Large Sample Properties Of Nearest Neighbor Estimates For LNQD Samples

The concept of LNQD is first introduced by Newman[1] in1984, and it is a class of dependent random variables including independent situation. LNQD random variables is widely applied not only in the multivariate statistical analysis, penetration theory and reliability theory, but also in communications, weather and many engineering fields and risk analysis. It aroused wide atten-tion and research interests from scholars of probability theory and mathematical statistics at home and abroad. The LNQD sequence is a class of wide dependent sequences composing of indepen-dent sequences and NA sequences. With the proof of the exponential inequalities and moment inequalities of LNQD sample, people have obtained a lot signigicant results. This thesis devote to the consistency and asymptotic of the nearest neighbor density estimator and regression function estimator under LNQD sample and its convergence rate.In chapter3, we research the weak, strong consistency and uniformly strong consistency of nearest neighbor density estimator fn(x) for LNQD sample. Meanwhile we research the uni-formly strong consistency of the hazard rate estimated rn(x). We prove that the rate of strong consistency of neighbor density estimate fn(x) is the n-1/4, and the uniform strong consistency is n-1/6, the results we get is the same as the results in literature[2]. In literature[2], it supposed the NA samples is in the special case of t=2.In chapter4, under the case of the smooth LNQD sample sequence, we give the strong con-sistency and convergence rate of the LNQD non-parametric regression function nearest neighbor estimate. For general weight of nearest neighbor, the strong convergent rate is basically n-1/2.In chapter5,we research the asymptotic normality of the LNQD nearest neighbor density esti-mator, we obtain the convergence rate and the way to select the kn for a large sample’s confidence interval.In chapter6, we discuss the asymptotic normality of LNQD nonparametric regression func-tion nearest neighbor estimator. Because of the randomness of weight function, we can not get the convergence rate, and it is worth to continue to explore. But we get the asymptotic normality and the convergence rate of Priestley-Chao weight function.Finally we select the ARMA (1,1) time series model. By numerical simulation we research the influence of the sample size of the dependent sequence, the dependent coefficient and the dependent complex structure on neighbors’s results. We apply actual financial data to the empir-ical analysis and the results show that the nearest neighbor has good practicability for the LNQD sequence and NA sequences of ARMA (1,1) structure.