部分线性模型中的广义似然比检验
摘要
本文研究了部分线性模型中的广义似然比检验。我们首先考虑了在一个模型下,零假设分别是非参函数为常数和非参函数为线性函数时的情形,用局部多项式方法估计函数分量,用传统的估计方法估计参数分量,讨论了相应的估计量的渐近性质,估计量在假设nh5 = O(1)成立时都是最优的。在此基础上导出了广义似然比统计量的表达式及渐近正态性质。本文还考虑了两个模型下非参函数的比较,在与一个模型同样的假设下,用相同的方法估计出了零假设下与全空间下的参数分量与非参数分量,它们都达到了最优收敛速度,并在此基础上导出了广义似然比统计量的表达式,并证明了它是渐近正态的。
Since Engle et al.(1986) proposed the partially linear models and ap-plied them to analyze the relation between electricity usage and average dailytemperature, the models have been extensively studied in literature, especiallythe estimation of the parametric and nonparametric components. Variousestimation of the parametric and nonparametric components have been pro-posed. Engle et al.(1986), Heckman(1986), and Rice(1986) developed spline-type methods by using smoothing spline; Speckman(1988), Opsomer and Rup-pert(1999) proposed kernel-type estimation. H¨ardle,Liang,Gao (2000) gave acomprehensive overview of statistical inference for the partially linear mod-els. The models have also been applied in economics(Scemalensee and Stoker1999), biometrics(Zeger and Diggle 1994, Liang et al. 2004).
Despite extensive developments on nonparametric estimation techniques,there are few generally applicable methods for nonparametric inferences. Instatistical inferences, the likelihood ratio test is a useful method that is gener-ally applicable to most parametric hypothesis-testing problem. But the like-lihood ratio test is not applicable to the problem with nonparametric mod-els. First, nonparametric maximum likelihood estimators usually do not exist. Even when they exist, they are hard to compute. Generalized likelihood ratiotest is proposed by Fan,Zhang and Zhang(2001). Its basic idea is to find a suit-able estimate for nonparametric component g(t) under H0 and H1, respectively,and then to form a likelihood ratio statistic. A reasonable nonparametric es-timator of g(t) is the local likelihood estimator g(t). We use the local linearregression estimator of g(t) in this paper.
In this paper, we are concerned about two problems: The first problem iswhether it is worthy to regard g(t) as a nonparametric function. The contri-bution of this paper is to study whether g(t) is constant and whether g(t) islinear. The second one is whether two nonparametric functions in two di?er-ent partially linear models are same. It is a very interesting question. In thispaper, we focus on the statistical inferences for partially linear models whencovariate X and T are independent.
The model assume that the relation between between the response variableand the covariates can be represented as
where X are row p-vector covariate, the expectation of X is 0, T is scalar co-variate defined in (0,1), T is independent of X. The function g(·) is unknown,and model errorεhas normal distribution N(0,σ~2),εis independent of (X,T).The first contribution of this paper is to study the following hypothesis:withαis some unknown parameter.
We make the following assumptions:
Assumption 1.1. The nonparametric function g(·) has continuous sec-ond derivative.
Assumption 1.2. The kernel K(·) is symmetric and bounded in [?1,1]density function.
Assumption 1.3. As n→∞, h→0, nh→∞, and nh5 = O(1),where 0 < h < 21.
Assumption 1.4. The fourth moments of X exist. E(X) = 0, E(XTX) =Σ,Σis positive matrix.
Put sλ= ?11 uλK(u)du,ωλ= ?11 uλK2(u)du,λis the nonnegativeinteger.
Suppose we have a independent random sample of size n, (X1,T1,Y1),(X2,T2,Y2),···,(Xn,Tn,Yn) from model(1.1). When model errorεhas standard nor-mal distribution N(0,1), the logarithm of generalized likelihood ratio teststatistic,λn, is defined as
where RSS0 and RSS1 are the residual sum of squares under H0 and the wholespace. Putwhere K ? K(·) denote the convolution of K(·).
