正倒向随机微分方程和高维模型的统计推断
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摘要
随着现代社会的发展与金融领域研究的日益深入,金融产品已经成为人们生活中不可或缺的组成部分,投资组合分析,资产定价及金融风险度量等辅助金融市场交易的数理模型和分析工具层出不穷,自上世纪九十年代以来频发的世界范围内的金融危机也凸现出这类研究的巨大意义,倒向随机微分方程(BSDE)正是在此大环境下被发掘出其旺盛的生命力,正倒向随机微分方程(FBSDE)也逐渐巩固了它在金融业界的地位.此外在数理经济,工程技术,生物科技等各个领域研究者遭遇了越来越多的大样本海量数据或复杂抽样数据,一方面维数的膨胀为数据信息的模式识别和规律发现布置了维数灾难,而另一方面高维数据中蕴藏的丰富信息也带来了维数福音,这就要求寻找更有效的方法处理高维数据满足统计建模和计量经济分析等方面的需求.
BSDE自问世以来已被广泛的应用于数理金融与生物动力系统等领域,它与正向随机微分方程(OSDE)的本质区别在于BSDE依赖于终端条件,这恰好符合某些金融市场或生态环境运行态势的典型特征,然而这类终端相依模型的统计推断工作仍悬而未决.本论文首次构建了FBSDE模型并提出了FBSDE终端相依的积分型半参数估计和终端控制变量估计,简要拓展了模型的贝叶斯分析法,三种方法均以终端条件为基础解决了上述目标相依的问题.由于引入了积分形式,控制变量与贝叶斯观点,新模型的估计与OSDE估计的经典推断技术大相径庭或更为复杂,但保留了估计的相合性与渐近正态性,数据模拟进一步验证了估计在有限样本内的良好表现.
为了降低多元非参数回归维数灾难的影响,本论文引进了一种基于数值模拟的两步估计法,具体的是受到模拟外推法的启发将多元非参数回归模型分解为两部分,第一部分作为模型的主体其估计可达到参数收敛速度,第二部分足够小到可利用截断参数也较小的正交基函数展开作出近似,这两部分的线性组合即构成了多元回归函数的两步估计.这一方法不需借助回归函数的任何结构性假设,且对较小的截断参数也能保证估计的相合性,与受到维数诅咒的一般非参数估计如核平滑方法和局部线性估计等相比,我们提出的两步估计更具优越性.
近年来模型误定问题在统计学与计量经济学界也日渐引起广泛关注,一个不容忽视的障碍是当模型存在全局误定,即使其覆盖了大量参数与预测变量的信息,误定导致的与真实模型的偏离不仅不会消除反而会被加剧.本文采用了广义矩方法(GMM)对发散维数误定模型进行推断,详细阐述了新估计在局部相合性,全局相合性和渐近正态性等方面的渐近表现.为了减小全局误定的偏差,一种可完善模型及估计的半参数修正方法在理论结果和数值试验均证实了它自身的有效性.
本论文共分为五个章节,全文组织与创新如下：
第一章为FBSDE模型,多元非参数回归模型和发散维参数模型述评,扼要回顾了各类统计建模过程与现有推断方法的进展,指出了它们存在的优势与不足,提出了FBSDE模型的三种终端相依估计,多元回归模型的两步估计与发散维误定模型的GMM与半参数误定纠偏方法的研究背景与理论基础.
第二章着重探索了如下FBSDE模型的终端相依统计推断,假设在观测时间区间[0,T]内的初始观测点为t1,记录等时间间隔的观测时间点为{ti=t1+(i-1)Δ,i=1,…,n},相应的观测数据序列为{Xi,Yi,i=1,…,n},终端条件ξ服从某已知分布,抽取样本{ξi,1≤i≤m}.借助积分型方程的离散化重新建立具有线性生成元的模型为我们分别提出了模型中未知成分Zt的非参数估计和生成元的半参数估计
Zt2在x0处的N-W型核估计其渐近性质满足下述定理.
