含指标项半参数回归模型的估计与检验
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摘要
含指标项半参数回归模型是非参数回归分析中一类非常重要的统计模型,主要包括单指标模型、部分线性单指标模型、变系数单指标模型。这类模型的重要特征是将一个多元向量转化为一元指标,不仅有效地避免了“维数祸根(Curse of Dimensionality)"问题,而且仍能捕捉到高维数据的重要特征。因此,对此类半参数模型的统计推断是多元非参数回归的重要问题,是当前研究的热点问题,也是本文要研究的主要问题。
首先,我们研究了单指标模型(Single-Index Model)指标参数的检验问题。单指标模型可表述为其中Y∈R是应变量,X=(X1,…,Xp)T∈Rq是协变量,α0=(α01,…,α0q)T是Rq上的未知指标参数且为了模型的可识别性满足‖α0‖=1,α0(·)是未知的可测函数,称为指标函数；误差ε独立于X且E(ε)=0和Var(ε)=σ2。在上述的半参数模型的检验问题中,通常的极大似然比检验可能不存在,主要是因为未知函数α0(·)的最大似然估计(Maximum Likelihood Estimator)并不存在,即便它是存在的,但其相应的极大似然比检验也不是最优的,为此,Fan et al.(2001)提出了广义似然比(Generalized Likelihood Ratio)检验,简记为GLR检验,并得到了非参数类型的Wilks定理,本文则利用GLR检验方法研究了单指标模型中指标参数α0的检验,建立了相应的GLR检验统计量,并证明了该统计量渐近服从χ2分布,不仅在含指标项半参数回归模型中揭示了新的Wilks现象,而且扩大了GLR检验的适用范围。我们的模拟研究表明所提出的检验统计量表现出了较优的功效。
其次,在本论文中我们研究了部分线性单指标模型(Partially Linear Single-Index Model)的估计与检验问题,主要包括模型的剖面最小二乘估计(Profile Least-Squared Estimators)、指标参数与线性部分参数的检验。部分线性单指标模型首先由Carroll etal.(1997)提出并研究的,它是上述单指标模型的推广,可表述为其中Z=(Z1,…,Zp)T∈Rp是协变量,β0=(β1,…,β0p)T是Rp上的未知参数,误差ε独立于X和Z,其他条件同上面的单指标模型。本文提出了模型中未知量的剖面最小二乘估计(Profile Least-Squares Estimators)，证明了所给估计渐近服从正态分布；建立在剖面最小二乘估计的基础上,利用GLR检验方法给出在一定限制条件下的指标参数α0和线性参β0的检验的检验统计量,并证明了该统计量渐近服从χ2分布,模拟例子表明剖面最小二乘估计表现较优,所提出的检验统计量的功效表现较好,揭示了新的Wilks现象。
再次,在本论文中我们将研究变系数单指标模型(Varying-Coefficient Single-Index Model)的检验问题,主要包括指标参数、指标函数以及函数系数的检验。为了研究环境污染对呼吸疾病的影响,同时考虑到呼吸疾病的发病与气候有一定关系,五种空气污染物(二氧化硫、二氧化氮、一氧化氮、臭氧、可吸入的空气尘埃)和两个气候指标(温度、湿度)被认为是引起呼吸疾病的主要因素,Wong et al.(2008)引入了变系数单指标模型,即这里a(·)=(α1(·),…,αp(·))T是未知的函数系数,U∈R是一协变量,误差变量￡与X、Z独立,其他条件同上面的单指标模型。他们探讨了这些因素对呼吸疾病的影响。由于单指标变系数模型中指标函数和系数函数具有不同的自变量,这些特点为模型的估计和检验带来了极大的困难。Wong et al. (2008)结合局部线性方法和回切技巧给出了该模型的参数和非参数估计以及估计的计算方法,并且讨论了它们的大样本和小样本性质,以及该模型在大众卫生方面的应用。而本文则利用GLR检验方法研究了关于此模型的指标函数α0(·)是否具有线性形式的检验问题,以及系数函数a(·)是否随协变量U可变的检验问题,也研究了指标参数α0的检验问题。本文证明了所提出的GLR检验统计量的渐近服从χ2分布,模拟与实际例子研究表明所提检验方法的有效性,揭示了一类新的Wilks现象。
The semiparametric regression models with indexes are a kind of important statis-tical models in multivariate nonparametric and semiparametric regression. They mainly include the single-index model, the partially linear single-index model and the varying-coefficient single-index model. By reducing the dimensionality from multivariate predictors to a univariate index, this kind of models avoid the so-called "curse of dimensionality" while still capturing important features in high-dimensionality data. Thus, statistical in-ference for this kind of models is a very important problem in multivariate nonparametric regression, and presently it is a hot topic. This forms the main theme of the study.
