相依数据下协变量调整回归模型及其在金融时间序列中的应用
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摘要
经典的统计学是建立在独立性假设之上的。独立随机变量的极限理论在20世纪30年代至40年代已经得到完善的发展,这些极限理论在统计学中起着至关重要的作用,是人们进行统计推断的理论基础。虽然独立性假设在某些时候是合理的,但是要验证一个样本的独立性是很困难的。而且在大部分的实际问题中,样本也并非是独立的观测值,因此,在20世纪50年代,随机变量的相依性概念引起了概率统计学家的研究兴趣,在概率论与数理统计的某些分支,如马氏链,随机场理论以及时间序列分析等学科中被相继提出,取得了大量的研究成果。在相依数据的研究中混合相依是广泛应用的概念。混合相依是指序列变量之间的相依关系是以时间或空间的距离衰退的,即当随机变量的指标只差趋于无穷时,随机变量是渐近独立的。
在经济金融,气象,水文,工程技术,自然科学和社会科学各个领域中,人们会遇到各种各样的数据,这些数据大多以时间序列的形式出现的。例如股票的每日收盘价格,产品的年销量,国民生产总值的年数据等等。因此,对时间序列进行研究,可以揭示各种现象变化和发展的内在规律,对于人们正确的认识事物并且由此作出科学的决策具有重要的现实意义。
协变量调整回归模型是最近新提出的一种统计分析方法。假设X和Y分别为预测变量和响应变量,在传统的回归模型中,通过(X,Y)的观测值来研究X和Y之间的关系。但是,在实际问题中,变量X和Y有可能会受到其他因素的干扰,如果在进行统计分析时没有把干扰因素考虑进来,就可能会得到不准确的或者是错误的统计推断。而协变量调整回归模型就是考虑干扰因素的影响,称干扰因素为协变量,研究在协变量影响下X和Y之间的关系。
协变量调整模型提出以后,由于其重要的现实意义和应用价值,受到了人们的广泛关注,出现了各种各样的推广,主要包括数据类型的推广和模型类型的推广。数据类型的推广大多是把独立同分布场合推广到纵向数据场合。模型类型的推广主要包括变系数模型,非线性模型和部分线性模型等等。本文中,我们在数据类型和模型类型两方面都做了推广。数据类型方面,我们把独立同分布场合推广到相依数据场合,从而应用到金融数据中。模型类型方面,我们分别讨论了相依数据下的参数回归模型和非参数回归模型。
1.相依数据下协变量调整参数回归模型
在第二章,我们讨论了相依数据下的协变量调整参数回归模型,其中Xi0=1,φ0(·)三1。假设不可观测数据{(Ui,Xi,Yi),i=1,2,...,n}为一个满足α-混合条件的严平稳过程。我们的目标是,基于观预测数据{(Ui,Xi,Yi),i=1,2,...,n}估计未知回归参数γr(r=0,1.2,...,p)并且研究估计的渐近性质。我们提出了一个两步估计方法
第一步.首先把协变量调整模型转换为其中这是一个函数型系数模型。我们采用局部线性平滑方法估计模型中的系数函数βr(·),r=0,…p。记最小化下面的加权平方和可以得到θ的最小二乘估计则系数函数βr(·)的估计为其中erT,2p+2为2p+2维向量,第r个元素为1,其他元素为零。
第二步.我们提出回归参数γrm=0,1,...,p)的估计为其中
我们讨论参数估计的渐进性质。定理2.1证明了估计的相合性,并且给出了收敛速度。定理2.2证明了参数估计的渐近正态性。
定理2.1.(相合性定理)假设模型满足§2..6节中的条件(C2-1)-(C2-9),则下面的结论成立
定理2.2.(渐近正态性)假设模型满足§2.6节中的条件(C2-1)-(C2-9),当n→∞时,下面的结论成立其中
为了比较协变量调整模型和一般线性模型对数据的拟合程度,我们提出了一种拟合优度检验。设协变量调整模型转换为下面的函数型系数模型如果函数βr(·)(r=0.1)为常数,即βr(U)三βr(r=0,1),则模型转换为一个简单的线性回归模型这说明线性回归模型与数据拟合地更好,否则,若βr(·)(r=0.1)不恒为常数,则函数型系数模型与数据拟合地更好。
设原假设为检验统计量为若Tn取值较大,则拒绝原假设。我们提出了一种非参Bootstrap方法来计算上述拟合优度检验的?)值。
为了阐明提出的方法,我们研究金融市场中铜现货价格CSP(响应变量)和铜期货价格CFP(预测变量)的关系。一个简单的线性回归关系为另外,沪深300股指期货(IF)对CSP和CFP之间的关系有显著地影响。因此,我们把IF作为协变量U,考虑下面的函数型系数回归模型为了对模型进行检验,我们采用§2.4中提出拟合优度检验。结果说明CSP和CFP之间存在非线性关系并且两者之间的关系随着IF的变化而变化。
