Grant: Discovery Grants Program - Individual

Different types of data such as count, binary, censored, and longitudinal often arise in public health, biology, epidemiology, and social sciences. These data may be low dimensional (when the number of observations n is greater than the number of predictors p) or high dimensional (n<<p). In data analysis, if we know the sparsity of predictors, then these coefficients become restricted by some constraints in the model. We may apply the pretest procedure to see whether the linear combinations of coefficients of the inactive predictors are zero. In this situation we may form either a full model which includes all the variables or a reduced model which restricts some variables based on the prior experience or auxiliary information. Shrinkage estimators combine the full and reduced model estimators in an optimal way in order to achieve an improved estimator for the remaining significant predictors. I developed the idea of shrinkage estimation methods for the case of semi-parametric, censored regression, and generalized linear models (GLMs). I am extending and developing shrinkage methods that have proven to be efficient in the context of the GLMs when p>>n, partial linear models, and generalized linear mixed models. Penalty methods, such as LASSO, Adaptive LASSO, SCAD, and GLMMLasso alternative to shrinkage for variable selection and parameter estimation simultaneously have emerged in the past decades. These methods heavily depend on tuning parameters. The shrinkage method, on the other hand does not depend on the tuning parameters, and it may be an alternative to penalty methods. The asymptotic properties of the shrinkage and pretest estimators including the derivation of asymptotic biases and risks will be established. Finally I will compare the relative performance of the penalty estimator the pretest and shrinkage estimators in terms of simulated relative efficiency. It is expected that the proposed research results will provide an efficient method to researchers and practitioners for inference after model selection and in assessing the predictive ability and interpretability of the model.