Finite mixture of skewed distributions
This monograph published in Springer Briefs in Statistics is on finite mixture models and it presents results on finite scale mixtures of skew-normal distributions abbreviated as SMSN distributions. This class is a subclass of the skew-elliptical class and contains the entire family of independent normal distributions known as scale mixtures of normal distributions, in addition to the skew-normal, the skew-\(t\) and the skew-slash distributions. These distributions have tails heavier than the normal and are suitable for robust inference accommodating skewness, outliers and multimodality. Out of the six chapters, the first chapter gives a couple of motivating examples. The second chapter reviews the general theory of maximum likelihood estimation in finite mixture models with multivariate normal components, using the Expectation-Maximization (EM) algorithm with an explanation on the method of obtaining the standard error estimates for the EM estimators. The third chapter is on the SMSN distributions and contains examples of scale mixtures of normal distributions, discusses the multivariate SMSN distributions and their properties like moments, kurtosis, linear transformations, marginal and conditional distributions with examples, a simulation study and maximum likelihood estimation. Chapter 4 is on the univariate mixture modelling using SMSN distributions which is a flexible class, convenient for modelling data with skewness and population heterogeneity. An EM-type algorithm is developed for maximum likelihood estimation and the observed information matrix is obtained. The procedures are discussed with emphasis on finite mixtures of skew normal, skew-\(t\), skew slash and skew contaminated normal distributions. The algorithm is implemented in the R package mixsmsn. Under simulation studies, the ability of the model in clustering observations is investigated, some asymptotic properties of the estimates obtained using the proposed EM-type algorithm are investigated, model selection is discussed with an application of the methodology proposed with real data. The fifth chapter is on a flexible class of models with elements that are finite mixtures of multivariate scale mixtures of skew normal distributions. A general EM-type algorithm is used for iteratively computing the parameter estimates and this is discussed with emphasis on finite mixtures of skew normal, skew-\(t\), skew slash, and skew contaminated normal distribution. A general information based method for approximating the asymptotic covariance matrix of the estimates is also presented in this chapter. The EM-type algorithm is implemented in the R package mixsmsn. Applications are discussed using real and simulated data. The sixth and last chapter is on a unified robust mixture regression modelling based on SMSN distributions. The proposed method allows in modelling data accommodating skewness and heavy tails and also easy implementation of inference via a EM-type algorithm. Three simulation experiments are discussed along with an application of the proposed methods to a real dataset. The monograph ends with a comprehensive list of references and a subject index. The monograph is well written with very few typographical errors and will be very useful to researchers using finite mixture models as it discusses contemporary methods used in such modelling.
scientific article; zbMATH DE number 6992858