This research concerns estimation in high dimensional models. In the first part the use of high dimensional models for incomplete data is addressed. Suppose the pattern of incompleteness is related to an high dimensional vector of observations. Then the dimensionality of the vector precludes straightforward maximum likelihood estimation. The research addresses two modifications of maximum likelihood. In the first modification, the dimension of the vector is reduced by a balancing score and in the second modification, a weighted likelihood is used. The second part of this research focuses on maximum likelihood estimation for the transformation model, a high dimensional generalization of the linear regression model. This model stipulates that an unknown transformation of the response follows a linear regression. In special cases, such as the proportional hazards model, the estimators found by maximum likelihood are well understood. Yet in general, estimators of parameters in this model, although extremely popular, are rather difficult to analyze. This research investigates the use of new techniques, such as empirical processes and empirical likelihood in order to understand the estimators.
Models in which there are many unknowns or unknown functions (called parameters here) appear throughout the sciences. Often these models are formulated to address inadequacies of the classical linear regression model. For example, social scientists use high dimensional models in event history analysis of the causes of variability in the timing of life events such as premarital births, initiation of drug abuse, timing of retirement and duration of poverty spells. High dimensional models attempt to allow the data to speak for itself and to minimize the addition of spurious information caused by imposing a low-dimensional model on the data. This research advances the understanding of these high dimensional models. Often high dimensional models are applied without any theoretical understanding of when they may or may not produce quality estimators. In particular, this research investigates the bias and variability of of estimators found by the use of a common estimation method, maximum likelihood estimation. This is important as understanding the causes of variability in the timing of life events is crucial to designing appropriate social policies and intervention/prevention programs.
