Generalized Signed-Rank Estimation for Certain Complex Regression Models

Abstract

This dissertation is mainly concerned with the Generalized signed-rank estimation of model parameters in complex regression models, specially nonlinear models with multidimensional indices and two-phase linear models. First we consider a nonlinear regression model when the index variable is multidimensional. Such models are useful in signal processing, texture modeling, and spatio-temporal data analysis. A generalized form of the signed-rank estimator of the nonlinear regression coefficient is proposed. This general form of the signed-rank estimator includes Lp estimators and hybrid variants. Sufficient conditions for strong consistency and asymptotically normality of the estimator are given. It is shown that the rate of convergence to normality can be different from the square root of n. The sufficient conditions are weak in the sense that they are satisfied by harmonic type functions for which results in the current literature are not applicable. A simulation study shows that the certain generalized signed-rank estimators (e.g.. signed-rank) perform better in recovering signals than others (e.g.. least squares) when the error distribution is contaminated or is heavy-tailed. For two-phase regression models, we consider two-phase random design linear models with arbitrary error densities and where the regression function has a fixed jump at the true change-point. We establish the consistency and the limiting distributions of signed-rank estimators of the model parameters. The left end point of the minimizing interval with respect to the change-point, herein called the signed-rank estimator hat{r}_n of the change-point parameter r, is shown to be n-consistent and the underlying process, of the standardized change-point parameter, is shown to converge weakly to a compound Poisson process. This process obtains maximum over a bounded interval and n(hat{r}_n-r) converges weakly to the left end point of this interval.