The skew normal model is a class of distributions that extends the Gaussian family by including a skew parameter. Although its remarkable properties, this model presents some inferential problems linked to the estimation of the skew 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 easily 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 can be relevant to estimate the distribution of extreme statistics. A theoretical result, found by Leoperfido (2002), proves that such extreme statistics have a skew-normal distribution with skew parameter which 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.

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

MAMELI, VALENTINA;MUSIO, MONICA;
2012-01-01

Abstract

The skew normal model is a class of distributions that extends the Gaussian family by including a skew parameter. Although its remarkable properties, this model presents some inferential problems linked to the estimation of the skew 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 easily 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 can be relevant to estimate the distribution of extreme statistics. A theoretical result, found by Leoperfido (2002), proves that such extreme statistics have a skew-normal distribution with skew parameter which 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.
2012
skew-normal distribution; skewness parameter
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11584/91612
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