An Extension of Sphericity Test to the Multi-Sample Problem with Monotone Incomplete Data

Abstract In this study, we extend the sphericity test for variance–covariance matrices with monotone incomplete data from the one-sample problem to the multi-sample problem. We derive the likelihood ratio (LR) and asymptotic expansions concerning the likelihood ratio test (LRT) statistic and its modified version when the null hypothesis holds. In addition, we propose approximate upper percentiles of the test statistics under the null hypothesis. To improve the accuracy of inference, we also introduce an unbiased estimator for a common variance parameter and assess its performance using the mean squared error (MSE). Similar theoretical results are also derived under complete data for comparison. Monte Carlo simulations are conducted to evaluate the empirical type I error rates and estimation accuracy under both monotone incomplete and complete data. Finally, we present numerical examples in the multi-sample setting, including the two-sample case, to illustrate the practical applicability of our approach.