We have proposed a mixture of skew-normal distribution as an alternative to change of the original scale including the Box-Cox transformation when there exists a cer- tain amount of skewness in the data. The number of components is estimated by the reversible jump MCMC algorithm which reports the posterior probability of the number of components. When there are negative values, the Box-Cox transformation needs another parameter to shift the data to make them all positive. The amount of shift has been reported to affect the entire estimation procedure and the infer- ence of the number of components. In application to some real data and simulated data, the mixture of skew-normal distribution have drawn same conclusion as that obtained from transformation without changing the scale, which produces much eas- ier interpretation of the results. The following subsections consider the prior on the skewness parameter in the skew-normal distribution and the integration method for future research.
2.5.1 Integration
When the conjugacy is acquired by the prior structure, then the marginal likelihood
p(Y |k) = p(θk|k)p(Y |k, θk) p(θk|k, Y )
can be calculated and the posterior probabilities p(k|Y ) can be also obtained since it is proportional to p(k)p(Y |k). In such a case, Bayesian inference about k and θk
given k could be conducted separately.
There is difficulty in attaining the conjugacy for the skew-normal distribution. As described in this article, it could not be problematic in the univariate case. When the extension to the multivariate case is considered, the dimension reduction of the unknowns by integrating out a part of the set of parameters could help implement
the reversible jump MCMC algorithm. In the mixture of multivariate skew-normal distributions, the scale and skewness parameters can be integrated out to reduce much of the dimensions to ease the dimension varying MCMC.
2.5.2 Moment matching condition
There are a few other schemes developed to incorporate the dimension varying sit- uation, for example, Stephens (2000) and Carlin and Chib (1995) among which we are solely dependent on the algorithm by Green (1995) It could be challenging to use other algorithms to move across the models with different number of components in the mixture model. Also, within the reversible jump algorithm, we can take different approach for the update of the skewness parameter in the moment matching condi- tion or it is possible to consider more efficient proposal choice, for example, Brooks et al. (Brooks et al., 2003) .
2.5.3 Prior on the skewness parameter
The normal distribution with mean zero has been taken as the prior for the skewness parameter λ providing that there exists a certain amount of skewness in the data. As we put another hierarchical level of prior for the scale parameter, a prior will be taken for the hyperparameter b3.
Liseo and Loperfido (2005) showed that the Jeffreys’ prior of the skewness pa- rameter in the skew-normal distribution is proper, which could be considered for the prior distribution for λ.
CHAPTER III
THE MULTIVARIATE SKEW-NORMAL MIXTURE
3.1 Introduction
Suppose that X = (X1, ...Xd)> is a d-dimensional multivariate random variable. A
multivariate extension of the Box-Cox transformation is generalized in the k compo- nent of mixture context by Schork and Schork (1988) as a multivariate generalization of the technique of MacLean et al. (1976). Such transformation is applied to each variable as ym = (xλm m − 1)/λm+ λm if λ 6= 0 log(xm) if λ = 0 ,
where ym is the transformed and xm the untransformed variable (m = 1, ..., d). As
noted in the previous chapter, the variables should be translated to make the data all positive before analysis in the presence of negative values. To allow such location parameter to vary in a numerical routine increases the instability of convergence while adding an unimportant parameter to be estimated (Schork and Schork, 1988). Thus, Schork and Schork (1988) suggested that the value be established with discretion by the user before analysis. We observed in the previous chapter that the amount of such translation has an effect on the inference of the mixture model, especially when the number of components is unknown.
While MacLean et al. (1976) and Schork and Schork (1988) modeled the skew- ness in the mixture context, McLachlan et al. (2002) and Peel and McLachlan (2000) considered the multivariate t-distribution to incorporate heavy tails along with mul-
tiple modes.
Azzalini and Capitanio (1999) pointed out the possible problems when the trans- formation of the variables is taken into account to achieve multivariate normality, these are
(a) the transformations are usually on each component separately, and achievement of joint normality is only hoped for;
(b) the transformed variables are more difficult to interpret, especially when each variable is transformed by using a different function; and
(c) when multivariate homoscedasticity is required, this often requires a transfor- mation which is different from the transformation for normality.
As an alternative to the Box-Cox transformation, we suggest a mixture model approach using the multivariate skew-normal distribution.
Azzalini and Capitanio (1999) demonstrate that the multivariate skew normal distribution has reasonable flexibility in real data fitting, while it maintains some con- venient formal properties of the normal density. Discriminant analysis was presented as an application of multivariate skew normal distribution in their paper.
In this chapter, we present a methodology for model-based clustering using the mixture of multivariate skew-normal distributions. In section 3.2, a Bayesian estima- tion procedure for the multivariate skew-normal distribution is considered, and it is extended to the mixture model in section 3.3 followed by some applications in section 3.4.