Statistical Inference For Partially Linear Single-index Models With Several Types Of Data

The partially linear single-index model (PLSIM) is a kind of semiparametric model that plugs a nonparametric component into the classical linear. It not only carried out clear explanation by linear part, but also avoid the "curse of dimensionality" problem by index combine the dimension of the covariate. Therefore, it has attracted many attention and has been extensively studied in recent years. Relevant studies about this model have been done by Yu and Rppert (2002,2004), Zhu and Xue (2006), Wang.et al. (2010), Liang and Liu (2010), Lian and Liang (2016) and so on.According to the different data be used to solve the problem in different area, therefore the data shows various characteristics. For example, the significance sample investigation data research will be applied to in economics, epidemiology, engineering and so on. Due to various complicated reasons and observe inaccurately, the sam-ple data has lots of measurement errors. Moreover, missing data is a widely existing phenomena in market research, public opinion survey, questionnaire survey, industrial experiment, medical research, biological genetics and so on. However, for various ele-ments lead to the data has been lost and miss, including experimentation cost is too expensive and the privacy will be protected that not allowed to the published. Thus, the standard statistical methods are no longer suitable for incomplete observational data. For sequentially collected economic data, which often exhibit dependence in errors, so the independence assumption for the errors is not always appropriate in ap-plications. Base on the the characteristics of the data, it is necessary that different methods are used to deal with the model.In this thesis, we are mainly concerned about the statistical inference of the par-tially linear single-index model with several data including measurement error data, missing data, NA sequence. The main content contains three parts. In the first part, the authors study the partially linear single-index model when the covariate X is mea-sured with additive error and the response variable Y is sometimes missing. Based on the least-squared technique, imputation method is proposed to estimate the regres-sion coefficients, single-index coefficients and the nonparametric function respectively. Thereafter, asymptotical normalities of the corresponding estimators are proved. In the second part, the empirical likelihood method is applied to the partially linear single-index model when the covariate X is measured with additive error and the response variable Y is sometimes missing. The authors focus on estimating the regression coef-ficients and single-index coefficients. The weighted empirical likelihood ratio statistic for the parameters of interest is proved to be asymptotically chi-squared. Hence it can be directly used to construct confidence regions for the parameters of interest. Fi-nally, the authors consider the application of blockwise empirical likelihood method to the partially linear single-index model when the errors are negatively associated. The blockwise empirical likelihood ratio statistic for the parameters of interest is proved to be asymptotically chi-squared. And hence it can be directly used to construct confi-dence regions for the parameters of interest.In what follows, we introduce the main results of this thesis.1. Estimation in a Partially Linear Single-Index Model with Missing Response Variables and Error-prone Covariates.Suppose that we obtain a random sample of incomplete data {(Yi, Zi, Xi:Vi, ei,?i),i= 1,2,…,n} from the partially linear single-index EV model which is as follows: where Yi is a response variable, (Zi,Xi)?Rp×Rq is covariate, g(-) is an unknown univariate measurable function, (?,?) is an unknown vector in Rp×Rq with ???=1, ?i is a random error with E(?i|Zi,Xi)=0 almost surely. The restriction ???