2.相依数据下协变量调整非参数回归模型
在第三章,我们提出了相依数据下协变量调整非参数回归模型,其样本形式为假设不可观测样本{(U,Xi,Yi),i=1,2,...,n}为联合严平稳α-混合序列。
为了估计回归函数,我们如下的两步估计方法：
第一步.干扰函数ψ(U)和φ(U)的估计为我们可以建立一个协变量调整模型的近似表达
第二步.我们提出回归函数的Nadaraya-Watson估计为其中
定理3.1证明了回归函数的估计m(χ)的渐近收敛性,并且给出了收敛速度。
定理3.1如果§3.5中条件(A3-1)-似3-3)以及(C3-1)-(C3-5)满足,则下面的结论成立.
我们通过模拟计算和实际数据应用表明了协变量调整非参数回归方法的优良性。
3.基于局部LRS方法的稀疏信号片段检测
稀疏信号检测问题一直是信号处理中的热点问题。在高维数据中进行稀疏信号检测时经常会面临会两个挑战,一是如何提高检测精度,二是如何降低计算复杂度。在第四章,我们提出了一个局部LRS方法。与一般的LRS方法相比,局部LRS方法能显著地提高检测精度,降低计算复杂度。
假设观测样本{Xi,i=1,2,...,n}来自于模型其中I,,I2,…Iq为不相交区间,表示位置未知的信号片段,μ1,,μ2,…,μq为未知的信号强度。q=q(n)为未知的信号片段的个数,会随着n的增加而增加。{乙,i=1,2,...,n}为噪声,令Ⅱ={I1,I2,...,Iq}表示所有的信号片段的集合。我们的目标是检测信号片段是否存在,如果存在,识别信号片段的位置。我们把上述信号片段的检测和识别问题看作下面的假设检验问题,其中Φ表示空集。如果H1为真,说明信号片段存在,从而我们要确定信号片段集合Ⅱ。
提出的检验统计量为
检验统计量的阈值为我们提出的LRSL算法首先从所有的点中选出观测值大于t1n的“重要的”点,然后再考虑每一个“重要的”点的L-邻域。那么信号片段的合理估计应该是对应的检验统计量大于t2n并且取到最大值的那些区间。
定理4.1证明了检验方法的渐近性质。
定理4.1设§4.3中的条件(C4-1),(C4-2)满足,另外假设其中γn=(?)－1且∈n>0.那么,如果满足则下面的结论成立
模拟结果表明,局部LRS方法可以有效地提高检测精度,降低计算复杂度。
Classical statistics base on the independence assumption. Limit theories of in-dependent random variables has get the perfect development in the30s-40s of20th century. These limit theories constitute the theory foundation of statistical inference and play an important role in statistics. Although the hypothesis of independence is sometimes rational. it is hard to check the independence of the samples. Moreover, in most of the practical problems, the samples are not independent observations. Since the50s of20th century, the dependence of random variable has attracted much inter-est of statistician. In some branch of probability and mathematical statistics, such as Markov chain, random field theory and time series analysis, the dependence concept has been proposed and developed. In the study of dependent data, mixing is a broadly applied concept. A mixing process can be viewed as a sequence of random variables for which the past and distant future are asymptotically independent.