定理2.1除了满足条件(2.1),(2.2)和(2.3),{Xi：Xi∈(x0-h,x0+h),i=1,…,n}来自平稳的p-混合马尔科夫过程,且对于0<ρ<1,其ρ-混合系数满足ρ(l)=ρl,假设它的概率密度函数p(x)在支撑上连续有界,p(x0)>0,Zx0>0,且p(x)和Zx在x0的邻域内是二阶连续可微的.当n→∞时,若有nh→∞,nh5→0和nh△2→0均成立,则其中
参数β=(b,c)τ的估计可借助常规的参数估计方法得到,例如最小二乘估计,最小化下式通过下面的定理我们明确了半参数估计的渐近正态性.
定理2.2除了(2.1)-(2.4),假设{Xi,i=1,…,n}来自p-混合系数满足p(l)=ρl的平稳p-混合马尔科夫过程,对于00,Zx0>0,且p(x)和Zx在x0的邻域内是二阶连续可微的.当n→∞时,若有nh→∞,nh5→0和nhΔ2→0,那么这里σ2=Var(ζ/T).
第三章里引入了终端控制变量模型,离散化倒向方程并对终端取条件期望,其中m(Xt,ξ)=E(Zt(Bt+Δ-Bt)|Xt,ξ),ut=Zt(Bt+Δ-Bt)-m(Xt,ξ),对样本观测间隔△取值的两种情况分别展开讨论.△趋于0且收敛速度很快时,不妨通过最小二乘法得到β的相合估计,最小化下式若△趋于0的速度较慢,可得关于参数β的估计方程为可通过常规方法得到半参数估计βTC的显式表示.这里给出△下降很快时估计的渐近性质.
定理3.1除了假设条件(2.1),(2.2),(2,3)和(3.1)成立,{Xi,i=1,…,n}来自平稳的ρ-混合马尔科夫过程,对于0<ρ<1,ρ(l)=ρl,(Xt,ξ)有概率密度函数pXt,ξ(x0,ξ0),此外,函数pXt,ξ(x0,ξ0),m(x0,ξ0)和Zx0,ξ0如在(x0,ξ0)的邻域内存在二阶连续导数.当n,m→∞,h→0时,若满足nmh2→∞,则有本章结束前我们还简要分析了FBSDE模型的贝叶斯推断方法,包括单一风险投资及多风险投资场合下参数的后验分布及贝叶斯推断方法的主要步骤,该课题将在今后的研究中被给予关注.
在不对回归函数强加任何结构性假设的前提下解决高维非参数估计的维数问题,是第四章的主要目的.我们的研究对象是如下多元回归模型提出基于数值模拟的两步估计法和相应实施的步骤,最终r(x)的两步估计为随后研究了估计量r的渐近性质.
定理4.1若期望E(Y2)和E(fU2(U,σU2))存在,U(1),…,U(d)表示U的独立分量,且对任意的z=x+u∈X∪u有fZ(z)>0,Pm(x)Pm'(x)的最大特征根有界,r(x)属于(4.2.13)定义的索伯列夫椭球集S(β,L),并且具有余弦基展开的形式,则估计的偏满足这里r特别的对任意j,当并且那么对有
在最后一章中,我们探索了发散维数误定模型的广义矩方法,估计函数向量g(x,θ)全局有偏时,广义矩估计在满足识别条件等前提下具有如下性质,
定理5.1假设5.1-5.2成立,n趋于无穷,若有则Q(θ)存在局部极小值θn,使得
定理5.2假设5.1-5.3成立,且存在某正常数C使得λmin∧>C,若有(13)成立,则θn满足(14).
定理5.3假设5.1-5.4成立,n趋于无穷,若有则D代表依分布收敛.
而对于有一个可加项被误定的可加模型,θn0处修正的估计函数为并得到了半参数纠偏的修正估计为下面的定理陈述了调整后估计的渐进无偏性与相合性.
定理5.4假设可加模型(16)只有r(xn2,θn2)被误定,r(xn2,θn2)和r0(xn2)对xn2存在二阶连续偏导,则对任意j=1,2,…,qn,0 定理5.5定理5.1的条件和假设5.5成立时,若有则必定存在Q(θ)的局部最小元θn,满足
数值模拟实验进一步阐释了上述各种方法.
