Likelihood-based inference for multivariate skew scale mixtures of normal distributions

Abstract

Scale mixtures of normal distributions are often used as a challenging class for statistical analysis of symmetrical data. Recently, Ferreira et al. (Stat Methodol 8:154–171, 2011) defined the univariate skew scale mixtures of normal distributions that offer much needed flexibility by combining both skewness with heavy tails. In this paper, we develop a multivariate version of the skew scale mixtures of normal distributions, with emphasis on the multivariate skew-Student-t, skew-slash and skew-contaminated normal distributions. The main virtue of the members of this family of distributions is that they are easy to simulate from and they also supply genuine expectation/conditional maximisation either algorithms for maximum likelihood estimation. The observed information matrix is derived analytically to account for standard errors. Results obtained from real and simulated datasets are reported to illustrate the usefulness of the proposed method.

Appendices

Appendix 1: Details of the observed information matrix

Considering ${\varvec{\alpha }}=\mathrm{Vech}({\mathbf {B}})$, where ${\varvec{\varSigma }}^{1/2}={\mathbf {B}}={\mathbf {B}}({\varvec{\alpha }})$, the first and second derivatives of $\log |{\varvec{\varSigma }}|$, $A_i$ and $d_i$ are obtained. The notation used is that of Sect. 2 and for a p-dimensional vector ${\varvec{\rho }}=(\rho _1\ldots ,\rho _p)^{\top }$, we will use the notation $\dot{{\mathbf {B}}}_r=\partial {{\mathbf {B}}({\varvec{\alpha }})}/\partial {\alpha _r}$, with $r=1,2,\ldots ,p(p+1)/2$. Thus,

• ${\varvec{\varSigma }}$

  $$\begin{aligned} \frac{\partial ^2 \log {|{\varvec{\varSigma }}|}}{\partial \alpha _k\partial \alpha _s}=-2 \text {tr}({\mathbf {B}}^{-1}\dot{{\mathbf {B}}}_{s}{\mathbf {B}}^{-1}\dot{{\mathbf {B}}}_{k}), \end{aligned}$$

• $A_i$

  $$\begin{aligned} \frac{\partial A_i}{\partial {\varvec{\mu }}}= & {} -{\mathbf {B}}^{-1}{\varvec{\lambda }},\quad \frac{\partial A_i}{\partial \alpha _{k}}=-{\varvec{\lambda }}^{\top }{\mathbf {B}}^{-1}\dot{{\mathbf {B}}}_k{\mathbf {B}}^{-1}({\mathbf {y}}_i-{\varvec{\mu }}),\quad \frac{\partial A_i}{\partial {\varvec{\lambda }}}={\mathbf {B}}^{-1}({\mathbf {y}}_i-{\varvec{\mu }}), \\ \frac{\partial ^2 A_i}{\partial {\varvec{\mu }}\partial {\varvec{\mu }}^{\top }}= & {} {\mathbf {0}},\quad \frac{\partial ^2 A_i}{\partial {\varvec{\mu }}\partial \alpha _k}={\mathbf {B}}^{-1}\dot{{\mathbf {B}}}_k{\mathbf {B}}^{-1}{\varvec{\lambda }},\quad \frac{\partial ^2 A_i}{\partial {\varvec{\mu }}\partial {\varvec{\lambda }}^{\top }}=-{\mathbf {B}}^{-1},\\ \frac{\partial ^2 A_i}{\partial \alpha _k\partial \alpha _s}= & {} -{\varvec{\lambda }}^{\top }{\mathbf {B}}^{-1} [\dot{{\mathbf {B}}}_s{\mathbf {B}}^{-1}\dot{{\mathbf {B}}}_k+\dot{{\mathbf {B}}}_k{\mathbf {B}}^{-1}\dot{{\mathbf {B}}}_s]{\mathbf {B}}^{-1}({\mathbf {y}}_i-{\varvec{\mu }}),\\ \frac{\partial ^2 A_i}{\partial \alpha _k\partial {\varvec{\lambda }}}= & {} -{\mathbf {B}}^{-1}\dot{{\mathbf {B}}}_k{\mathbf {B}}^{-1}({\mathbf {y}}_i-{\varvec{\mu }}),\quad \frac{\partial ^2 A_i}{\partial {\varvec{\lambda }}\partial {\varvec{\lambda }}^{\top }}={\mathbf {0}}, \end{aligned}$$