There exist a wide variety of data in economy and finance field, meteorology, hydrology, project technology and science, and so on. Most of these data appear in the form of time series, such as the stock closing price, annual turnover, the gross national product, and so on. Thus, analysis to time series is of growing importance to help participants make scientific decisions
Covariate-adjusted regression model is a new method which studied recently. As-sume X and Y are predictor and response, respectively. The traditional regression model study the relationship between X and Y based on the observations of (X,Y). In practical problems, however, variables X and Y might be distorted by other factors. If the distorting factors has not been considered, inaccurate statistic inference may oc- cur. Covariate-adjusted regression model precisely take into account the distortion of the other factors to analysis the relationship between X and Y. The distorting factor is called covariate.
Covariate-adjusted regression model has get a lot of attention because of its im-portant practical significance and application value. The extent of covariate-adjusted regression model include data type and model type. Most extent of data type extend the model from independent data to longitudinal data, and the extent of model type include varying-coefficient model, nonlinear model, partial linear model, and so on. In this thesis, the extent of the model in both data type and model type are considered. In data type, we consider the covariate-adjusted regression model in dependent data with application in financial data. In model type, we study the covariate-adjusted parametric and nonparametric regression on dependent data.
1. covariate-adjusted parametric regression model on dependent data
In Chapter2, we discuss the covariate-adjusted parametric regression model on dependent data where Xio=1,Φo(·)=1. We assume the unobservable data{(Ui, Xi, Yi), i=1,2,..., n} is a jointly strictly stationary α-mixing sequence. The main objective is to estimate the unknown regression parameters γr(r=0,1,2,...,p) and to consider the asymptotic property based on the observable data{(Ui. Xi, Yi), i=1,2,...,n}. To achieve this goal, we propose a two-step procedure.
Step1. Firstly, we transform the covariate-adjusted model into where This is a functional-coefficient time series model. In order to estimate coefficient func-tions βr(·)(r=0,1,...,p), a local linear smoothing method is employed. Set To minimize the sum of weighted squares it follows from the least square theory that the estimate of θ is The local least-square estimate of coefficient function βr(·) is given by where er.2p+2T is an unit vector of length2p+2with1at r-th position and0elsewhere.
Step2. The estimates of γ0and γr are given by where
we consider the asymptotic property of the proposed estimator. Theorem2.1proves that estimates are consistent and gives the consistency rate. The asymptotic normality of estimates are presented in Theorem2.2.
Theorem2.1Under the technical conditions given in§2.6, it holds that
Theorem2.2Under the technical conditions given in§2.6. as n→∞, it holds that where
In order to test the goodness-of-fit between the covariate-adjusted regression model and the general linear regression model, we propose a goodness-of-fit test. Suppose the covariate-adjusted regression model can be transformed into a functional-coefficient regression model. If βr(U)(r=0,1) are constant, that is βr(U)=βr{r=0,1), we get this implies that the simple linear regression model fit the data well. Otherwise, if βr(U)(r=0,1) are not constant, this implies that functional-coefficient regression model is more appropriate for the data.
Consider the null hypothesis The test statistic is defmed as and we reject the null hypothesis for a large value of Tn. We propose a nonparametric bootstrap approach to evaluate the p-value of the goodness-of-fit test.
To illustrate our methods, we consider the relationship between Copper Spot Price (CSP)(response) and Copper Futures Price (CFP)(predictor). A simple linear re-gression model would be Shanghai and Shenzhen300Stock Index Futures (IF), as a stock index futures, affects the relationship between CSP and CFP. Then we choose IF as the covariate U and consider the functional-coefficient regression model We apply the goodness-of-fit test described in§2.4. The conclusion indicate the functional-coefficient regression model is more appropriate for the data than the linear regression model. This provide the evidence that the relationship between CSP and CFP changes with IF.
