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Large sample confidence intervals for the skewness parameter of the skew-normal distribution based on Fisher's transformation

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  • Valentina Mameli
  • Monica Musio
  • Erik Sauleau
  • Annibale Biggeri

Abstract

The skew-normal model is a class of distributions that extends the Gaussian family by including a skewness parameter. This model presents some inferential problems linked to the estimation of the skewness parameter. In particular its maximum likelihood estimator can be infinite especially for moderate sample sizes and is not clear how to calculate confidence intervals for this parameter. In this work, we show how these inferential problems can be solved if we are interested in the distribution of extreme statistics of two random variables with joint normal distribution. Such situations are not uncommon in applications, especially in medical and environmental contexts, where it can be relevant to estimate the distribution of extreme statistics. A theoretical result, found by Loperfido [7], proves that such extreme statistics have a skew-normal distribution with skewness parameter that can be expressed as a function of the correlation coefficient between the two initial variables. It is then possible, using some theoretical results involving the correlation coefficient, to find approximate confidence intervals for the parameter of skewness. These theoretical intervals are then compared with parametric bootstrap intervals by means of a simulation study. Two applications are given using real data.

Suggested Citation

  • Valentina Mameli & Monica Musio & Erik Sauleau & Annibale Biggeri, 2012. "Large sample confidence intervals for the skewness parameter of the skew-normal distribution based on Fisher's transformation," Journal of Applied Statistics, Taylor & Francis Journals, vol. 39(8), pages 1693-1702, February.
    • Handle: RePEc:taf:japsta:v:39:y:2012:i:8:p:1693-1702
    DOI: 10.1080/02664763.2012.668177
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    References listed on IDEAS

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    1. Loperfido, Nicola, 2002. "Statistical implications of selectively reported inferential results," Statistics & Probability Letters, Elsevier, vol. 56(1), pages 13-22, January.
    2. A. Azzalini & A. Capitanio, 1999. "Statistical applications of the multivariate skew normal distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 579-602.
    3. Loperfido, Nicola, 2008. "A note on skew-elliptical distributions and linear functions of order statistics," Statistics & Probability Letters, Elsevier, vol. 78(18), pages 3184-3186, December.
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    1. M. Teimourian & T. Baghfalaki & M. Ganjali & D. Berridge, 2015. "Joint modeling of mixed skewed continuous and ordinal longitudinal responses: a Bayesian approach," Journal of Applied Statistics, Taylor & Francis Journals, vol. 42(10), pages 2233-2256, October.

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