• $d_i$

  $$\begin{aligned} \frac{\partial d_i}{\partial {\varvec{\mu }}}= & {} -2{\mathbf {B}}^{-2}({\mathbf {y}}_i-{\varvec{\mu }}),\quad \frac{\partial d_i}{\partial \alpha _k}=-({\mathbf {y}}_i-{\varvec{\mu }})^{\top }{\mathbf {B}}^{-1} [\dot{{\mathbf {B}}}_k{\mathbf {B}}^{-1}+{\mathbf {B}}^{-1}\dot{{\mathbf {B}}}_k]{\mathbf {B}}^{-1}({\mathbf {y}}_i-{\varvec{\mu }}),\\ \frac{\partial d_i}{\partial {\varvec{\lambda }}}= & {} {\mathbf {0}}, \frac{\partial ^2 d_i}{\partial {\varvec{\mu }}\partial {\varvec{\mu }}^{\top }}=2{\mathbf {B}}^{-2},\quad \frac{\partial ^2 d_i}{\partial {\varvec{\mu }}\partial \alpha _k}=2{\mathbf {B}}^{-1} [\dot{{\mathbf {B}}}_k{\mathbf {B}}^{-1}+{\mathbf {B}}^{-1}\dot{{\mathbf {B}}}_k]{\mathbf {B}}^{-1}({\mathbf {y}}_i-{\varvec{\mu }}),\\ \frac{\partial ^2 d_i}{\partial {\varvec{\mu }}\partial {\varvec{\lambda }}^{\top }}= & {} {\mathbf {0}},\quad \frac{\partial ^2 d_i}{\partial \alpha _k\partial {\varvec{\lambda }}^{\top }}={\mathbf {0}},\quad \frac{\partial ^2 d_i}{\partial {\varvec{\lambda }}\partial {\varvec{\lambda }}^{\top }}={\mathbf {0}},\\ \frac{\partial ^2 d_i}{\partial \alpha _k\partial \alpha _s}= & {} ({\mathbf {y}}_i-{\varvec{\mu }})^{\top }{\mathbf {B}}^{-1} [\dot{{\mathbf {B}}}_s{\mathbf {B}}^{-1}\dot{{\mathbf {B}}}_k{\mathbf {B}}^{-1}+\dot{{\mathbf {B}}}_k{\mathbf {B}}^{-1}\dot{{\mathbf {B}}}_s{\mathbf {B}}^{-1}+\dot{{\mathbf {B}}}_k{\mathbf {B}}^{-2}\dot{{\mathbf {B}}}_s+\dot{{\mathbf {B}}}_s{\mathbf {B}}^{-2}\dot{{\mathbf {B}}}_k\\&+{\mathbf {B}}^{-1}\dot{{\mathbf {B}}}_s{\mathbf {B}}^{-1}\dot{{\mathbf {B}}}_k+{\mathbf {B}}^{-1}\dot{{\mathbf {B}}}_k{\mathbf {B}}^{-1}\dot{{\mathbf {B}}}_s]{\mathbf {B}}^{-1}({\mathbf {y}}_i-{\varvec{\mu }}). \end{aligned}$$

Appendix 2: Joint, conditional and marginal distributions of $({\mathbf {Y}},U,T)$

Note first that from (7), it follows that

$$\begin{aligned} \begin{array}{rcl} {\mathbf {Y}}|T=t, U= u&{}\sim &{} N_p({\varvec{\mu }}+ \frac{t}{u^{1/2}}{\varvec{\varSigma }}^{1/2}{\varvec{\delta }}_u,\frac{1}{u}{\varvec{\varSigma }}^{1/2}({\mathbf {I}}_p+{\varvec{\lambda }}_u{{\varvec{\lambda }}_u}^\top )^{-1}{\varvec{\varSigma }}^{1/2}),\\ U&{} \sim &{}H({\varvec{\tau }}),\quad T\sim TN(0,1;(0,+\infty )),\end{array} \end{aligned}$$

(23)

with U and T independent, ${\varvec{\delta }}_u=\frac{{\varvec{\lambda }}}{\sqrt{u+{\varvec{\lambda }}^{\top }{\varvec{\lambda }}}}$, ${\varvec{\lambda }}_u={\varvec{\lambda }}/\sqrt{u}$.

Using some results given in Lachos et al. (2010), it follows that the joint distribution of $({\mathbf {Y}},U,T)$ is given by

$$\begin{aligned} f({\mathbf {y}},u,t)= & {} 2\phi _p\left( {\mathbf {y}}|{\varvec{\mu }}+{\mathbf {A}}t,{\varvec{\varSigma }}_a\right) \phi _1(t|0,1)h(u;{\varvec{\tau }})\\= & {} 2\phi _p({\mathbf {y}}|{\varvec{\mu }},{\varvec{\varSigma }}_a+{\mathbf {A}}{\mathbf {A}}^\top )\phi _1(t|\varLambda {\mathbf {A}}^\top {\varvec{\varSigma }}_a^{-1}({\mathbf {y}}-{\varvec{\mu }}),\varLambda )h(u;{\varvec{\tau }}),\\&{\mathbf {y}}\in {\mathbb {R}}^p,\;t >0,\;u>0, \end{aligned}$$