With the development of modern society and the deepening of financial field, financial products have become the indispensable part of people's life, and various mathematical models and analysis tools for dealing with the fi-nancial market transactions, which including investment portfolio analysis, asset pricing and financial risk measure, also emerge in an endless stream. S-ince the nineteen nineties, the worldwide financial crises happened frequently have highlighted the significance of such research to prevent these disaster-s. Backward stochastic differential equation (BSDE) has been well devel-oped in financial mathematics, and Forward-backward stochastic differential equations (FBSDE) is also playing its increasingly important role. In addi-tion, among the fields of the economic and financial, engineering technology, biotechnology and others, we encounter more large sample complicated data, especially the high-dimension data, which bring not only the model specifica-tion great challenges by the curse of dimensionality, and also rich information hidden behind just like the gospel of dimensionality. All these above require us finding effective way managing multivariate and high-dimensional data in order to cope with their applications in statistical modeling and econometric analysis, etc.
Backward Stochastic Differential Equation (BSDE) has been well stud-ied and widely applied in mathematical finance. The main difference from the original stochastic differential equation (OSDE) is that the BSDE is designed to depend on a terminal condition, which is a key factor in some financial and ecological circumstances. However, to the best of our knowledge, the terminal-dependent statistical inference for such a model has not been ex-plored in the existing literature. This paper proposes two terminal-dependent estimation methods via a terminal control variable model and the integral form of Forward-backward Stochastic Differential Equation (FBSDE). The reasons why we do so are that the resulting models contain terminal condition as model variable and therefore the newly proposed inference procedures in-herit the terminal-dependent characteristic. In this paper, the FBSDE is first rewritten as the regression versions and then the semi-parametric estimation procedures are proposed. Because of the control variable and integral form, the newly proposed regression versions are more complex than the classical ones and thus the inference methods are somewhat different from which de-signed for the OSDE. Even so, the statistical properties of the new methods are similar to the classical ones. Simulations are conducted to demonstrate finite sample behaviors.
To reduce the curse of dimensionality arising from nonparametric esti-mation procedure for multiple nonparametric regression, in this paper we sug-gest a simulationbased two-stage estimation. We first introduce a simulation-based method to decompose the multiple nonparametric regression into two parts. The first part can be estimated with the parametric convergence rate and the second part is small enough so that it can be approximated by or-thogonal basis functions with a small trade-off parameter. Then the linear combination of the first and second step estimators results in a two-stage estimator for multiple regression function. Our method does not need any specified structural assumption on regression function and it is proved that the newly proposed estimation is always consistent even if the trade-off pa-rameter is designed to be small. Thus when the common nonparametric estimator such as local linear smoothing collapses because of the curse of dimensionality, our estimator still works well.
Misspecified models have attracted much attention in some fields such as statistics and econometrics. When a global misspecification exists, even the model contains a large number of parameters and predictors, the mis-specification cannot disappear and sometimes it instead goes further away from the true one. Then the inference and correction for such a model are of very importance. In this paper we use the generalized method of moments (GMM) to infer the misspecified model with diverging numbers of param-eters and predictors, and to investigate its asymptotic behaviors, such as local and global consistency, and asymptotic normality. Furthermore, we suggest a semiparametric correction to reduce the global misspefication and, consequently, to improve the estimation and enhance the modeling. The theoretical results and the numerical comparisons show that the corrected estimation and fitting are better than the existing ones.
This dissertation consists of five chapters. Its main conclusions and innovations are organized as follows:
In Chapter1, after reviewing the FBSDE model, multiple nonparametric model and the misspecified model with diverging numbers of parameters and predictors, we survey briefly various statistical modelings and existing infer-ence methods, and point out their relative merits. We put forward research backgrounds and theory foundations for three kinds terminal-dependent sta-tistical inference for the FBSDE, simulation-based two-stage estimation for multiple nonparametric regression, GMM and misspecification correction for misspecified models with diverging number of parameters.