2. covariate-adjusted nonparametric regression model on dependent data
In chapter3, we propose the covariate-adjusted nonparametric regression model on dependent data. The sample version is We assume the unobservable data{(Ui, Xi, Yi), i=1,2,..., n} is a jointly strictly sta-tionary α-mixing sequence. To estimate the regression function, a two-step estimate procedure is proposed as follows:
Step1. The nonparametric estimators of ψ(U) and Φ(U) are proposed as We may construct the approximate formula of covariate-adjusted model as
Step2. We propose the Nadaraya-Watson estimator of regression function as where
Theorem3.1proves the asymptotic convergence of the estimated regression func-tion m(·) and gives the convergence rate.
Theorem3.1. If conditions (A3-1)-(A3-3) and (C3-1)-(C3-5) in§3.5are satisfied, the following result holds: Both real data and simulated examples are provided for illustration.
3. Detecting Sparse Signal Segments by Local LRS Method
Sparse signal detection is an important problem in signal processing. Two impor-tant challenges in detecting sparse signals are how to improve the detection accuracy and reduce the computational complexity. In Chapter4, we propose a Local LRS method. Compare with conditional LRS method, the proposed procedure can greatly reduce the computational complexity and improve the improve the detection accuracy.
Suppose the data{Xi,i=1,2,...,n} satisfy the model where I1,I2,...,Iq are disjoint intervals which presenting signal segments with unknown locations,μ1,μ2,...,μq are unknown signal strength, q=q(n) is the unknown number of the signal segment, possibly increasing with n.{Zi,i=1,2,...,n} are noise data. Let Ⅱ={I1,I2,..., Iq} denote the collection of all the signal segments. Our goals are to detect whether signal segments exist and identify the locations of these segments when they do exist. The detection and identification of the signal segments can be regard as a statistical testing problem where Φ denote empty set. If H1is true, it means that there exist some signal segments, and then to identify the set of signal segments Ⅱ. The test statistic is proposed as where Ⅰ(?){1,2,...,n} is any interval, and|Ⅰ|denote the length of Ⅰ.
