A general class of scale-shape mixtures of skew-normal distributions: properties and estimation

Computational Statistics volume 32, pages451–474(2017)Cite this article

Abstract

This paper introduces the scale-shape mixtures of skew-normal (SSMSN) distributions which provide alternative candidates for modeling asymmetric data in a wide variety of settings. We obtain the moments and study some characterizations of the SSMSN distributions. Instead of resorting to numerical optimization procedures, two variants of EM algorithms are developed for carrying out maximum likelihood estimation. Our algorithms are analytically simple because closed-form expressions of conditional expectations in the E-step as well as the updating estimators in the M-step can be explicitly obtained. The observed information matrix is derived for approximating the asymptotic covariance matrix of parameter estimates. A simulation study is conducted to examine the finite sample properties of ML estimators. The utility of the proposed methodology is illustrated by analyzing a real example.

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Acknowledgments

We gratefully acknowledge the chief editor, the associate editor and two anonymous referees for their insightful comments and suggestions, which led to a much improved version of this article. This research was supported by MOST 105-2118-M-005-003-MY2 awarded by the Ministry of Science and Technology of Taiwan.

Author information

Affiliations

  1. Department of Statistics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran

    Ahad Jamalizadeh

  2. Institute of Statistics, National Chung Hsing University, Taichung 402, Taiwan

    Tsung-I Lin

  3. Department of Public Health, China Medical University, Taichung 404, Taiwan

    Tsung-I Lin

Authors
  1. Ahad Jamalizadeh
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  2. Tsung-I Lin
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Corresponding author

Correspondence to Tsung-I Lin.

Appendix

Appendix

A. R function double.int()

figurea

B. Proof of Eq. (16)

Let \(\tau _1\,{\sim }\, \varGamma (\nu _1/2,\nu _1/2)\) and \(\tau _2\,{\sim }\, \varGamma (\nu _2/2,\nu _2/2)\) be two independent random variables. Also, let and \(Z_1\) and \(Z_2\) be two independent N(0, 1) random variables, which are independent of \(\tau _1\) and \(\tau _2\). If \(Y_0\,{\sim }\,\textit{STT}(0,1,\lambda ,\nu _1,\nu _2)\), then

$$\begin{aligned} Y_0\mathop {=}\limits ^\mathrm{d} \left[ (\tau _1^{-1/2} Z_1)\mid (\tau _2^{-1/2}Z_2<\lambda \tau _1^{-1/2} Z_1) \right] \mathop {=}\limits ^\mathrm{d} \left[ W_1\mid W_2< \lambda W_1 \right] , \end{aligned}$$

where \(W_1\,{=}\,\tau _1^{-1/2} Z_1\,{\sim }\, t(0,1,\nu _1)\) and \(W_2\,{=}\,\tau _2^{-1/2} Z_2\,{\sim }\, t(0,1,\nu _2)\), and they are independent. By Bayes’ theorem, the pdf of \(Y_0\) is

$$\begin{aligned} f(y_0)= & {} \frac{f_{W_1}(y_0)\Pr (W_2<\lambda W_1\mid W_1=y_0)}{\Pr (W_2<\lambda W_1)=0.5}\\= & {} 2 f_{W_1}(y_0) F_{W_2}(\lambda y_0)\\= & {} 2 t(y_0;\nu _1) T(\lambda y_0;\nu _2). \end{aligned}$$

Now, if we make the location and scale transformation \(Y\,{=}\,\xi +\sigma Y_0\). The pdf of Y is

$$\begin{aligned} f(y)=\frac{1}{\sigma }f_{Y_0}\left( \frac{y-\xi }{\sigma }\right) = \frac{2}{\sigma } t(u;\nu _1)T(\eta ;\nu _2), \end{aligned}$$

where \(u\,{=}\,(y-\xi )/\sigma \) and \(\eta \,{=}\,\lambda u\).

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Jamalizadeh, A., Lin, T. A general class of scale-shape mixtures of skew-normal distributions: properties and estimation. Comput Stat 32, 451–474 (2017). https://doi.org/10.1007/s00180-016-0691-1

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Keywords

  • Asymmetry
  • Bessel function
  • Kurtosis
  • Observed information
  • Skew distributions
  • Truncated normal