Mean mixtures of normal distributions: properties, inference and application

The skew normal (SN) distribution of Azzalini (Scand J Stat 12:171–178, 1985) is one of the widely used probability distributions for modelling skewed data. In this article, we introduce a general class of skewed distributions based on mean mixtures of normal distributions, which includes the SN distribution as a special case. Some properties of this new class, such as expressions for mean, variance, skewness and kurtosis coefficients and characteristic function, are derived. Also, estimates of the model parameters are first obtained by the method of moments. Two special cases of this new class are studied in detail. It is shown that the range of skewness and kurtosis coefficients for the special cases is wider than that of the SN distribution, and in addition, unlike the SN distribution, one of these models is an infinitely divisible distribution. For carrying out the maximum likelihood (ML) estimation, an ECM algorithm is developed. This algorithm is analytically simple because closed-form expressions of conditional expectations in the E-step as well as the updating estimators in the CM-step are in explicit form. The observed information matrix is provided for approximating the asymptotic covariance matrix of the ML estimators of the parameters. The usefulness of the proposed distribution is illustrated through simulated as well as two real data sets. Next, a new extension of regression models is constructed by assuming the proposed distributions for the error term. Finally, a multivariate version of the proposed model is discussed.