Sphericity Test on Variance-Covariance Matrix with Monotone Missing Data

• Published in: Journal of statistical theory and practice Vol. 19; no. 2
• Main Authors: Sato, Tetsuya, Yagi, Ayaka, Seo, Takashi
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2025
• Subjects: Mathematics and Statistics; Original Article; Probability Theory and Stochastic Processes; Statistical Theory and Methods; Statistics; Maximum likelihood estimator; Distribution function; Likelihood ratio test statistic; Modified likelihood ratio test statistic; 62D10; 62E20
• ISSN: 1559-8608; 1559-8616
• DOI: 10.1007/s42519-025-00431-9

This study considers the sphericity test, a specific test of variance-covariance matrix under monotone missing data for a one-sample problem. We provide the likelihood ratio (LR) and derive an asymptotic expansion of the likelihood ratio test (LRT) statistic and modified LRT statistic for the null distribution. We also derive the upper percentiles of the LRT statistic and modified LRT statistic when the null hypothesis holds, and provide approximate upper percentiles. Furthermore, we prove that the LR under monotone missing data is affine invariant under the null hypothesis. For complete data, we provide an asymptotic expansion of the LRT statistic and modified LRT statistic for the null distribution. Furthermore, we numerically evaluate the actual type I error rates for the approximate upper percentiles using Monte Carlo simulation and provide examples of the LRT statistic and modified LRT statistic and approximate upper percentiles under monotone missing data.