Summary
In recent years, the skew-normal models introduced in Azzalini (1985) have enjoyed an amazing success, although an important literature has reported that they exhibit, in the vicinity of symmetry, singular Fisher information matrices and stationary points in the profile log-likelihood function for skewness, with the usual unpleasant consequences for inference. For general multivariate skew-symmetric and skew-elliptical models, the open problem of determining which symmetric kernels lead to each such singularity has been solved in Ley and Paindaveine (2010). In the present paper, we provide a simple proof that, in generalized skew-elliptical models involving the same skewing scheme as in the skew-normal distributions, Fisher information matrices, in the vicinity of symmetry, are singular for Gaussian kernels only. Then we show that if the profile log-likelihood function for skewness always has a point of inflection in the vicinity of symmetry, the generalized skew-elliptical distribution considered is actually skew-(multi)normal. In addition, we show that the class of multivariate skew-t distributions (as defined in Azzalini and Capitanio 2003), which was not covered by Ley and Paindaveine (2010), does not suffer from singular Fisher information matrices in the vicinity of symmetry. Finally, we briefly discuss the implications of our results on inference.
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References
Arellano-Valle, R. B. and Azzalini, A. (2008) The centred parametrization for the multivariate skew-normal distribution, Journal of Multivariate Analysis, 99, 1362–1382.
Azzalini, A. (1985) A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12, 171–178.
Azzalini, A. (2005) The skew-normal distribution and related multivariate families (with discussion), Scandinavian Journal of Statistics, 32, 159–188.
Azzalini, A. and Capitanio, A. (1999) Statistical applications of the multivariate skew-normal distributions, Journal of the Royal Statistical Society B, 61, 579–602.
Azzalini, A. and Capitanio, A. (2003) Distributions generated by perturbation of symmetry with emphasis on a multivariate skew-t distribution, Journal of the Royal Statistical Society B, 65, 367–389.
Azzalini, A. and Dalla Valle, A. (1996) The multivariate skew-normal distribution, Biometrika, 83, 715–726.
Azzalini, A. and Genton, M.G. (2007) On Gauss’s characterization of the normal distribution, Bernoulli, 13, 169–174.
Azzalini, A. and Genton, M.G. (2008) Robust likelihood methods based on the skew-t and related distributions, International Statistical Review, 76, 106–129.
Barndorff-Nielsen, O. E. and Cox, D. R. (1994) Inference and asymptotics, Chapman and Hall, London.
Bottai, M. (2003) Confidence regions when the Fisher information is zero, Biometrika, 90, 73–84.
Branco, M. D. and Dey, D. K. (2001) A general class of multivariate skew-elliptical distributions, Journal of Multivariate Analysis, 79, 99–113.
Chiogna, M. (2005) A note on the asymptotic distribution of the maximum likelihood estimator for the scalar skew-normal distribution, Statistical Methods and Applications, 14, 331–341.
DiCiccio, T. J. and Monti, A. C. (2004) Inferential aspects of the skew exponential power distribution, Journal of the American Statistical Association, 99,439–450.
DiCiccio, T. J. and Monti, A. C. (2011) Inferential aspects of the skew t-distribution, Manuscript in preparation.
Genton, M. G. and Loperfido, N. (2005) Generalized skew-elliptical distributions and their quadratic forms, Annals of the Institute of Statistical Mathematics, 57, 389–401.
Hallin, M. and Paindaveine, D. (2006) Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity, Annals of Statistics, 34, 2707–2756.
Gomez, H. W., Venegas, O. and Bolfarine, H. (2007) Skew-symmetric distributions generated by the distribution function of the normal distribution, Environmetrics, 18, 395–407.
Ley, C. and Paindaveine, D. (2010) On the Singularity of Multivariate Skew-Symmetric Models, Journal of Multivariate Analysis, 101, 1434–1444.
Paindaveine, D. (2008) A canonical definition of shape, Statistics and Probability Letters, 78, 2240–2247.
Patefield, W. M. (1977) On the maximized likelihood function, Sankhyā Series B, 39, 92–96.
Pewsey, A. (2000) Problems of inference for Azzalini’s skew-normal distribution, Journal of Applied Statistics, 27, 859–870.
Rotnitzky, A., Cox, D. R., Bottai, M. and Roberts, J. (2000) Likelihood-based inference with singular information matrix, Bernoulli, 6, 243–284.
Wang, J., Boyer, J. and Genton, M. G. (2004) A skew-symmetric representation of multivariate distribution, Statistica Sinica, 14, 1259–1270.
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Ley, C., Paindaveine, D. On Fisher information matrices and profile log-likelihood functions in generalized skew-elliptical models. METRON 68, 235–250 (2010). https://doi.org/10.1007/BF03263537
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DOI: https://doi.org/10.1007/BF03263537