google Scholar 

  13. W. K. Newey and D. L. Mcfadden, Estimation in large samples and hypothesis testing, Handbook of Econometrics, Vol. 4, R. F. Engle and D. L. McFadden (eds.), North-Holland, Amsterdam, 1993, 2111–2245.

    Google Scholar 

  14. A. Rotnitzky, D. R. Cox, M. Bottai and J. Robins, Likelihood-based inference with singular information matrix, Bernoulli, 6 (2000), 243–284.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Shao, Mathematical Statistics, Springer, New York, 1999.

    MATH  Google Scholar 

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Correspondence to Rolando Cavazos-Cadena.

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Communicated by István Berkes

Dedicated to the memory of Don José María Morelos y Pavón

This work was supported by the PSF Organization under Grant No. 2007-4, and by CONA-CYT under Grant 105657 /CB2008

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Cavazos-Cadena, R., González-Farías, G.M. Optimal reparametrization and large sample likelihood inference for the location-scale skew-normal model. Period Math Hung 64, 181–211 (2012). https://doi.org/10.1007/s10998-012-5259-4

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  • DOI: https://doi.org/10.1007/s10998-012-5259-4

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