We now describe our result as follows.
Theorem 1.1. Suppose that Assumption 1.1～1.4 hold. Then underH0, as h→0,
When we study the following hypothesis under the same conditions as thefirst hypothesiswhereα0,α1 are two unknown parameters. We obtain the same result asTheorem1.1 .
If the variance of errorε,σ2, is unknown, we study whether it is regardg(t) as a nonparametric function. We test the linearity of the g(t) and definethe generalized likelihood ratio test statisticThe second main reslut of this paper is the following theorem:
Theorem 1.2. Suppose that Assumption 1.1～1.4 hold. Then underH0, as h→0,whereμn andσn are defined as above.
Remark 1.1. Whenβ= 0,μn andσn both are free of parameters. Theconclusion of this theorem coinside with Fan et al.(2001).
The other models we study areWhere X1 and X2 are row vector of p1-dimensional covariates and p2-dimensionalcovariates, respectively. Tl,l = 1,2 is scalar covariate, defined on (0,1). Thefunction gl(t),l = 1,2 is unknown. Xl and Tl are independent.εl is indepen-dent of (Xl,Tl) andεl has a normal distribution with mean 0 and varianceσl2 ,namely, N(0,σl2 ). Suppose that we have two independent random samples ofsize n, (X1i,T1i,Y1i),i = 1,···,n, from model(1.2), and (X2i,T2i,Y2i),i = 1,···,n, from model(1.3).Now we consider semiparametric hypothesis versus semiparametric hypothesistesting problem:
Assumption:
Assumption 1.5. The nonparametric function g(·) has continuous thirdderivative.
Assumption 1.6. The kernel K(·) is Lipschitz continuous, symmetricand bounded in [?1,1] density function.
Assumption 1.7. The fourth moments of Xl exists. E(Xl) = 0, E(XlT Xl) =Σl,Σl is positive matrix, l = 1,2.Denote where K~2(u)du,The third main result of this paper is the following theorem :
Theorem 1.3. Suppose that Assumption 1.3 and Assumption 1.5～1.7hold. Then under H0, as h→0,
where Rn0,μn andσn are defined in equations (1.4),(1.5) and (1.6).
Remark 1.2. Whenσ12 =σ22=σ2, thenμl,r =μl,l,l = 1,2,r≠l. We have The conclusion of theorem 1.3 coincide with the conclusion of theorem 1.2.Moreover, ifβl = 0,l = 1,2, the asymptotic null distribution are independentof parametric and nonparametric components. It is similar to the conclusionof Fan et al.(2001).
The last contribution of this paper is studying the following hypothesis:whereαis some unknown parameter. We derive the asymptotic property ofthe generalized likelihood ratio test statistic is similar to Theorem1.3. Onlyreplace Rn0 aswhere
Since Engle et al.(1986) proposed the partially linear models and ap-plied them to analyze the relation between electricity usage and average dailytemperature, the models have been extensively studied in literature, especiallythe estimation of the parametric and nonparametric components. Variousestimation of the parametric and nonparametric components have been pro-posed. Engle et al.(1986), Heckman(1986), and Rice(1986) developed spline-type methods by using smoothing spline; Speckman(1988), Opsomer and Rup-pert(1999) proposed kernel-type estimation. H¨ardle,Liang,Gao (2000) gave acomprehensive overview of statistical inference for the partially linear mod-els. The models have also been applied in economics(Scemalensee and Stoker1999), biometrics(Zeger and Diggle 1994, Liang et al. 2004).
Despite extensive developments on nonparametric estimation techniques,there are few generally applicable methods for nonparametric inferences. Instatistical inferences, the likelihood ratio test is a useful method that is gener-ally applicable to most parametric hypothesis-testing problem. But the like-lihood ratio test is not applicable to the problem with nonparametric mod-els. First, nonparametric maximum likelihood estimators usually do not exist. Even when they exist, they are hard to compute. Generalized likelihood ratiotest is proposed by Fan,Zhang and Zhang(2001). Its basic idea is to find a suit-able estimate for nonparametric component g(t) under H0 and H1, respectively,and then to form a likelihood ratio statistic. A reasonable nonparametric es-timator of g(t) is the local likelihood estimator g(t). We use the local linearregression estimator of g(t) in this paper.
