Some Limit Theorems Of Empirical Processes And Empirical Likelihood

Random phenomena exists in almost every branch of the fields of science and engineering as well as permeates throughout all aspects of an ordinary person's modern life. Probability theory is a science of quantitaticely studying regularity of random phenomena, probability is a way of thinking about the world.The probability limit theory is one of the main branches of probability the-ory. The famous probability scholars Kolmogorov and Gnedenko said:" Only probability limit theory can reveal the epistemological value of probability the-ory. Without it, you couldn't understand the real meaning of the fundamental conceptions in probability." The probability limit theory is also the important foundation of large sample theory in statistics. When estimating parameters by statistics based on sample, people are all interested in whether the estimator is consistent to the true value of the parameter as long as the samples size tends to infinity. It is so-called the consistency in the large sample theory. Furthermore, people consider what the rate of the convergence is and what the limit distri-bution is? The probability limit theory is needed to solve these large sample problem in statistics.This thesis is mainly based on the empirical distribution function. First, we study a class of limit theory of the uniform empirical processes. Second, the empirical likelihood which is associated closely with the empirical distribution function is also studied. We use blockwise empirical Euclidean likelihood method to the parameter estimation and the statistical inference in the general estimating equations model under the association dependence.The empirical distribution function plays a very important role in statistics. Although it is a piece-wise function and is not so graceful, as a nonparametric estimator of distribution function it is unbiased, consistent and follow the normal distribution asymptotically. Empirical process is constructed on the empirical distribution function, of which the uniform empirical process is a special and significant one. As one of the sources of "invariance principle", Doob (1949) suggested that the limit properties of the uniform empirical process should agree with the corresponding properties of a Brownian bridge. Donsker (1952) was the first one justifying Doob's conjecture. On the other hand, the empirical distribution function is the nonparametric maximum likelihood estimator of the distribution function. The empirical likelihood is in essence a nonparametric likelihood method, and the empirical likelihood ratio is built on the empirical distribution function.In Chapter 1, the precise asymptotics of the uniform empirical process is studied. This kind of limit theorem is based on the complete convergence, and was presented by Heyde (1975) first. Suppose that{X,Xn;n≥1} is a sequence of nondegenerate independent and identically distributed random variables, Sn= (?) Xi, n≥1. Here is a well-known result: Based on this, Heyde (1975) provides the precise rate of the above infinite series asε(?)0. If EX=0, EX2<∞, then Spataru (1999), Gut and Spataru (2000a), Gut and Spataru (2000b) got several theorems of this kind for the partial sum Sn. Their results received wide at-tentions, and were called the "precise asymptotics". After that, many authors entered on this topic. It is worthy to mention that Zhang (2001a), Zhang (2002), Jiang and Zhang (2006) achieved a serial of more intensive and extensive results by the method of strong approximation. Inspired by these literature, we suppose to extend precise asymptotics to the uniform empirical processes.In section 1.2, basing on the result that the uniform empirical process conver-gence weakly to the Brownian bridge, refering to the classical method used by Gut and Spataru, using some nice inequalities and so on, we obtain the second moment convergence rates of the uniform empirical process. For 0<β≤2,δ> 2/β-1, one has Here we suppose f(t) is a function defined on the interval [0,1], define the norm as‖f‖= sup0≤t≤1 |f(t)|, let B(t) be the Brownian bridge. In section 1.3, using the method of strong approximation that is the uniform empirical processαn(·) can be approximated by the Brownian bridge B(·), we get the precise asymptotic results whenεtends to a positive constant. Suppose that a> -1, b> -1, bn(ε) is a function ofε, and bn(ε)(?), n→∞. Then, one has By using the method of strong approximation again, we get the moment conver-gence rates for the uniform empirical process whenεtends to a positive constant in section 1.4. For a> -1, one has In the last two sections, the related results for the uniform quantile process un(t) are also obtained.Following the limit theorems in Chapter 1, we obtained the precise asymp-totic theorems for the self-normalized sums in Chapter 2. Suppose that {X, Xn; n≥1} is also a sequence of nondegenerate independent and identically distributed random variables, (?), n≥1. (?), then Sn/Vn is what so-called self-normalized sums. So far, there are abundant results of limit theory for the self-normalized sums, including the laws of iterated logarithm provided by Griffin and Kuelbs (1989,1991), the necessary and sufficient condition for the asymptotic normality presented by Gine et al. (1997a), the large deviation re-sults given by Shao(1997) and so on. Recently, de la Pena et al. researched about the limit theorems of the self-normalized process which is based on dependent random variables. The main result in this chapter is that, if X is attracted to the normal distribution and some mild conditions are added, then for -1< b< 0, one has and here Mn:=maxk≤n Sk.Because of the practical requirement, statisticians usually feel more inter-ested in the dependent random samples. Therefore, positive and negative asso-ciation are considered in the last chapter. They are widely exist in real life and engineering, such as reliability theory, statistical mechanics and so on. From the past twenty to thirty years, the limit properties of association dependent random variables are explored thoroughly. In this chapter, we mainly refer the results, such as the law of the iterated logarithm and the law of large numbers for the as-sociation random variables, which are presented by Peligrad and Sureash (1995), Shao and Su (1999) and Zhang (2001b). We get the conclusions including the asymptotic properties of the estimator and the testing statistics for the model and the parameter.In section 3.2, we introduce the general estimating equations model un-der association dependence and the method of the blockwise empirical Euclidean likelihood. Let Y1,…, YN be d-variate random variables with an unknown distri-bution function F(y,θ), in whichθis a p-dimensional unknown parameter vector. The information aboutθand F(y,θ) is available in the form of r functionally in-dependent unbiasd estimate functions. Let g(Yi,θ)=[g1(Yi,θ),…,gr(Y1,θ)]T, we have E[g(Y1,θ)]= 0, here r≥p. We adopt the blocking technique similar to Kitamura (1997). Let L be the window-width, denote obviously, ETi(θ)=0. The blockwise empirical Euclidean log-likelihood ratio is By Lagrange multiplier, one can get where Since r>p, we are unable to get the solution ofθfrom (?) Suppose thatθmaximize lEQ(θ), we call it the blockwise maximum empirical Euclidean likelihood estimator ofθ. After the introduction of the model and the method, we get the rates of the T(θ0) under the positive and the negative association respectively, whereθ0 is the true value of the parameter. Under several mild conditions, we obtain that T(θ0)=(?) under the positive association, (?) under the negative association. These are preparation for the asymptotic properties.In section 3.3, we prove thatθis a strong consistent estimator under cer-tain conditions. Furthermore, we obtain that (?) follows the normal distribution asymptotically. We prove that both R1=2lEQ(θ) - 2lEQ(θ0) and R2= -2lEQ(θ) follow the chi-square distribution asymptotically. R1, R2 are two statistics for testing the parameter and the model.This thesis include part of the author's work in recent years. All in all, due to the limited knowledge of the author, minor errors many incur in this article, so your criticism would be greatly appreciated. Thank you!...