Maximum likelihood estimation for scale-shape mixtures of flexible generalized skew normal distributions via selection representation

Abstract

A scale-shape mixtures of flexible generalized skew normal (SSMFGSN) distributions is proposed as a novel device for modeling asymmetric data. Computationally feasible EM-type algorithms derived from the selection mechanism are presented to compute maximum likelihood (ML) estimates of SSMFGSN distributions. Some characterizations and probabilistic properties of the SSMFGSN distributions are also studied. Monte Carlo simulations show that the proposed estimating procedures can provide desirable asymptotic properties of the ML estimates and demand less computational burden in comparison with other existing algorithms based on convolution representations. The usefulness of the proposed methodology is illustrated by analyzing a real dataset.

Appendix A: Proof of Eq. (27)

Without loos of generality, we assume that $\xi =0$ and $\sigma =1$. Let $Y\overset{d}{=}(\tau ^{-1/2})Z_1\vert (\tau ^{-1/2}Z_2<\lambda _1(\tau ^{-1/2}Z_1)+\cdots +\lambda _m(\tau ^{-1/2}Z_1)^{2m-1})$, where $Z_1$ and $Z_2$ are two independent N(0, 1) variables and $\tau \sim \varGamma (\nu /2,\nu /2)$. Clearly, $(X_1,X_2)^\top \overset{d}{=}\tau ^{-1/2}(Z_1,Z_2)^\top \sim t_2({\mathbf {0}},I_2,\nu )$, where following a bivariate t distribution $I_2$ is an identity matrix of order 2. Therefore, we have $X_1\sim t(0,1,\nu )$, $X_2\sim t(0,1,\nu )$ and $\sqrt{\frac{\nu +1}{\nu +z^2}}X_2\vert (X_1=z)\sim t(0,1,\nu +1)$. By Bayes’ theorem, the pdf of $Y\overset{d}{=}X_1\vert (X_2<\lambda _1X_1+\cdots +\lambda _mX_1^{2m-1})$ is

$$\begin{aligned} f_Y(y)= & {} \frac{f_{X_1}(y)Pr(X_2<\lambda _1X_1+\cdots +\lambda _mX_1^{2m-1}\vert X_1=y)}{\Pr (X_2<\lambda _1X_1+\cdots +\lambda _mX_1^{2m-1})}\\= & {} 2f_{X_1}(y)Pr\bigg (\sqrt{\frac{\nu +1}{\nu +y^2}}X_2 <\sqrt{\frac{\nu +1}{\nu +y^2}}(\lambda _1y+\cdots +\lambda _my^{2m-1})\vert X_1=y\bigg )\\= & {} 2t(y;\nu )T\bigg (\sqrt{\frac{\nu +1}{\nu +y^2}}(\lambda _1y+\cdots +\lambda _my^{2m-1});\nu +1\bigg ). \end{aligned}$$