Firstly we are concerned with statistical inference for the index parameter ao in the single-index model, which is written as where Y (?) R is a response variable, X= (X\1,…,Xq)T (?) Rq are covariates, the un-known index parameter vectorα0= (a01,…,α0q)T is in Rq and‖α0‖= 1 for model identifiability, ao(·) is an unknown univariable measurable function, which is called as index function, the errorεis independent of X with E(ε)= 0 and Var(ε)=σ2. The con-ventional maximum likelihood ratio (MLR) test may not exist in a semiparametric versus another semiparametric regression setting. This is because the nonparametric MLE for the unknown function ao(·) does not exist. Even if the MLE exists, the corresponding MLR test is not optimal. To end this, Fan et al. (2001) proposed a family of test called generalized likelihood ratio test, which is written as GLR test. And they obtained non-parametric version Wilks theorems. In this paper, we extend the GLR test to the index parameter ao in the single-index model, and establish the GLR statistics and demonstrate that its limiting null distribution follows aχ2-distribution. This not only unveils a new type of Wilks phenomenon for the semiparametric regression models with indexes, but also enlarges the application of the GLR test. A simulated example is used to illustrate the good performance of the testing approach.
Secondly statistical estimating and testing for the partially linear single-index model is considered in this paper. It mainly includes the profile least-squared estimators of the model and the checking for the index parameter and the index function of the model. The partially linear single-index model proposed by Carroll et al. (1997) is the generalization of the single-index model, and it can be written as where Z= (Z1,…,ZP)T (?) Rp are covariates, the unknown parameter vectorβ0= (β01，…，β0p)T is in RP, the errorεis independent of X and Z, and the other conditions is the same as that used in the above single-index model. A profile least-squares technique for estimating the parametric part is proposed and the asymptotic normality of the profile least-squares estimator is given. Based on the estimator, the GLR test is proposed to test whether parameters on linear part and the index parameter for the model are under a contain linear restricted condition. Under the null model, the proposed GLR statistic follows asymptotically the x2-distribution, which unveils the Wilks phenomenon. The simulated data example shows that our proposed methods is very well.
Thirdly tests for the varying-coefficient single-index model are considered in this paper. It mainly includes the tests for the index parameter, the index function and the coefficient functions of the model. To study the relationship between the levels of chemical pollutants and the number of daily total hospital admissions for respiratory diseases, and consider the five pollutants and two environmental factors, i.e. sulphur dioxide, nitrogen dioxide, nitrogen oxide, respirable suspended particulate, ozone, temperature and relative humidity, Wong et al. (2008) proposed the varying-coefficient single-index model where a(·)= (α1(·),…,αp(·))T is an unknown function vector, U (?) R is a covariate，the errorεis independent of X and Z, and the other conditions is the same as that used in the above single-index model. They discussed the relationship between the levels of chemical pollutants and the number of daily total hospital admissions for respiratory diseases. Note that the model has two different covariates in the index function and functional coefficient, which makes it very difficult to estimate and test the model. Wong et al. (2008) employed the local linear method, the average method and the back-fitting technique to obtain the estimates of the unknown parameters and the unknown functions of the model, and gave their asymptotic distribution. They also employed the model to the application of the public health. In this paper, we consider the GLR test on the model, which answers the questions that whether the index function is the linear function and the coefficient functions are not varying. And the inference for the index parameterα0 is also considered. We prove that the proposed test statistics follow theχ2-distribution asymptotically, which unveils a new type of Wilks phenomenon, and both simulated and real examples are used to show the good performance of the testing approach.
