The Statistical Analysis of Monotone Incomplete Multivariate Normal Data
Open Access
- Author:
- Romer, Megan M
- Graduate Program:
- Statistics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- May 01, 2009
- Committee Members:
- Donald Richards, Dissertation Advisor/Co-Advisor
Rana Arnold, Committee Chair/Co-Chair
Vernon Michael Chinchilli, Committee Member
Diane Marie Henderson, Committee Member
Donald Richards, Committee Member - Keywords:
- monotone imcomplete data
multivariate normal - Abstract:
- We consider problems in finite-sample inference with monotone incomplete data drawn from N_d(μ, Σ), a multivariate normal population with mean μ and covariance matrix Σ. In the case of two-step, monotone incomplete data, we show that μ-hat and Σ-hat, the maximum likelihood estimators of μ and Σ, respectively, are equivariant and obtain a new derivation of a stochastic representation for μ-hat. Our new derivation allows us to identify explicitly in terms of the data the independent random variables that arise in that stochastic representation. Again, in the case of two- step, monotone incomplete data, we derive a stochastic representation for the exact distribution of a generalization of Hotelling's T^2, and therefore obtain ellipsoidal confidence regions for μ. We then derive probability inequalities for the T^2-statistic. We apply these results to construct confidence regions for linear combinations of μ, and provide a numerical example in which we analyze a data set consisting of cholesterol measurements on a group of Pennsylvania hospital patients. . In the case of three-step, monotone incomplete data, we examine the independence properties and joint distribution of subvectors of μ-hat, the maximum likelihood estimator of μ. In our examination of the joint distribution of μ-hat, we first establish that μ-hat is equivariant and then identify the distribution of μ-hat up to a certain set of conditioning variables.