Chapter2investigates the following FBSDE model, expresses the FBSDE as a statistical framework, assuming g(t, Yt, Zt)=bYt+cZt.
Let{Xi,Yi,i=1,...,n} be the observed time series data. Since the distribution of ξ is supposed known, we can get its sample as{ξi,1≤i≤m} for m≥1/Δn, then the original model can be approximately rewritten through integral discretization as
Then address the proper estimator of g and Zt. We might adopt the N-W kernel nonparametric method to estimate Zt as Its asymptotic property is shown as below.
Theorem2.1Besides the conditions (2.1),(2.2) and (2.3), suppose that Xi∈(x0-h, x0+h) is a stationary ρ-mixing Markov process with the p-mixing coefficients satisfying ρ(l)=ρl for0<ρ<1, and has a common probability density p(x) satisfying p(x0)>0. Furthermore, functions p(x) and Zx have continuous two derivatives in a neighborhood of X0. As n→∞, if nh→∞, nh5→0and nhΔ2→0, then
From the above, it is simple to deduce the estimator of β=(b, c)τ with common parametric methods. For example, the least square (LS) estimator is obtained by minimizing The following theorem states the result aomost standard as the asymptotic normality with the convergence rate of (n)
Theorem2.2Besides the conditions(2.1),(2.2),(2.3)and(2.4), suppose that{Xi,i=1,…,n}is a stationary ρ-mixing Markou process with the ρ-mixing coefficients satisfying ρ(l)=ρ for0<ρ<1and has a common probability density p(x)satisfying p(x0)>0. Furthermore,functions p(x) and Zx have continuous two derivatives in a neighborhood of x0.As n→∞, if nh→∞,nh5→0and nhΔ2→0,then where σ2=Var(ξ/T).
In chapter3,we first introduce the terminal control variable model, where m(Xt,ξ)=E(Zt(B+Δ-Bt)|Xt,ξ),ut=Zt(Bt+Δ-Bt)-m(Xt,ξ). When Δ tends to zero quite fast,we can derive the estimator of β by mini-mizing Otherwise,the flollowing estimating equation could be used to derive the estimator of β: Then the estimators have the closed representations of βTC,whose asymp-totic distribution is as follows.
Theorem3.1Besides the conditions(2.1),(2.2),(2.3)and(3.1),sup-pose that {Xi,i=1,…,n}is a stationary ρ-mixing Markou process with the ρ-mixing coefficients satisfying ρ(l)=ρl for0 By the end of this Chapter we also briefly analyzes the FBSDE mod-el by means of the Bayesian method in the cases including one single risk investment, and K candidates instead, and infer the posterior distributions and major estimation procedures.
To reduce the curse of dimensionality arising from nonparametric esti-mation procedures for multiple nonparametric regression without any speci-fied structural assumption on the regression function, in Chapter4we suggest a simulation-based two-stage estimation, then the resultant estimator is and we present its asymptotic behavior as follow,
Theorem4.1If E(Y2) and E(fU2(U, σU2)) both exist, the components U(1),...U(d) of U are designed to be independent, fz(z)>0for all z=x+u∈∈X∪U the maximum eigenvalue of Pm(x)P'm(x) is bounded, and r(x) belongs to the Soblev ellipsoid S(β,L) defined in (4.2.13), and can be expressed by a series expansion of the cosine basis functions as above, then Particulary, if βj=β0and mj=O(nδ) for all j and then with ρ=β0d/(2(β0d+1))-log(γU(m)Ld)/(β0d+1) log n).
The last Chapter we use the generalized method of moments (GMM) to infer the misspecified model with diverging numbers of parameters and predictors, which means g{x,θ) is globally biased such as, We only consider the estimator of GMM defined by and investigate its asymptotic behaviors, such as local and global consistency, and asymptotic normality.
Theorem5.1Suppose the Assumptions5.1-5.2hold. When n tends to infinity, if then there is a local minimizer θn of Q(θ) such that
Theorem5.2Under the Assumptions5.1-5.3, if λminΛ>C for a positive constant C, and (13) holds, then θn satisfies (14).
