| Varying coefficient model, also known as functional coefficient model, is auseful extension of the general linear model. Since varying coefficient model cancover a lot of other models, it has attracted the interest of many statistical researchers.Varying coefficient models not only retains partially the characteristics ofnonparametric regression, but is also simple in structure and easy to explain, whichhas led to its extensive application in statistical analysis in biology, medicine andmedicinal research, economics and finance.This paper conducts research on varying coefficient model via wavelet method.As a technique, wavelet method was first used in the analysis of seismic data. With itstheoretical development, the method has been also found extensive application inimage processing, signal processing and data compressing. The early 1990s haswitnessed the application of wavelet method in statistical studies. Since waveletmethod provides a powerful new technique for statistical researchers and hasdemonstrated its various advantages, it has been highly valued, and recent years, it hasbeen rapid increase in the use of wavelet method in applied statistical studies.This paper first studies the varying coefficient model by Hastie and Tibshirani(1993), and establishes unknown function coefficients and variance error estimator, and discusses asymptotic properties of estimators. The paper also discusses anothertype of nonparametric estimation. This research was financed by Fudan-SwitzerlandReinsurance Fund and the National Natural Science Foundation of China, and ourresearch findings have provided a practical method for research in mutual insurancefor the elderly. This paper tries to combine theory with practice, and realizes the crossconnection of different disciplines. The major contents of each chapter are as follows:Chapter One summarizes the development and evolution of statistical models, withemphasis on variations of varying coefficient models and related estimation methods.Research achievements are outlined, and feasibility and extensiveness of the application of varying coefficient model are documented. Compared with othermethods, wavelet method has many advantages. For example, compared withorthogonal series estimation, wavelet method has a low demand on function, while thelarge sample resulting is more satisfactory. Wavelet method can accurately describethe local features of complex curves, which has caught the attention of manyengineers and researchers in mathematics and statistics. This chapter will give ageneral instruction to wavelet method so as to lay a basis for Chapters Two and Threein which wavelet method will be employed in analysis. At the end of this chapter, major research contents and results are elaborated, and a summary is given.Chapter Two discusses the following varying coefficient model: y=x1β1(t)+…+xpβp(t)+ewhere y is response variable, x=(x1,…, xp)T and t are covariables, e is randomerror, whereas E(e)=0, E(e2)=σ2. {β-j(t), j=1,…, p} are unknownnonparametric functions, without losing generality, let the domain ofβj(t) be [0, 1].σ2 is unknown parameter. This model can be seen as an extension of general linearmodel, which allows regression coefficient to be the function of a regression variablet.βj(·) is able to describe in detail the relationship between xj and t. In addition, sinceβj(·) is a general nonparametrie function, the error of regression model isnoticeably reduced, hence strongly avoid the phenomenon of "curse of dimension". Inthis paper, let x be random design, while t be fixed design. Apply nonparametricregression wavelet estimation to the afore-mentioned varying coefficient model, wecan establish wavelet estimation of function coefficients {βj(t), j=1,…, p}, and canget strong uniform convergence and asymptotic normality.Chapter Three discusses with error variance estimation of varying coefficientmodel mentioned previously, proposes wavelet estimation (?)n2 for error varianceσ2, and proves its large sample nature. It also establishes wavelet estimation (?)n2 for var(e2), and proves the weak consistence of (?)n2, from whichn1/2((?)n2-σ2)/(?)n(?)N(0, 1) is deduced. This result can be used to construct largesample interval estimation or conduct large sample test forσ2. For same model, compared with existing research results, this method has a weak demand for functionsto be estimated while achieving more satisfactory estimation property. In addition, theresearch results here can be used for statistical purposes, so it has practicalapplications.Chapter Four discusses the estimation of survival functions in "mutual insurancefor the elderly project". The project stems from intensive computation of insurance.In mutual insurance for the elderly, we need to conduct estimation for survivalfunctions of self-sustaining life span, semi self-sustaining life span andnon-self-sustaining life span. The only know information is the sample of individualstate after the age of 60. What needs to be estimated is the survival functions of agiven life period prior to 60. What is special is that the sample-taking interval isdifferent from that for survival function, therefore traditional empirical distribution isnot totally applicable.Let X1, X2,…, Xn be independent sample from F(x), when functional form ofF(x) is unknown but it can be approximated with polynomial on [0, 1]θ1X±…±θrXr, Zheng Zukang (1995) proposed the mixed moment estimation andestablished strong consistence of estimators. But Zheng Zukang's (1995) approachassumes that the support set for distribution function is on [0, 1], this has limited itsscope of application. This paper discusses unknown function form F(x) which canbe approximated after logit conversion, thereby eliminates the limit of support set[0, 1]. Let F(x) has density function f(x), and there exists a certain r≥1, (r isunknown) which satisfies log F(x)/1-F(x)=a0+a1x+…+arxr(?)Jr(x)where x∈(a, b), (a, b)(?)(a0, b0). According to least square estimation theory, wecan get the estimation of approximating polynomial coefficients {a0, a1,…, ar}, onwhich we get the estimators for distribution function F(x), and prove the uniformconvergence of coefficient estimation {(?)0, (?)1,…, (?)r} and distribution function (?)(x).While discussing distribution function F(x), we also get the nonparametricestimation and corresponding large sample results for its density function f(x)andfailure functionλ(x). The simulation results indicate that estimation is satisfactory.Last chapter outlines research achievements and assumes related studies in thefuture.
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