=1 assures identifiability, and the first nonzero component of ? is positive. Xi is not a direct observation of the potential linear covariates, and Vi is a substitute of Xi, ei is a measurement error with E(ei)= 0 and Cov(e)=?e. At first, it is assumed that ?e is known. If it is unknown, it can be estimated with partial replication (Liang et al. (1999)).?i=0 if Yi is missing, ?i= 1 otherwise. Throughout this paper, the authors assume the data missing mechanism is as follows: for some unknown ?(Zj,Xi). In addition, P(?i=1|Yi, Zi, Xi,Vi)=?(Zi,Xi), this is because the measurement error ei is independent from (Yi, Zi,Xi,?i).Firstly, we get ?n,?n and gn(·) by Complete Method. Let Yio=?iYi+(1-?i)[g(ZiT?)+Vi??]. But the Xi is measured with error, we could not obtain the exact data of Yio. Let Yi*=?iYI+(1-?i) [gn(ZiT?n)+ViT?n], and Yi* is the estimator of Yio. By local least-square method, the imputation estimator of ? can be defined as where and. are the estimators of E(V|t) and E(Y*|t) respectively. Bsed on the local linear method by Fan and Gijbels (1996), the imputation estimators of g(t) and g'(t) are defined asIt is a must to estimate ? firstly by minimizing the sum of square errors say ?n. The above formula can not be applied directly in practice, since it contains the estimators of ? and g(·), so we need to make iteration. Theorem 1 Assume that conditions C2.1-C2.7 (Page 11) are satisfied, then we obtain whereTheorem 2 Suppose the conditions C2.1-C2.7 (Page 11) are satisfied, then we have where with Q and P given in (2.4.30) and (2.4.31) of appendix respectively.Theorem 3 Suppose the conditions C2.1-C2.7(Page 11)are satisfied,then we have where ?(t0)=E(?|ZT?=t0),,?2(K4)=?K42(u)du,?g=?2+?T?e?.2.Empirical Likelihood for Partially Linear Single-Index Models with Missing Response Variables and Error-prone Covariates.With the first part of the model assumptions,the "delete-one-component" method is adopted,which is widely used in semi-parameter model,then the empirical likelihood ratio of (?(r),?)is constructed.Let where g'(·)is the derivative of g(·)with respect to ?(r),J?(r) is the Jacobi matrix,?(·)is an bounded nonnegative weight function with a compact support,s(z)=E(?|Z=z), ?1(t)=E(Z|ZT?=t),?2(t):E(X|ZT?=t)=E(V|ZT?=t).If(?(r),?)is the true parameter,the bias-corrected empirical log-likelihood is defined as: where ?i(?(r),?)is the estimator of ?i(?(r),?),and ??Rp+q-1 is determined byTheorem 4 Assume that C3.1-C3.10(Page 33)are satisfied.If(?(r),?)is the true value of the parameter,and ?r>0,then where ?p+q-12 means the chi-square variable with p+g-1 degrees of freedom.3.Empirical Likelihood for Partially Linear Single-Index Models Under Negatively Associated Errors.Suppose that we obtain a random sample {(Xi,Zi,Yi),i=1,2,...,n} from the partially linear single-index model which is as follows: where Yi is a response variable,(Zi,Xi)?Rp×Rq is covariate,g(·)is an unknown univariate measurable function,(?,?)is an unknown vector Rp×Rq with ???=1, {?i,i=1,…,n].are NA errors with E(?i|Zi,Xi)=0,var(?i|Zi,Xi)=?i2<?.The restriction ? ??=1 assures identifiability,and the first nonzero component of ? is positive. The "delete-one-component" method is adopted,which is widely used in semi-parameter model,the empirical likelihood ratio of (?(r),?)is constructed.Let where g'(·)is the derivative of g(·)with respect to ?(r),J?(r)is the Jacobi matrix, ?1(t)=E(Z|ZT?=t)and ?2(t)=E(X|ZT?=t).However,?i(?(r),?)(i=1,…,n)Imay be dependent when the error satisfies NA. For this,we apply the blockwise method to construct the empirical likelihood ratio. Let k=k=[n/(d+s)],where[a]denotes the integral part of a,and d=d(n) and s=s(n)are positive integers satisfying d+s?n.?i(?(r),?)is the estimator of ?i(?(r),?).For m=1,.…,k,let where By using the Lagrange multiplier method,the bia.s-corrected blockwise empirical like-lihood ratio statistic is where ??Rp+q-1 is determined byTheorem 5 Assume that C4.1-C4.8(Page 50)are satisfied.If(?(r),?)is the true value of the parameter,and ?r>0,when n??,then where ?p+q-12 means the chi-square variable with p+q-1 degrecs of freedom.