Theorem5.3Suppose the Assumptions5.1-5.4hold. When n tends to infinity, if then where D stands for the convergence in distribution.
Furthermore, the additive regression model is defined by finally an corrected version of estimating function valued at θn0is given by consequently, we suggest an corrected estimator as The theoretical results show that the corrected estimation and fitting are better than the existing ones.
Theorem5.4Suppose that in additive regression model (16) only the term r(xn2,θn2) is misspecified, r(xn2,θn2) and r0(xn2) have two continuous derivatives with respect to xn2. Then for j=1,2,...,qn and0 Theorem5.5Under the condition of Theorem5.1and Assumption5.5, then there is a local minimizer θn of Q(θ) such that, if then
Simulations are used to illustrate various methods.
BSDE自问世以来已被广泛的应用于数理金融与生物动力系统等领域,它与正向随机微分方程(OSDE)的本质区别在于BSDE依赖于终端条件,这恰好符合某些金融市场或生态环境运行态势的典型特征,然而这类终端相依模型的统计推断工作仍悬而未决.本论文首次构建了FBSDE模型并提出了FBSDE终端相依的积分型半参数估计和终端控制变量估计,简要拓展了模型的贝叶斯分析法,三种方法均以终端条件为基础解决了上述目标相依的问题.由于引入了积分形式,控制变量与贝叶斯观点,新模型的估计与OSDE估计的经典推断技术大相径庭或更为复杂,但保留了估计的相合性与渐近正态性,数据模拟进一步验证了估计在有限样本内的良好表现.
为了降低多元非参数回归维数灾难的影响,本论文引进了一种基于数值模拟的两步估计法,具体的是受到模拟外推法的启发将多元非参数回归模型分解为两部分,第一部分作为模型的主体其估计可达到参数收敛速度,第二部分足够小到可利用截断参数也较小的正交基函数展开作出近似,这两部分的线性组合即构成了多元回归函数的两步估计.这一方法不需借助回归函数的任何结构性假设,且对较小的截断参数也能保证估计的相合性,与受到维数诅咒的一般非参数估计如核平滑方法和局部线性估计等相比,我们提出的两步估计更具优越性.
近年来模型误定问题在统计学与计量经济学界也日渐引起广泛关注,一个不容忽视的障碍是当模型存在全局误定,即使其覆盖了大量参数与预测变量的信息,误定导致的与真实模型的偏离不仅不会消除反而会被加剧.本文采用了广义矩方法(GMM)对发散维数误定模型进行推断,详细阐述了新估计在局部相合性,全局相合性和渐近正态性等方面的渐近表现.为了减小全局误定的偏差,一种可完善模型及估计的半参数修正方法在理论结果和数值试验均证实了它自身的有效性.
本论文共分为五个章节,全文组织与创新如下：
第一章为FBSDE模型,多元非参数回归模型和发散维参数模型述评,扼要回顾了各类统计建模过程与现有推断方法的进展,指出了它们存在的优势与不足,提出了FBSDE模型的三种终端相依估计,多元回归模型的两步估计与发散维误定模型的GMM与半参数误定纠偏方法的研究背景与理论基础.
第二章着重探索了如下FBSDE模型的终端相依统计推断,假设在观测时间区间[0,T]内的初始观测点为t1,记录等时间间隔的观测时间点为{ti=t1+(i-1)Δ,i=1,…,n},相应的观测数据序列为{Xi,Yi,i=1,…,n},终端条件ξ服从某已知分布,抽取样本{ξi,1≤i≤m}.借助积分型方程的离散化重新建立具有线性生成元的模型为我们分别提出了模型中未知成分Zt的非参数估计和生成元的半参数估计
Zt2在x0处的N-W型核估计其渐近性质满足下述定理.
