Goodness-of-fit tests for multivariate distributions

We propose three new statistics, Z(p), C-p, and R-p for testing a p-variate (p >= 2) normal distribution and compare them with the prominent test statistics. We show that C-p is overall most powerful and is effective against skew, long-tailed as well as short-tailed symmetric alternatives. We show that Z(p) and R-p are most powerful against skew and long-tailed alternatives, respectively. The Z(p) and R-p statistics can also be used for testing an assumed p-variate nonnormal distribution.