| In statistics, frequently, one wants to know how and to what extent a certain response variable is related to a set of explanatory variables. In most cases, the relationships are not known and, furthermore, are complicated. These models are statistical in nature. The purpose of my dissertation is to study the fixed design regression model yni = g(xni) + &egr;ni, where &egr;ni is a negative associated random field and where g is a nonparametric function. The function g is estimated by a nonparametric estimator gn(x). Our results establish the asymptotic normality of gn(x), under very general conditions. The method used here is to decompose the summation of weighted negatively associated random variables into big blocks and small blocks. The asymptotic normality of the estimate is then obtained by the Lindeberg central limit theorem.; The results are illustrated by many examples. The well known Priestley and Chao's estimators serve as examples of our estimators. The bias, E(g n(x)-g(x)), is then studied in order to establish the asymptotic normality for gn(x)-g(x). The convergence rate of the Priestley and Chao's kernel weight estimate is obtained. Here we are interested in how fast |g n(x)-g(x)| converges to zero. Under certain assumptions, we are able to establish that the convergence rate is n-1/3(log n) 1/3 in the one dimensional case, which is the optimal rate. Analogues of results for the one dimensional case are then obtained for high dimensional cases. |
