Modified Nonparametric Kernel Estimates of a Regression Function and their Consistencies with Rates

摘要

The theory of regression is concerned with the prediction of the value of a variable, called the response or dependent variable, at a given value of another (correlated) variable, called the predictor or independent variable. Prediction is needed in several practical situations. For example, an agriculturist wants to know the yield of wheat at an amount of a specified fertilizer, a meteorologist wants to forecast weather several hours ahead on the basis of previous atmospheric measurements and a physician is interested in determining the weight of a patient in terms of the number of weeks he or she has been on a diet. In this document, two sets of modified kernel estimates of a regression function are proposed: one when a bound on the regression function is known and the other when nothing of this sort is at hand. Explicit bounds on the mean square errors of the estimators are obtained. Pointwise as well as uniform consistency in mean square and consistency in probability of the estimators are proved. Speed of convergence in each case is investigated.