Dependent Data Of The Statistical Model And Analysis

The data gathered from the study of some modern sciences such as biology and information science is usually large, high dimensional and dependent. So, the theory of dependent random variables is an important subject in statistics. It is well known that one has obtained some theoretical and practical achievements in the field of dependent random variables recently (Lu and Lin (1997) and the literature listed in the reference). And some powerful tools for treating dependent random variables, for example, Bootstrap, blockwise Bootstrap, Jackknife and blocks of blocks jackknife, have been introduced in recent literature (Lahiri (1999), Demitris and Joseph (1992)). So far, however, the theory that studies specially the statistical models with dependent data such as nonlinear regression and nonparametric regression models is not very satisfactory. In order to develop and unprove the theory of dependent random variables and related statistical models, this paper mainly focuses on the statistical theory of parameter estimation and fitting of function in the statistical models with dependent data involving linear regression model, nonlinear regression model, semi-parametric model and nonparametric regression model.In Chapter 1, we consider the following fixed design or random design nonparametric model with dependent noises:where {c^, e2,.?.} is a stationary stochastic process with mean 0 and variaiice a2. Under some weak assumptions on auto-covariance R(k) = E(ⁱci), we define two Bootstrap wavelet estimators of gQ) in the fixed design model as1 NNg*(t) t⁾n(t⁾ and g^t = Y7 J Em(t, s)dsi=1j=iand a Bootstrap wavelet estimator of g(.) in the random design model asi= 1where Y7, t^, X^, and A are respectively the Bootstrap samples of observations, design points and design intervals. Under weaker assumptions on auto-covariance than that oflXdata i1l fixed desigIl model, we define two blockwise BootstraP wave1et estimators of g(.)in the fixed design model as1 + K*Em(t, t:)K(t:) and g*(t) = f K* l Em(t, s)d8g*(t) = ai F K*Em(t, t:)K(t:) and g*(t) = Z K* l* Em(t, s)d8l=1 i=l iA:and a blockwise Bootstrap wavelet estimators of g(.) in the random design model asblj(t) = (bl)--' Z K*Em(t, X;)/i(t),i=lwhere K*, t: 3 X; 5 and A: are re8pectively the blockwise Bootstfap samples of observations,design points and design interals, l is the width of data block. In the fixed design model,the asymptotic bounds of the biases and variances fOr the Bootstrap wavelet and blockwiseBootstrap wave1et estimators g*(t) and g*(t) are obtained, their asymptotic normality fora modified version are establi8hed. In the random design model, the consistency and theasymptotic bound of the bias for the Bootstrap wavelet and blockwise Bootstrap waveletestimator g(t) are proved and a principle of selecting the width of data block is given.These re8ults show that both the BootstraP wavelet and blockwise BootstraP waveletmethods are valid in the models with weakly dependent processes.In Chapter 2, we study the blockwise quasi--regression in the linear models with weaklydependent data. Quasi-regression, a new method motivated by the problerns arising incomputer experiments, mainly focuses on speeding up evaluation (Owen (2000), An andOwen (2001)). For i.i.d. observations, the quasi-regression method is valid not only i11speeding l1p evaluation, but also in parametric statistical inference and fitting of function.However, fOr dePendent observations, this method is invalid. To improve quasi--regression,in this ChaPter, a blockwise quasi--regression estimator of the regression coefficient fij isdefined asQ1,QPj = 6 gTf5j = 0,1,...,P-- l,where Ti is the block of observations and Q is the flumber of the blocks. Some small sam-ple and large sample superiority of this estimate 8uch as the unbiasedness, convergence,asymptotic norlnality and the property of simulation, are obtained. We also find soIne de-fects of blockwise quasi-r...