Rank-Based Regression for Nonlinear and Missing Response Models

Abstract

This dissertation is mainly concerned with the rank-based estimation of model parameters in complex regression models: a general nonlinear regression model and a semi-parametric regression model with missing responses. For the estimation of nonlinear regression parameters, we consider weighted generalized-signed-rank estimators. The generalization allows us to study rank estimators as well as popular estimators such as the least squares and least absolute deviations estimators. However, the generalization by itself does not give bounded influence estimators. Added weights provide estimators with bounded influence function. We establish conditions needed for the consistency and asymptotic normality of the proposed estimator and discuss how weight functions can be chosen to achieve bounded influence function of the estimator. Real life examples and Monte Carlo simulation experiments demonstrate that the proposed estimator is robust, efficient, and useful in detecting outliers in nonlinear regression. For the estimation of the linear regression parameter of a semi-parametric model with missing response, we propose imputed rank estimators under simple imputation and imputation by inverse probability. It is shown that these rank estimators have favorable asymptotic properties. Moreover, it is demonstrated that the rank estimators perform better than the classical least squares estimator under heavy tailed error distributions and cases containing contamination while they are generally comparable to the least squares estimator under normal error. Moreover, rank estimators with inverse probability imputation are superior than their least squares counterpart when the proportion of missing data is large. This makes rank estimation extremely appealing for situations where we encounter high rates of missing information.