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

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.

Appendix

1.1 A. R function double.int()

1.2 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 $\eta \,{=}\,\lambda u$.