By applying techniques from constrained optimization and information theory, this research enhances statistical inference, makes new developments and gains new insights into existing methodology. PI concentrates on: (1) Considering the covariance selection problem of multivariate normal distributions, PI develops its Fenchel dual formulation and shows that the dual solutions are themselves elements of the solution covariance matrix. This insightful observation yields mechanics to calculate direct estimates under decomposable models. This understanding coupled with tools from convex duality help PI to generalize the covariance selection to multivariate dependence, which includes MTP2 and trends popularly used in longitudinal studies using covariance pattern matrices. The iterative proportional scaling algorithm, used for estimation in covariance selection problems, may not lead to the correct solution under such dependence. Addressing this situation, PI presents a new algorithm for dependence models and shows that it converges correctly using tools from Fenchel duality. Results concerning the speed of convergence of the new algorithm are addressed. The methodology will be applied on a real data set involving decreasing CD4+ cell numbers from an AIDS study. (2) PI develops constrained conditional log-linear (or multinomial response) models for panel data. Information theoretic tools are used to show these models are I-projections on certain moment equalities. This observations leads to construction of prediction functions which can incorporate restrictions on parameters by considering moment inequalities. Constrained versions of conventional Markov models, independence models, distance models are developed by using previous responses and present and past covariates when predicting the current response. This approach has advantages over the marginal modelling approach used in longitudinal studies. Statistical properties of these prediction functions are investigated from a population and as well as a sampling perspective. Problems such as existence and uniqueness of optimal prediction functions are addressed. Basic properties of measures of prediction quality are examined using information theory tools. Estimation, consistency and asymptotic distributions of the estimators are studied. The results will be applied on data from National Longitudinal Study of Youth. (3) A parametric model is presented for the analysis of square contingency tables where there is a one-to-one correspondence betweeen the categories of the row, column variables. Parametric scores are assigned to the rows/columns to reflect the ordinality of the categories. Efficient estimation under order restrictions of these scores is a difficult problem as they are functions of each other. Instead of using unrestricted estimates, PI proposes functions which can be used when pooling the adjacent violators (PAV). For these functions, PI proposes results that derive their asymptotic normality, consistency, and show that they have same order relations as the score parameters. For computational purposes, several algorithms are proposed, including PAV, complete search and an active-set method. Goodness-of-fit testing of the model has a chi-bar-square distribution. Parametric bootstrap procedure is proposed for yielding reliable p-value.
New constrained statistical inferential methods are developed using information theoretic techniques by exploring connections between covariance selection and multivariate dependence; constrained conditional log-linear models for panel data are presented and their statistical properties are investigated; and in square tables with ordered parameters, statistical inference under order restrictions are proposed. The results are directly useful in analyzing longitudinal data arising from medicine, reliability, sociology, econometrics, etc. The proposed activities will involve training graduate students for future researchers in statistics and providing selected undergraduate students with research experience.