The threshold for the test statistic can be set as Our procedure first finds all "important" points whose observed data greater than tln, and then considers the L-neighborhood of each selected point. The proper estimates of signal segments should be intervals whose corresponding test statistic is greater than t2n and achieve the maximum.
Theorem4.1present the consistency of the proposed procedure.
Theorem4.1Assume (C4-1) and (C4-2) in§4.3hold. Additionally, assume where Then if we have
The simulation results indicate that the proposed procedure has high detection accuracy and computational efficiency.
在经济金融,气象,水文,工程技术,自然科学和社会科学各个领域中,人们会遇到各种各样的数据,这些数据大多以时间序列的形式出现的。例如股票的每日收盘价格,产品的年销量,国民生产总值的年数据等等。因此,对时间序列进行研究,可以揭示各种现象变化和发展的内在规律,对于人们正确的认识事物并且由此作出科学的决策具有重要的现实意义。
协变量调整回归模型是最近新提出的一种统计分析方法。假设X和Y分别为预测变量和响应变量,在传统的回归模型中,通过(X,Y)的观测值来研究X和Y之间的关系。但是,在实际问题中,变量X和Y有可能会受到其他因素的干扰,如果在进行统计分析时没有把干扰因素考虑进来,就可能会得到不准确的或者是错误的统计推断。而协变量调整回归模型就是考虑干扰因素的影响,称干扰因素为协变量,研究在协变量影响下X和Y之间的关系。
协变量调整模型提出以后,由于其重要的现实意义和应用价值,受到了人们的广泛关注,出现了各种各样的推广,主要包括数据类型的推广和模型类型的推广。数据类型的推广大多是把独立同分布场合推广到纵向数据场合。模型类型的推广主要包括变系数模型,非线性模型和部分线性模型等等。本文中,我们在数据类型和模型类型两方面都做了推广。数据类型方面,我们把独立同分布场合推广到相依数据场合,从而应用到金融数据中。模型类型方面,我们分别讨论了相依数据下的参数回归模型和非参数回归模型。
1.相依数据下协变量调整参数回归模型
在第二章,我们讨论了相依数据下的协变量调整参数回归模型,其中Xi0=1,φ0(·)三1。假设不可观测数据{(Ui,Xi,Yi),i=1,2,...,n}为一个满足α-混合条件的严平稳过程。我们的目标是,基于观预测数据{(Ui,Xi,Yi),i=1,2,...,n}估计未知回归参数γr(r=0,1.2,...,p)并且研究估计的渐近性质。我们提出了一个两步估计方法
第一步.首先把协变量调整模型转换为其中这是一个函数型系数模型。我们采用局部线性平滑方法估计模型中的系数函数βr(·),r=0,…p。记最小化下面的加权平方和可以得到θ的最小二乘估计则系数函数βr(·)的估计为其中erT,2p+2为2p+2维向量,第r个元素为1,其他元素为零。
第二步.我们提出回归参数γrm=0,1,...,p)的估计为其中
我们讨论参数估计的渐进性质。定理2.1证明了估计的相合性,并且给出了收敛速度。定理2.2证明了参数估计的渐近正态性。
定理2.1.(相合性定理)假设模型满足§2..6节中的条件(C2-1)-(C2-9),则下面的结论成立
定理2.2.(渐近正态性)假设模型满足§2.6节中的条件(C2-1)-(C2-9),当n→∞时,下面的结论成立其中
为了比较协变量调整模型和一般线性模型对数据的拟合程度,我们提出了一种拟合优度检验。设协变量调整模型转换为下面的函数型系数模型如果函数βr(·)(r=0.1)为常数,即βr(U)三βr(r=0,1),则模型转换为一个简单的线性回归模型这说明线性回归模型与数据拟合地更好,否则,若βr(·)(r=0.1)不恒为常数,则函数型系数模型与数据拟合地更好。
设原假设为检验统计量为若Tn取值较大,则拒绝原假设。我们提出了一种非参Bootstrap方法来计算上述拟合优度检验的?)值。
为了阐明提出的方法,我们研究金融市场中铜现货价格CSP(响应变量)和铜期货价格CFP(预测变量)的关系。一个简单的线性回归关系为另外,沪深300股指期货(IF)对CSP和CFP之间的关系有显著地影响。因此,我们把IF作为协变量U,考虑下面的函数型系数回归模型为了对模型进行检验,我们采用§2.4中提出拟合优度检验。结果说明CSP和CFP之间存在非线性关系并且两者之间的关系随着IF的变化而变化。
2.相依数据下协变量调整非参数回归模型
在第三章,我们提出了相依数据下协变量调整非参数回归模型,其样本形式为假设不可观测样本{(U,Xi,Yi),i=1,2,...,n}为联合严平稳α-混合序列。
为了估计回归函数,我们如下的两步估计方法：
第一步.干扰函数ψ(U)和φ(U)的估计为我们可以建立一个协变量调整模型的近似表达
第二步.我们提出回归函数的Nadaraya-Watson估计为其中
定理3.1证明了回归函数的估计m(χ)的渐近收敛性,并且给出了收敛速度。
定理3.1如果§3.5中条件(A3-1)-似3-3)以及(C3-1)-(C3-5)满足,则下面的结论成立.
