Efficient Rank Regression with Wavelet Estimated Scores

Abstract

Wavelets have been widely used lately in many areas such as physics, astronomy, biological sciences and recently to statistics. The main goal of this dissertation is to provide a new contribution to an important problem in statistics and particularly nonparametric statistics, namely estimating the optimal score function from the data with unknown underlying distribution. This problem naturally arises in nonparametric linear regression models and could be important in order to have a better insight on more important and actual problems in longitudinal and repeated measures analysis through mixed models. Our approach in estimating the score function is to use suitable compactly supported wavelets like the Daubechies, Symlets or Coi ets family of wavelets. The smoothness and time-frequency properties of these wavelets allow us to nd an asymptotically e - cient estimator of the slope parameter of the linear model. Consequently, we are also able to provide a consistent estimator of the asymptotic variance of the regression parameter. For related mixed models, asymptotic relative efficiency is also discussed.