首先,我们研究了单指标模型(Single-Index Model)指标参数的检验问题。单指标模型可表述为其中Y∈R是应变量,X=(X1,…,Xp)T∈Rq是协变量,α0=(α01,…,α0q)T是Rq上的未知指标参数且为了模型的可识别性满足‖α0‖=1,α0(·)是未知的可测函数,称为指标函数；误差ε独立于X且E(ε)=0和Var(ε)=σ2。在上述的半参数模型的检验问题中,通常的极大似然比检验可能不存在,主要是因为未知函数α0(·)的最大似然估计(Maximum Likelihood Estimator)并不存在,即便它是存在的,但其相应的极大似然比检验也不是最优的,为此,Fan et al.(2001)提出了广义似然比(Generalized Likelihood Ratio)检验,简记为GLR检验,并得到了非参数类型的Wilks定理,本文则利用GLR检验方法研究了单指标模型中指标参数α0的检验,建立了相应的GLR检验统计量,并证明了该统计量渐近服从χ2分布,不仅在含指标项半参数回归模型中揭示了新的Wilks现象,而且扩大了GLR检验的适用范围。我们的模拟研究表明所提出的检验统计量表现出了较优的功效。
其次,在本论文中我们研究了部分线性单指标模型(Partially Linear Single-Index Model)的估计与检验问题,主要包括模型的剖面最小二乘估计(Profile Least-Squared Estimators)、指标参数与线性部分参数的检验。部分线性单指标模型首先由Carroll etal.(1997)提出并研究的,它是上述单指标模型的推广,可表述为其中Z=(Z1,…,Zp)T∈Rp是协变量,β0=(β1,…,β0p)T是Rp上的未知参数,误差ε独立于X和Z,其他条件同上面的单指标模型。本文提出了模型中未知量的剖面最小二乘估计(Profile Least-Squares Estimators)，证明了所给估计渐近服从正态分布；建立在剖面最小二乘估计的基础上,利用GLR检验方法给出在一定限制条件下的指标参数α0和线性参β0的检验的检验统计量,并证明了该统计量渐近服从χ2分布,模拟例子表明剖面最小二乘估计表现较优,所提出的检验统计量的功效表现较好,揭示了新的Wilks现象。
再次,在本论文中我们将研究变系数单指标模型(Varying-Coefficient Single-Index Model)的检验问题,主要包括指标参数、指标函数以及函数系数的检验。为了研究环境污染对呼吸疾病的影响,同时考虑到呼吸疾病的发病与气候有一定关系,五种空气污染物(二氧化硫、二氧化氮、一氧化氮、臭氧、可吸入的空气尘埃)和两个气候指标(温度、湿度)被认为是引起呼吸疾病的主要因素,Wong et al.(2008)引入了变系数单指标模型,即这里a(·)=(α1(·),…,αp(·))T是未知的函数系数,U∈R是一协变量,误差变量￡与X、Z独立,其他条件同上面的单指标模型。他们探讨了这些因素对呼吸疾病的影响。由于单指标变系数模型中指标函数和系数函数具有不同的自变量,这些特点为模型的估计和检验带来了极大的困难。Wong et al. (2008)结合局部线性方法和回切技巧给出了该模型的参数和非参数估计以及估计的计算方法,并且讨论了它们的大样本和小样本性质,以及该模型在大众卫生方面的应用。而本文则利用GLR检验方法研究了关于此模型的指标函数α0(·)是否具有线性形式的检验问题,以及系数函数a(·)是否随协变量U可变的检验问题,也研究了指标参数α0的检验问题。本文证明了所提出的GLR检验统计量的渐近服从χ2分布,模拟与实际例子研究表明所提检验方法的有效性,揭示了一类新的Wilks现象。
The semiparametric regression models with indexes are a kind of important statis-tical models in multivariate nonparametric and semiparametric regression. They mainly include the single-index model, the partially linear single-index model and the varying-coefficient single-index model. By reducing the dimensionality from multivariate predictors to a univariate index, this kind of models avoid the so-called "curse of dimensionality" while still capturing important features in high-dimensionality data. Thus, statistical in-ference for this kind of models is a very important problem in multivariate nonparametric regression, and presently it is a hot topic. This forms the main theme of the study.