我们通过模拟计算和实际数据应用表明了协变量调整非参数回归方法的优良性。
3.基于局部LRS方法的稀疏信号片段检测
稀疏信号检测问题一直是信号处理中的热点问题。在高维数据中进行稀疏信号检测时经常会面临会两个挑战,一是如何提高检测精度,二是如何降低计算复杂度。在第四章,我们提出了一个局部LRS方法。与一般的LRS方法相比,局部LRS方法能显著地提高检测精度,降低计算复杂度。
假设观测样本{Xi,i=1,2,...,n}来自于模型其中I,,I2,…Iq为不相交区间,表示位置未知的信号片段,μ1,,μ2,…,μq为未知的信号强度。q=q(n)为未知的信号片段的个数,会随着n的增加而增加。{乙,i=1,2,...,n}为噪声,令Ⅱ={I1,I2,...,Iq}表示所有的信号片段的集合。我们的目标是检测信号片段是否存在,如果存在,识别信号片段的位置。我们把上述信号片段的检测和识别问题看作下面的假设检验问题,其中Φ表示空集。如果H1为真,说明信号片段存在,从而我们要确定信号片段集合Ⅱ。
提出的检验统计量为
检验统计量的阈值为我们提出的LRSL算法首先从所有的点中选出观测值大于t1n的“重要的”点,然后再考虑每一个“重要的”点的L-邻域。那么信号片段的合理估计应该是对应的检验统计量大于t2n并且取到最大值的那些区间。
定理4.1证明了检验方法的渐近性质。
定理4.1设§4.3中的条件(C4-1),(C4-2)满足,另外假设其中γn=(?)－1且∈n>0.那么,如果满足则下面的结论成立
模拟结果表明,局部LRS方法可以有效地提高检测精度,降低计算复杂度。
Classical statistics base on the independence assumption. Limit theories of in-dependent random variables has get the perfect development in the30s-40s of20th century. These limit theories constitute the theory foundation of statistical inference and play an important role in statistics. Although the hypothesis of independence is sometimes rational. it is hard to check the independence of the samples. Moreover, in most of the practical problems, the samples are not independent observations. Since the50s of20th century, the dependence of random variable has attracted much inter-est of statistician. In some branch of probability and mathematical statistics, such as Markov chain, random field theory and time series analysis, the dependence concept has been proposed and developed. In the study of dependent data, mixing is a broadly applied concept. A mixing process can be viewed as a sequence of random variables for which the past and distant future are asymptotically independent.
There exist a wide variety of data in economy and finance field, meteorology, hydrology, project technology and science, and so on. Most of these data appear in the form of time series, such as the stock closing price, annual turnover, the gross national product, and so on. Thus, analysis to time series is of growing importance to help participants make scientific decisions
Covariate-adjusted regression model is a new method which studied recently. As-sume X and Y are predictor and response, respectively. The traditional regression model study the relationship between X and Y based on the observations of (X,Y). In practical problems, however, variables X and Y might be distorted by other factors. If the distorting factors has not been considered, inaccurate statistic inference may oc- cur. Covariate-adjusted regression model precisely take into account the distortion of the other factors to analysis the relationship between X and Y. The distorting factor is called covariate.
Covariate-adjusted regression model has get a lot of attention because of its im-portant practical significance and application value. The extent of covariate-adjusted regression model include data type and model type. Most extent of data type extend the model from independent data to longitudinal data, and the extent of model type include varying-coefficient model, nonlinear model, partial linear model, and so on. In this thesis, the extent of the model in both data type and model type are considered. In data type, we consider the covariate-adjusted regression model in dependent data with application in financial data. In model type, we study the covariate-adjusted parametric and nonparametric regression on dependent data.
1. covariate-adjusted parametric regression model on dependent data
In Chapter2, we discuss the covariate-adjusted parametric regression model on dependent data where Xio=1,Φo(·)=1. We assume the unobservable data{(Ui, Xi, Yi), i=1,2,..., n} is a jointly strictly stationary α-mixing sequence. The main objective is to estimate the unknown regression parameters γr(r=0,1,2,...,p) and to consider the asymptotic property based on the observable data{(Ui. Xi, Yi), i=1,2,...,n}. To achieve this goal, we propose a two-step procedure.
Step1. Firstly, we transform the covariate-adjusted model into where This is a functional-coefficient time series model. In order to estimate coefficient func-tions βr(·)(r=0,1,...,p), a local linear smoothing method is employed. Set To minimize the sum of weighted squares it follows from the least square theory that the estimate of θ is The local least-square estimate of coefficient function βr(·) is given by where er.2p+2T is an unit vector of length2p+2with1at r-th position and0elsewhere.
Step2. The estimates of γ0and γr are given by where
we consider the asymptotic property of the proposed estimator. Theorem2.1proves that estimates are consistent and gives the consistency rate. The asymptotic normality of estimates are presented in Theorem2.2.
