Remarks for the singular multivariate skew-normal distribution and its quadratic forms

Under the singular multivariate skew normal (SMSN) setting, we showed that, in this paper, the necessary and sufficient conditions for independence of two sub-vectors given in Young et al. (2017) are equivalent to the results in Wang et al. (2009). In addition, the distribution of quadratic form of SMSN random vector is derived, with this new definition of the noncentral skew chi-square distribution. Several examples are given to illustrate our main results.