Inference for High-dimensional Left-censored Linear Model and High-dimensional Precision Matrix

In the first two chapters, we consider inference for high-dimensional left-censored linear models. Left-censored data arises from measurement limits in scientific devices and social science data. We consider the problem of constructing confidence intervals for the parameters in left-censored linear models. In Chapter 1, we present smoothed estimating equations (SEE) and smoothed robust estimating equations(SREE) frameworks that are adaptive to censoring level and are more robust to misspecification of the error distribution. In Chapter 2, we study inference problem for parameters in high-dimensional left-censored quantile regression model. We modify the quantile loss to accommodate the left-censored nature of the problem, by extending the idea of redistribution of mass. Furthermore, applying the de-biasing technique to the initial estimator leads to an improved estimator suitable for high-dimensional inference under left-censored quantile regression setting. For both problems, asymptotic properties have been investigated.In Chapter 3, we devise a projection pursuit testing procedure for generalized hypotheses on high-dimensional precision matrix. We illustrate the procedure under specific examples of hypotheses: testing for row sparsity, minimum signal strength, bandedness and generalized bandedness. We demonstrate the performance of the testing procedure through extensive numerical experiments, and present the findings for two real datasets