Estimation Theory And Variable Selection Methods Under Relative Error Criterion

Relative error|(Y-Y)/Y rather than the error Y-Y itself is of the main interest in many practical applications. Criteria based on minimizing the sum of absolute rela-tive errors (MRE), and the sum of squared relative errors (RLS) were proposed in the different areas. Motivated by Chen etc.’s recent work (JASA105,1104-1112) on the least absolute relative error (LARE) estimation minimizing two types of relative errors|(Y-Y)|Y|+|(Y-Y)|Y|for the accelerated failure time (AFT) model, in this paper we es-tablish the connection between relative error estimators and the M-estimation in the linear model. This connection allows us to deduce the asymptotic properties of many relative error estimators (e.g. LARE) by the well-developed M-estimation theories. On the other hand, the asymptotic properties of some important estimators (e.g. MRE and RLS) cannot be established directly. So in this thesis, we first propose a general relative error criteri-on (GREC) for estimating the unknown parameter in the AFT model. Then we develop the approaches to deal with the asymptotic normalities for M-estimators with non-convex and non-differentiable loss functions in the linear model. The simulation studies are con-ducted to evaluate the performance of the proposed estimates for the different scenarios, and the simulation results suggest that our method is feasible. Illustration with a real data example is also provided.Then we study the AFT model with a diverging number of parameters (which goes to infinity as the sample size goes to infinity) and obtain the-(?)n/pn-consistencies and asymptotical normalities of some GREC estimators. A new consistent estimator of the covariance matrix is also given. Results of random simulations suggest that our estimates of the regression coefficients, and the covariance matrix is practical; the RLS estimator is sensitive to the distribution of the random error, and the least product absolute relative errors (LPARE) estimator is more stable comparing with the RLS estimator.On the base of these works, we consider the variable selection problem via the pe-nalized general relative error criterion (PGREC) estimators, and prove their "Oracle" properties under certain regular conditions. The estimators select the proper model with probability tending to1, and the estimators of nonzero coefficients have the same asymp-totic distribution that they would if the zero coefficients were known in advance. At last, we propose the estimators of the asymptotic covariance matrix and the bias of the PGREC estimator, and prove their consistencies in the sense of the matrix norm.