In this paper, we are concerned about two problems: The first problem iswhether it is worthy to regard g(t) as a nonparametric function. The contri-bution of this paper is to study whether g(t) is constant and whether g(t) islinear. The second one is whether two nonparametric functions in two di?er-ent partially linear models are same. It is a very interesting question. In thispaper, we focus on the statistical inferences for partially linear models whencovariate X and T are independent.
The model assume that the relation between between the response variableand the covariates can be represented as
where X are row p-vector covariate, the expectation of X is 0, T is scalar co-variate defined in (0,1), T is independent of X. The function g(·) is unknown,and model errorεhas normal distribution N(0,σ~2),εis independent of (X,T).The first contribution of this paper is to study the following hypothesis:withαis some unknown parameter.
We make the following assumptions:
Assumption 1.1. The nonparametric function g(·) has continuous sec-ond derivative.
Assumption 1.2. The kernel K(·) is symmetric and bounded in [?1,1]density function.
Assumption 1.3. As n→∞, h→0, nh→∞, and nh5 = O(1),where 0 < h < 21.
Assumption 1.4. The fourth moments of X exist. E(X) = 0, E(XTX) =Σ,Σis positive matrix.
Put sλ= ?11 uλK(u)du,ωλ= ?11 uλK2(u)du,λis the nonnegativeinteger.
Suppose we have a independent random sample of size n, (X1,T1,Y1),(X2,T2,Y2),···,(Xn,Tn,Yn) from model(1.1). When model errorεhas standard nor-mal distribution N(0,1), the logarithm of generalized likelihood ratio teststatistic,λn, is defined as
where RSS0 and RSS1 are the residual sum of squares under H0 and the wholespace. Putwhere K ? K(·) denote the convolution of K(·).
We now describe our result as follows.
Theorem 1.1. Suppose that Assumption 1.1～1.4 hold. Then underH0, as h→0,
When we study the following hypothesis under the same conditions as thefirst hypothesiswhereα0,α1 are two unknown parameters. We obtain the same result asTheorem1.1 .
If the variance of errorε,σ2, is unknown, we study whether it is regardg(t) as a nonparametric function. We test the linearity of the g(t) and definethe generalized likelihood ratio test statisticThe second main reslut of this paper is the following theorem:
Theorem 1.2. Suppose that Assumption 1.1～1.4 hold. Then underH0, as h→0,whereμn andσn are defined as above.
Remark 1.1. Whenβ= 0,μn andσn both are free of parameters. Theconclusion of this theorem coinside with Fan et al.(2001).
The other models we study areWhere X1 and X2 are row vector of p1-dimensional covariates and p2-dimensionalcovariates, respectively. Tl,l = 1,2 is scalar covariate, defined on (0,1). Thefunction gl(t),l = 1,2 is unknown. Xl and Tl are independent.εl is indepen-dent of (Xl,Tl) andεl has a normal distribution with mean 0 and varianceσl2 ,namely, N(0,σl2 ). Suppose that we have two independent random samples ofsize n, (X1i,T1i,Y1i),i = 1,···,n, from model(1.2), and (X2i,T2i,Y2i),i = 1,···,n, from model(1.3).Now we consider semiparametric hypothesis versus semiparametric hypothesistesting problem:
Assumption:
Assumption 1.5. The nonparametric function g(·) has continuous thirdderivative.
Assumption 1.6. The kernel K(·) is Lipschitz continuous, symmetricand bounded in [?1,1] density function.
Assumption 1.7. The fourth moments of Xl exists. E(Xl) = 0, E(XlT Xl) =Σl,Σl is positive matrix, l = 1,2.Denote where K~2(u)du,The third main result of this paper is the following theorem :
Theorem 1.3. Suppose that Assumption 1.3 and Assumption 1.5～1.7hold. Then under H0, as h→0,
where Rn0,μn andσn are defined in equations (1.4),(1.5) and (1.6).