定理2.1除了满足条件(2.1),(2.2)和(2.3),{Xi：Xi∈(x0-h,x0+h),i=1,…,n}来自平稳的p-混合马尔科夫过程,且对于0<ρ<1,其ρ-混合系数满足ρ(l)=ρl,假设它的概率密度函数p(x)在支撑上连续有界,p(x0)>0,Zx0>0,且p(x)和Zx在x0的邻域内是二阶连续可微的.当n→∞时,若有nh→∞,nh5→0和nh△2→0均成立,则其中
参数β=(b,c)τ的估计可借助常规的参数估计方法得到,例如最小二乘估计,最小化下式通过下面的定理我们明确了半参数估计的渐近正态性.
定理2.2除了(2.1)-(2.4),假设{Xi,i=1,…,n}来自p-混合系数满足p(l)=ρl的平稳p-混合马尔科夫过程,对于0
第三章里引入了终端控制变量模型,离散化倒向方程并对终端取条件期望,其中m(Xt,ξ)=E(Zt(Bt+Δ-Bt)|Xt,ξ),ut=Zt(Bt+Δ-Bt)-m(Xt,ξ),对样本观测间隔△取值的两种情况分别展开讨论.△趋于0且收敛速度很快时,不妨通过最小二乘法得到β的相合估计,最小化下式若△趋于0的速度较慢,可得关于参数β的估计方程为可通过常规方法得到半参数估计βTC的显式表示.这里给出△下降很快时估计的渐近性质.
定理3.1除了假设条件(2.1),(2.2),(2,3)和(3.1)成立,{Xi,i=1,…,n}来自平稳的ρ-混合马尔科夫过程,对于0<ρ<1,ρ(l)=ρl,(Xt,ξ)有概率密度函数pXt,ξ(x0,ξ0),此外,函数pXt,ξ(x0,ξ0),m(x0,ξ0)和Zx0,ξ0如在(x0,ξ0)的邻域内存在二阶连续导数.当n,m→∞,h→0时,若满足nmh2→∞,则有本章结束前我们还简要分析了FBSDE模型的贝叶斯推断方法,包括单一风险投资及多风险投资场合下参数的后验分布及贝叶斯推断方法的主要步骤,该课题将在今后的研究中被给予关注.
在不对回归函数强加任何结构性假设的前提下解决高维非参数估计的维数问题,是第四章的主要目的.我们的研究对象是如下多元回归模型提出基于数值模拟的两步估计法和相应实施的步骤,最终r(x)的两步估计为随后研究了估计量r的渐近性质.
定理4.1若期望E(Y2)和E(fU2(U,σU2))存在,U(1),…,U(d)表示U的独立分量,且对任意的z=x+u∈X∪u有fZ(z)>0,Pm(x)Pm'(x)的最大特征根有界,r(x)属于(4.2.13)定义的索伯列夫椭球集S(β,L),并且具有余弦基展开的形式,则估计的偏满足这里r特别的对任意j,当并且那么对有
在最后一章中,我们探索了发散维数误定模型的广义矩方法,估计函数向量g(x,θ)全局有偏时,广义矩估计在满足识别条件等前提下具有如下性质,
定理5.1假设5.1-5.2成立,n趋于无穷,若有则Q(θ)存在局部极小值θn,使得
定理5.2假设5.1-5.3成立,且存在某正常数C使得λmin∧>C,若有(13)成立,则θn满足(14).
定理5.3假设5.1-5.4成立,n趋于无穷,若有则D代表依分布收敛.
而对于有一个可加项被误定的可加模型,θn0处修正的估计函数为并得到了半参数纠偏的修正估计为下面的定理陈述了调整后估计的渐进无偏性与相合性.
定理5.4假设可加模型(16)只有r(xn2,θn2)被误定,r(xn2,θn2)和r0(xn2)对xn2存在二阶连续偏导,则对任意j=1,2,…,qn,0
数值模拟实验进一步阐释了上述各种方法.
