A Maximum Likelihood Mixture Approach for Multivariate Hypothesis Testing in case of Incomplete Data

Multivariate hypothesis testing becomes more and more necessary when data is in the process of changing from scalar and univariate format to multivariate format, especially financial and biological data is often constituted of n-dimension vectors. Likelihood ratio test is the best method that applies the test on mean of multivariate sample with known or unknown covariance matrix but it is impossible to use likelihood ratio test in case of incomplete data when the data incompletion gets popular because of many reasons in reality. Therefore, this research proposes a new approach that gives an ability to apply likelihood ratio test into incomplete data. Instead of replacing missing values in incomplete sample by estimated values, this approach classifies incomplete sample into groups and each group is represented by a potential or partial distribution. All partial distributions are unified into a mixture model which is optimized via expectation maximization (EM) algorithm. Finally, likelihood ratio test is performed on mixture model instead of incomplete sample. This research provides a thorough description of proposed approach and mathematical proof that is necessary to such approach. The comparison of mixture model approach and filling missing values approach is also discussed in this research.