Donald Richards

We consider problems in statistical inference with two-step, monotone incomplete data drawn from a multivariate normal population. We derive stochastic representations for the distributions of the maximum likelihood estimators of the population mean vector and covariance matrix and obtain results for inference on the mean vector and covariance matrix, including: lower bounds on the level of confidence associated with ellipsoidal confidence regions for the mean, confidence regions for linear combinations of the components of the mean, and unbiasedness results for several testing problems on the mean vector and covariance matrix. In testing the normality of monotone incomplete data, we construct Mardia-type statistics for testing kurtosis, derive their asymptotic distributions, and provide an application to a well-known cholesterol data set featured in the Minitab Handbook. With regard to shrinkage estimation for the mean, we extend to the case of two-step monotone incomplete samples some classical results on the reduced risk of estimators of James-Stein type, and we comment on difficulties arising in the case of higher-step monotone incompleteness. These results were obtained in collaborations with Wan-Ying Chang (NSF), Megan Romer (Penn State University), and Tomoya Yamada (Sapporo Gakuin University).