An Asymptotic Two-Sided Test in a Family of Multivariate Distribution

In the present paper, a two-sided test in a family of multivariate distribution according to the Mahalanobis distance with mean vector and positive definite matrix is considered. First, a family of multivariate distribution is introduced, then using the likelihood ratio method a test statistic is computed. The distribution of the test statistic is proposed for different sample sizes and fixed dimension. We study the distribution approximation computed using the likelihood ratio test and an efficient algorithm to compute the density functions can be derived according to Witkovsk ́y, J. Stat. Plan. Inference. 94 (2001), 1–13. Also, a simulation study is presented on the sample sizes and powers to compare the performance of tests and show that the proposed distribution approximation is better than the classical distribution approximation.