where ${\mathbf {A}}=\frac{{\varvec{\varSigma }}^{1/2}{\varvec{\delta }}_u}{u^{1/2}}$, ${\varvec{\varSigma }}_a=\frac{1}{u}{\varvec{\varSigma }}^{1/2}({\mathbf {I}}_p+{\varvec{\lambda }}_u{\varvec{\lambda }}_u^\top )^{-1}{\varvec{\varSigma }}^{1/2}$ and $\varLambda =(1+{\mathbf {A}}^\top {\varvec{\varSigma }}_a^{-1}{\mathbf {A}})^{-1}$. Using the results given in Harville (1997), and after some algebraic manipulations, it follows that ${\varvec{\varSigma }}_a+{\mathbf {A}}{\mathbf {A}}^\top =\frac{1}{u}{\varvec{\varSigma }}$, $\varLambda =\frac{u}{u+{\varvec{\lambda }}^\top {\varvec{\lambda }}}$ and $\varLambda {\mathbf {A}}^\top {\varvec{\varSigma }}_a^{-1}=\varLambda ^{1/2}{\varvec{\lambda }}^\top {\varvec{\varSigma }}^{-1/2}$.

Thus, the marginal distribution of ${\mathbf {Y}}\sim \mathrm{SSMN}_p({\varvec{\mu }},{\varvec{\varSigma }},{\varvec{\lambda }};H)$ is given by

$$\begin{aligned} f({\mathbf {y}})= & {} 2\int _0^{+\infty }\int _0^{+\infty }\phi _p \left( {\mathbf {y}}|{\varvec{\mu }},\frac{{\varvec{\varSigma }}}{u}\right) \phi _1 (t|\varLambda {\mathbf {A}}^\top {\varvec{\varSigma }}_a^{-1}({\mathbf {y}}-{\varvec{\mu }}),\varLambda )h(u;{\varvec{\tau }})\mathrm{d}t\mathrm{d}u\\= & {} 2\int _0^{+\infty }\phi _p\left( {\mathbf {y}}|{\varvec{\mu }},\frac{{\varvec{\varSigma }}}{u}\right) h(u;{\varvec{\tau }})\int _0^{+\infty }\phi _1(t|\varLambda ^{1/2}{\varvec{\lambda }}^\top {\varvec{\varSigma }}^{-1/2}({\mathbf {y}}-{\varvec{\mu }}),\varLambda )\mathrm{d}t\mathrm{d}u\\= & {} 2\int _0^{+\infty }\int _0^{+\infty }\phi _p\left( {\mathbf {y}}|{\varvec{\mu }},\frac{{\varvec{\varSigma }}}{u}\right) h(u;{\varvec{\tau }})\phi _1(t|{\varvec{\lambda }}^\top {\varvec{\varSigma }}^{-1/2}({\mathbf {y}}-{\varvec{\mu }}),1)\mathrm{d}t\mathrm{d}u\\= & {} 2\int _0^{+\infty }\phi _p\left( {\mathbf {y}}|{\varvec{\mu }},\frac{{\varvec{\varSigma }}}{u}\right) h(u;{\varvec{\tau }})\mathrm{d}u\varPhi _1({\varvec{\lambda }}^\top {\varvec{\varSigma }}^{-1/2}({\mathbf {y}}-{\varvec{\mu }})). \end{aligned}$$

Then the joint distribution of $({\mathbf {Y}},T)$ is given by

$$\begin{aligned} f({\mathbf {y}},t)=2f_0({\mathbf {y}}|{\varvec{\mu }},{\varvec{\varSigma }})\phi _1 (t|{\varvec{\lambda }}^\top {\varvec{\varSigma }}^{-1/2}({\mathbf {y}}-{\varvec{\mu }}),1),\quad {\mathbf {y}}\in {\mathbb {R}}^p,\;t>0, \end{aligned}$$

(24)

and

$$\begin{aligned} f(t|{\mathbf {y}})=\frac{\phi _1(t|{\varvec{\lambda }}^\top {\varvec{\varSigma }}^{-1/2} ({\mathbf {y}}-{\varvec{\mu }}),1)}{\varPhi _1({\varvec{\lambda }}^\top {\varvec{\varSigma }}^{-1/2}({\mathbf {y}}-{\varvec{\mu }}))}, \end{aligned}$$

(25)

so that, $T|{\mathbf {Y}}={\mathbf {y}}\sim TN({\varvec{\lambda }}^\top {\varvec{\varSigma }}^{-1/2}({\mathbf {y}}-{\varvec{\mu }}),1;(0,+\infty ))$.