Theorem2.1Under the technical conditions given in§2.6, it holds that
Theorem2.2Under the technical conditions given in§2.6. as n→∞, it holds that where
In order to test the goodness-of-fit between the covariate-adjusted regression model and the general linear regression model, we propose a goodness-of-fit test. Suppose the covariate-adjusted regression model can be transformed into a functional-coefficient regression model. If βr(U)(r=0,1) are constant, that is βr(U)=βr{r=0,1), we get this implies that the simple linear regression model fit the data well. Otherwise, if βr(U)(r=0,1) are not constant, this implies that functional-coefficient regression model is more appropriate for the data.
Consider the null hypothesis The test statistic is defmed as and we reject the null hypothesis for a large value of Tn. We propose a nonparametric bootstrap approach to evaluate the p-value of the goodness-of-fit test.
To illustrate our methods, we consider the relationship between Copper Spot Price (CSP)(response) and Copper Futures Price (CFP)(predictor). A simple linear re-gression model would be Shanghai and Shenzhen300Stock Index Futures (IF), as a stock index futures, affects the relationship between CSP and CFP. Then we choose IF as the covariate U and consider the functional-coefficient regression model We apply the goodness-of-fit test described in§2.4. The conclusion indicate the functional-coefficient regression model is more appropriate for the data than the linear regression model. This provide the evidence that the relationship between CSP and CFP changes with IF.
2. covariate-adjusted nonparametric regression model on dependent data
In chapter3, we propose the covariate-adjusted nonparametric regression model on dependent data. The sample version is We assume the unobservable data{(Ui, Xi, Yi), i=1,2,..., n} is a jointly strictly sta-tionary α-mixing sequence. To estimate the regression function, a two-step estimate procedure is proposed as follows:
Step1. The nonparametric estimators of ψ(U) and Φ(U) are proposed as We may construct the approximate formula of covariate-adjusted model as
Step2. We propose the Nadaraya-Watson estimator of regression function as where
Theorem3.1proves the asymptotic convergence of the estimated regression func-tion m(·) and gives the convergence rate.
Theorem3.1. If conditions (A3-1)-(A3-3) and (C3-1)-(C3-5) in§3.5are satisfied, the following result holds: Both real data and simulated examples are provided for illustration.
3. Detecting Sparse Signal Segments by Local LRS Method
Sparse signal detection is an important problem in signal processing. Two impor-tant challenges in detecting sparse signals are how to improve the detection accuracy and reduce the computational complexity. In Chapter4, we propose a Local LRS method. Compare with conditional LRS method, the proposed procedure can greatly reduce the computational complexity and improve the improve the detection accuracy.
Suppose the data{Xi,i=1,2,...,n} satisfy the model where I1,I2,...,Iq are disjoint intervals which presenting signal segments with unknown locations,μ1,μ2,...,μq are unknown signal strength, q=q(n) is the unknown number of the signal segment, possibly increasing with n.{Zi,i=1,2,...,n} are noise data. Let Ⅱ={I1,I2,..., Iq} denote the collection of all the signal segments. Our goals are to detect whether signal segments exist and identify the locations of these segments when they do exist. The detection and identification of the signal segments can be regard as a statistical testing problem where Φ denote empty set. If H1is true, it means that there exist some signal segments, and then to identify the set of signal segments Ⅱ. The test statistic is proposed as where Ⅰ(?){1,2,...,n} is any interval, and|Ⅰ|denote the length of Ⅰ.
The threshold for the test statistic can be set as Our procedure first finds all "important" points whose observed data greater than tln, and then considers the L-neighborhood of each selected point. The proper estimates of signal segments should be intervals whose corresponding test statistic is greater than t2n and achieve the maximum.
Theorem4.1present the consistency of the proposed procedure.
Theorem4.1Assume (C4-1) and (C4-2) in§4.3hold. Additionally, assume where Then if we have
The simulation results indicate that the proposed procedure has high detection accuracy and computational efficiency.
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