With the development of modern society and the deepening of financial field, financial products have become the indispensable part of people's life, and various mathematical models and analysis tools for dealing with the fi-nancial market transactions, which including investment portfolio analysis, asset pricing and financial risk measure, also emerge in an endless stream. S-ince the nineteen nineties, the worldwide financial crises happened frequently have highlighted the significance of such research to prevent these disaster-s. Backward stochastic differential equation (BSDE) has been well devel-oped in financial mathematics, and Forward-backward stochastic differential equations (FBSDE) is also playing its increasingly important role. In addi-tion, among the fields of the economic and financial, engineering technology, biotechnology and others, we encounter more large sample complicated data, especially the high-dimension data, which bring not only the model specifica-tion great challenges by the curse of dimensionality, and also rich information hidden behind just like the gospel of dimensionality. All these above require us finding effective way managing multivariate and high-dimensional data in order to cope with their applications in statistical modeling and econometric analysis, etc.
Backward Stochastic Differential Equation (BSDE) has been well stud-ied and widely applied in mathematical finance. The main difference from the original stochastic differential equation (OSDE) is that the BSDE is designed to depend on a terminal condition, which is a key factor in some financial and ecological circumstances. However, to the best of our knowledge, the terminal-dependent statistical inference for such a model has not been ex-plored in the existing literature. This paper proposes two terminal-dependent estimation methods via a terminal control variable model and the integral form of Forward-backward Stochastic Differential Equation (FBSDE). The reasons why we do so are that the resulting models contain terminal condition as model variable and therefore the newly proposed inference procedures in-herit the terminal-dependent characteristic. In this paper, the FBSDE is first rewritten as the regression versions and then the semi-parametric estimation procedures are proposed. Because of the control variable and integral form, the newly proposed regression versions are more complex than the classical ones and thus the inference methods are somewhat different from which de-signed for the OSDE. Even so, the statistical properties of the new methods are similar to the classical ones. Simulations are conducted to demonstrate finite sample behaviors.
To reduce the curse of dimensionality arising from nonparametric esti-mation procedure for multiple nonparametric regression, in this paper we sug-gest a simulationbased two-stage estimation. We first introduce a simulation-based method to decompose the multiple nonparametric regression into two parts. The first part can be estimated with the parametric convergence rate and the second part is small enough so that it can be approximated by or-thogonal basis functions with a small trade-off parameter. Then the linear combination of the first and second step estimators results in a two-stage estimator for multiple regression function. Our method does not need any specified structural assumption on regression function and it is proved that the newly proposed estimation is always consistent even if the trade-off pa-rameter is designed to be small. Thus when the common nonparametric estimator such as local linear smoothing collapses because of the curse of dimensionality, our estimator still works well.
Misspecified models have attracted much attention in some fields such as statistics and econometrics. When a global misspecification exists, even the model contains a large number of parameters and predictors, the mis-specification cannot disappear and sometimes it instead goes further away from the true one. Then the inference and correction for such a model are of very importance. In this paper we use the generalized method of moments (GMM) to infer the misspecified model with diverging numbers of param-eters and predictors, and to investigate its asymptotic behaviors, such as local and global consistency, and asymptotic normality. Furthermore, we suggest a semiparametric correction to reduce the global misspefication and, consequently, to improve the estimation and enhance the modeling. The theoretical results and the numerical comparisons show that the corrected estimation and fitting are better than the existing ones.
This dissertation consists of five chapters. Its main conclusions and innovations are organized as follows:
In Chapter1, after reviewing the FBSDE model, multiple nonparametric model and the misspecified model with diverging numbers of parameters and predictors, we survey briefly various statistical modelings and existing infer-ence methods, and point out their relative merits. We put forward research backgrounds and theory foundations for three kinds terminal-dependent sta-tistical inference for the FBSDE, simulation-based two-stage estimation for multiple nonparametric regression, GMM and misspecification correction for misspecified models with diverging number of parameters.
Chapter2investigates the following FBSDE model, expresses the FBSDE as a statistical framework, assuming g(t, Yt, Zt)=bYt+cZt.
Let{Xi,Yi,i=1,...,n} be the observed time series data. Since the distribution of ξ is supposed known, we can get its sample as{ξi,1≤i≤m} for m≥1/Δn, then the original model can be approximately rewritten through integral discretization as
Then address the proper estimator of g and Zt. We might adopt the N-W kernel nonparametric method to estimate Zt as Its asymptotic property is shown as below.