Firstly we are concerned with statistical inference for the index parameter ao in the single-index model, which is written as where Y (?) R is a response variable, X= (X\1,…,Xq)T (?) Rq are covariates, the un-known index parameter vectorα0= (a01,…,α0q)T is in Rq and‖α0‖= 1 for model identifiability, ao(·) is an unknown univariable measurable function, which is called as index function, the errorεis independent of X with E(ε)= 0 and Var(ε)=σ2. The con-ventional maximum likelihood ratio (MLR) test may not exist in a semiparametric versus another semiparametric regression setting. This is because the nonparametric MLE for the unknown function ao(·) does not exist. Even if the MLE exists, the corresponding MLR test is not optimal. To end this, Fan et al. (2001) proposed a family of test called generalized likelihood ratio test, which is written as GLR test. And they obtained non-parametric version Wilks theorems. In this paper, we extend the GLR test to the index parameter ao in the single-index model, and establish the GLR statistics and demonstrate that its limiting null distribution follows aχ2-distribution. This not only unveils a new type of Wilks phenomenon for the semiparametric regression models with indexes, but also enlarges the application of the GLR test. A simulated example is used to illustrate the good performance of the testing approach.
Secondly statistical estimating and testing for the partially linear single-index model is considered in this paper. It mainly includes the profile least-squared estimators of the model and the checking for the index parameter and the index function of the model. The partially linear single-index model proposed by Carroll et al. (1997) is the generalization of the single-index model, and it can be written as where Z= (Z1,…,ZP)T (?) Rp are covariates, the unknown parameter vectorβ0= (β01，…，β0p)T is in RP, the errorεis independent of X and Z, and the other conditions is the same as that used in the above single-index model. A profile least-squares technique for estimating the parametric part is proposed and the asymptotic normality of the profile least-squares estimator is given. Based on the estimator, the GLR test is proposed to test whether parameters on linear part and the index parameter for the model are under a contain linear restricted condition. Under the null model, the proposed GLR statistic follows asymptotically the x2-distribution, which unveils the Wilks phenomenon. The simulated data example shows that our proposed methods is very well.
Thirdly tests for the varying-coefficient single-index model are considered in this paper. It mainly includes the tests for the index parameter, the index function and the coefficient functions of the model. To study the relationship between the levels of chemical pollutants and the number of daily total hospital admissions for respiratory diseases, and consider the five pollutants and two environmental factors, i.e. sulphur dioxide, nitrogen dioxide, nitrogen oxide, respirable suspended particulate, ozone, temperature and relative humidity, Wong et al. (2008) proposed the varying-coefficient single-index model where a(·)= (α1(·),…,αp(·))T is an unknown function vector, U (?) R is a covariate，the errorεis independent of X and Z, and the other conditions is the same as that used in the above single-index model. They discussed the relationship between the levels of chemical pollutants and the number of daily total hospital admissions for respiratory diseases. Note that the model has two different covariates in the index function and functional coefficient, which makes it very difficult to estimate and test the model. Wong et al. (2008) employed the local linear method, the average method and the back-fitting technique to obtain the estimates of the unknown parameters and the unknown functions of the model, and gave their asymptotic distribution. They also employed the model to the application of the public health. In this paper, we consider the GLR test on the model, which answers the questions that whether the index function is the linear function and the coefficient functions are not varying. And the inference for the index parameterα0 is also considered. We prove that the proposed test statistics follow theχ2-distribution asymptotically, which unveils a new type of Wilks phenomenon, and both simulated and real examples are used to show the good performance of the testing approach.
引文
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