Finite mixtures of skew Laplace normal distributions with random skewness
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Finite mixtures of skew Laplace normal distributions with random skewness Fatma Zehra Dogru ˘ 1
· Olcay Arslan2
Received: 29 November 2019 / Accepted: 14 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, the shape mixtures of the skew Laplace normal (SMSLN) distribution is introduced as a flexible extension of the skew Laplace normal distribution which is also a heavy-tailed distribution. The SMSLN distribution includes an extra shape parameter, which controls skewness and kurtosis. Some distributional properties of this distribution are derived. Besides, we propose finite mixtures of SMSLN distributions to model both skewness and heavy-tailedness in heterogeneous data sets. The maximum likelihood estimators for parameters of interests are obtained via the expectation–maximization algorithm. We also give a simulation study and examine a real data example for the numerical illustration of proposed estimators. Keywords EM algorithm · Finite mixture model · ML · SMSLN
1 Introduction Finite mixture models are a most popular analytical tool for modeling heterogeneous data sets which are used in many application areas such as classification, cluster and latent class analysis, density estimation, data mining, image analysis, pattern recognition, etc. [see for more detailed explanations, Titterington et al. (1985), McLachlan and Basford (1988), McLachlan and Peel (2000), Bishop (2006), Frühwirth-Schnatter (2006)]. In general, distributions of components are assumed to be normal since it has a comprehensive application area and ease of computation. In practice, it is not easy to find a data set, which behaves normally since the measured component den-
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Fatma Zehra Do˘gru [email protected] Olcay Arslan [email protected]
1
Department of Statistics, Faculty of Arts and Sciences, Giresun University, 28200 Giresun, Turkey
2
Department of Statistics, Faculty of Science, Ankara University, 06100 Ankara, Turkey
123
F. Z. Dogru, ˘ O. Arslan
sities may contain asymmetric observations or heavier tail than normal. Some studies have been proposed to get rid of these problems in literature. For instance, Lin et al. (2007b) introduced the mixture model using the skew normal (SN) (Azzalini 1985, 1986) distribution; Lin et al. (2007a) proposed a robust mixture model based on the skew t (ST) (Azzalini and Capitanio 2003) distribution to cope with both skewness and heavy-tailedness; Cabral et al. (2008) considered a Bayesian density estimation based on skew Student-t-normal (STN) mixtures; Basso et al. (2010) examined the robust mixture modeling based on scale mixtures of SN distributions; Ho et al. (2012) studied mixtures of the STN (Gómez et al. 2007) distributions to model heavy-tailed data with strong degrees of asymmetry; Do˘gru and Arslan (2017) proposed finite mixtures of skew Laplace normal (SLN) (Gómez et al. 2007) distributions and also applied in mixture regression modeling, and Tamandi and Jamalizadeh (2020) investigated finite mixture modeling based on sha