Remark 1.2. Whenσ12 =σ22=σ2, thenμl,r =μl,l,l = 1,2,r≠l. We have The conclusion of theorem 1.3 coincide with the conclusion of theorem 1.2.Moreover, ifβl = 0,l = 1,2, the asymptotic null distribution are independentof parametric and nonparametric components. It is similar to the conclusionof Fan et al.(2001).
The last contribution of this paper is studying the following hypothesis:whereαis some unknown parameter. We derive the asymptotic property ofthe generalized likelihood ratio test statistic is similar to Theorem1.3. Onlyreplace Rn0 aswhere
引文
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[5] Carroll, R.J., Fan, J., Gijbels, I. and Wand, M.P. Generalized Partially Linear Single-Index Models. Journal of American Statistical Association, 1997, 92: 477-489.
[6] Chen Hung. Convergence Rate for Parametric Components in a Partly LinearModel.Ann.Stat. 1988, 16(1), 136-146.
[7] Chiu, S.T. Why Do Bandwidth Selectors Tend to Choose Smaller Bandwidths and aRemedy. Biometrika, 1990, 77: 222-226.
[8] Choi, Sungsub; Hall, W. J.; Schick, Anton. Asymptotically Uniformly Most PowerfulTests in Parametric and Semiparametric Models. Ann. Statist. 1996, 24, no. 2: 841–861.
[9] Choi, Suktae; Jin, Zhezhen; Ying, Zhiliang. Goodness-of-fit Tests for SemiparametricModels with Multiple Event-time Data. Statist. Sinica. 2001, 11, no. 3, 723–736.
[10] Cozalo,P. A Consistent Model Specification Test for Nonparametric Estimation of Re-gression Function Models. Econometric Theory, 1993, 9, 451-477.
[11] Cuzick,J. Semiparametric Additive Regression. Journal of the Royal Statistical Society.Series B, 1992a, 54: 831-843.
[12] Cuzick,J. E?cient Estimates in Semiparametric Additive Regression Models with Un-known Error Distribution. Annals of Statistic, 1992b, 20: 1129-1136.
[13] De Jong,P. A Central Limit Theorem for Generalized Quadratic Forms. Probab. TheoryRelated Fields. 1987, 75: 261-277.
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[15] Eubank, R. L.; Hart, J. D.; Speckman, Paul Trigonometric Series Regression Estimatorswith an Application to Partially Linear Models. J. Multivariate Anal. 1990, 32: no. 1,70–83.
[16] Fan, J. Local Linear Regression Smoothers and Their Minimax E?ciency. Annals ofStatistics, 1993, 21: 196-216.
[17] Fan, J., Zhang, C. and Zhang, J. Generalized Likelihood Ratio Statistics and WilksPhenomenon. The Annals of Statistics, 2001, 29: 153-193.
[18] Fan, J. and Huang, T. Profile Likelihood Inferences on Semiparametric Varying-coe?cient Partially Linear Models. Bernoulli, 2005, 11: 1031-1057
[19] 高集体. 半参数回归模型中估计的大样本理论. 中国科技大学研究生院(合肥). 博士论文, 1992.
[20] 高集体,赵林城. 部分线性回归模型中的自适应估计. 中国科学, 1992A, 8, 791-803.
[21] Golubev,G. and H¨ardle,W. On Adaptive in Partial Linear Models. Disscussion Paper,No 371. Weierstresse-Institut fu¨r Angewandte Analysis und Stochastik zu Berlin. 1997.
[22] Gozalo,P. A Consistent Model Specification Test for Nonparametric Estimation of Re-gression Function Models. Econometric Theory, 1993, 9, 451-477.
[23] Hall,P., Marron,J.S. and Park,B.U. Smoothed Cross-Validation. Probability Theoryand Related Fields, 1992, 92: 1-20.
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