Theorem2.1Besides the conditions (2.1),(2.2) and (2.3), suppose that Xi∈(x0-h, x0+h) is a stationary ρ-mixing Markov process with the p-mixing coefficients satisfying ρ(l)=ρl for0<ρ<1, and has a common probability density p(x) satisfying p(x0)>0. Furthermore, functions p(x) and Zx have continuous two derivatives in a neighborhood of X0. As n→∞, if nh→∞, nh5→0and nhΔ2→0, then
From the above, it is simple to deduce the estimator of β=(b, c)τ with common parametric methods. For example, the least square (LS) estimator is obtained by minimizing The following theorem states the result aomost standard as the asymptotic normality with the convergence rate of (n)
Theorem2.2Besides the conditions(2.1),(2.2),(2.3)and(2.4), suppose that{Xi,i=1,…,n}is a stationary ρ-mixing Markou process with the ρ-mixing coefficients satisfying ρ(l)=ρ for0<ρ<1and has a common probability density p(x)satisfying p(x0)>0. Furthermore,functions p(x) and Zx have continuous two derivatives in a neighborhood of x0.As n→∞, if nh→∞,nh5→0and nhΔ2→0,then where σ2=Var(ξ/T).
In chapter3,we first introduce the terminal control variable model, where m(Xt,ξ)=E(Zt(B+Δ-Bt)|Xt,ξ),ut=Zt(Bt+Δ-Bt)-m(Xt,ξ). When Δ tends to zero quite fast,we can derive the estimator of β by mini-mizing Otherwise,the flollowing estimating equation could be used to derive the estimator of β: Then the estimators have the closed representations of βTC,whose asymp-totic distribution is as follows.
Theorem3.1Besides the conditions(2.1),(2.2),(2.3)and(3.1),sup-pose that {Xi,i=1,…,n}is a stationary ρ-mixing Markou process with the ρ-mixing coefficients satisfying ρ(l)=ρl for0
To reduce the curse of dimensionality arising from nonparametric esti-mation procedures for multiple nonparametric regression without any speci-fied structural assumption on the regression function, in Chapter4we suggest a simulation-based two-stage estimation, then the resultant estimator is and we present its asymptotic behavior as follow,
Theorem4.1If E(Y2) and E(fU2(U, σU2)) both exist, the components U(1),...U(d) of U are designed to be independent, fz(z)>0for all z=x+u∈∈X∪U the maximum eigenvalue of Pm(x)P'm(x) is bounded, and r(x) belongs to the Soblev ellipsoid S(β,L) defined in (4.2.13), and can be expressed by a series expansion of the cosine basis functions as above, then Particulary, if βj=β0and mj=O(nδ) for all j and then with ρ=β0d/(2(β0d+1))-log(γU(m)Ld)/(β0d+1) log n).
The last Chapter we use the generalized method of moments (GMM) to infer the misspecified model with diverging numbers of parameters and predictors, which means g{x,θ) is globally biased such as, We only consider the estimator of GMM defined by and investigate its asymptotic behaviors, such as local and global consistency, and asymptotic normality.
Theorem5.1Suppose the Assumptions5.1-5.2hold. When n tends to infinity, if then there is a local minimizer θn of Q(θ) such that
Theorem5.2Under the Assumptions5.1-5.3, if λminΛ>C for a positive constant C, and (13) holds, then θn satisfies (14).
Theorem5.3Suppose the Assumptions5.1-5.4hold. When n tends to infinity, if then where D stands for the convergence in distribution.
Furthermore, the additive regression model is defined by finally an corrected version of estimating function valued at θn0is given by consequently, we suggest an corrected estimator as The theoretical results show that the corrected estimation and fitting are better than the existing ones.
Theorem5.4Suppose that in additive regression model (16) only the term r(xn2,θn2) is misspecified, r(xn2,θn2) and r0(xn2) have two continuous derivatives with respect to xn2. Then for j=1,2,...,qn and0
Simulations are used to illustrate various methods.
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