半参模型的经验似然推断
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摘要
本文研究了经验似然方法在负相伴和删失数据下半参模型的统计推断.主要包含了以下几方面内容.
其一,我们在带有负相伴随机误差条件下的部分线性模型中,通过使用大小分块的经验似然方法,给出了基于估计方程的经验似然比统计量,并证明了该统计量的渐近分布是自由度和未知参数维度相同的卡方分布.我们可以通过该结果建立未知参数的置信域,以及进行假设检验.最后,我们通过模拟说明了分块经验似然比普通经验似然的置信域的覆盖率更高.
其二,我们对带有负相伴随机误差的单指标回归模型进行大样本统计推断.我们使用的得分函数是经过偏差矫正的.正是由于偏差矫正,使得我们构造的经验似然比统计量收敛于一个具有标准卡方分布的随机变量,而不是未经过偏差矫正的卡方分布的加权和形式.通过我们的结果可以对单指标模型中的指标进行置信域的估计.
其三,对于生存分析中经常遇到的删失数据,我们在随机右删失的情形下,针对部分线性模型进行了统计推断.基于Buckley和James估计量,我们构造了得分函数.通过将得分函数的鞅表示,利用计数过程鞅和Doob鞅的关系式,我们将得分函数表示为计数过程鞅的部分和.利用线性秩统计量的一般结论,结合Rebolledo鞅的中心极限定理,我们证明了经验似然比统计量的渐近分布是卡方分布.该结果可以用于构造未知参数的置信域.我们同时做了模拟,从数据上说明了BJ估计量优于KSV估计量的结论.
This dissertation studies statistical inferences on semiparametric models based on either negatively associated random errors or censored data based on empirical likelihood method. The main contents include the following three as-pects.
Firstly, we obtained empirical likelihood ratio statistics based on estimating equations for partial linear models under negatively associated random errors. Using the blockwise empirical likelihood with large block and small block, we proved the statistics is asymptotically chi-squared distributed with freedom equal to the dimension of the parameter of interest. We can build the confidence region for the parameter using this theorem, or making hypothesis tests on the parameter. Finally, we conducted a small sample simulation. The simulation results showed that the blockwise empirical likelihood are more attractable than the ordinary empirical likelihood in terms of higher coverage probability of the confidence region.
Secondly, we made large sample statistical inferences on single-index re-gression models with negatively associated random errors. The score function we use is bias corrected. It is because of the bias correction, that the empirical likelihood ratio statistics we constructed converges to a standard chi-squared dis-tributed random variable, instead of the weighted sum of chi-squared distribution form due to the score function without bias correction. By our conclusion, the estimation of the confidence region of the index in the single-index model can be conducted.
Thirdly, we deal with the censored data often appeared in the survival anal-ysis. We made statistical inferences on the partial linear model under random right censoring. We constructed the score function based on the Buckley-James estimator. Using the martingale representation of the score function, we can rewrite the score function as a partial sum of the counting process martingale by means of the relationship between the counting process martingale and the Doob's expectation martingale. Utilizing the common theory of the linear rank statistics and the Rebolledo martingale central limit theorem, we proved the em-pirical likelihood ratio statistics has asymptotic chi-squared distribution. This conclusion can be used to construct the confidence region of the parameter of in-terest. We also conducted a simulation which illustrated that the Buckley-James estimator is superior to the KSV estimator.
其一,我们在带有负相伴随机误差条件下的部分线性模型中,通过使用大小分块的经验似然方法,给出了基于估计方程的经验似然比统计量,并证明了该统计量的渐近分布是自由度和未知参数维度相同的卡方分布.我们可以通过该结果建立未知参数的置信域,以及进行假设检验.最后,我们通过模拟说明了分块经验似然比普通经验似然的置信域的覆盖率更高.
其二,我们对带有负相伴随机误差的单指标回归模型进行大样本统计推断.我们使用的得分函数是经过偏差矫正的.正是由于偏差矫正,使得我们构造的经验似然比统计量收敛于一个具有标准卡方分布的随机变量,而不是未经过偏差矫正的卡方分布的加权和形式.通过我们的结果可以对单指标模型中的指标进行置信域的估计.
其三,对于生存分析中经常遇到的删失数据,我们在随机右删失的情形下,针对部分线性模型进行了统计推断.基于Buckley和James估计量,我们构造了得分函数.通过将得分函数的鞅表示,利用计数过程鞅和Doob鞅的关系式,我们将得分函数表示为计数过程鞅的部分和.利用线性秩统计量的一般结论,结合Rebolledo鞅的中心极限定理,我们证明了经验似然比统计量的渐近分布是卡方分布.该结果可以用于构造未知参数的置信域.我们同时做了模拟,从数据上说明了BJ估计量优于KSV估计量的结论.
This dissertation studies statistical inferences on semiparametric models based on either negatively associated random errors or censored data based on empirical likelihood method. The main contents include the following three as-pects.
Firstly, we obtained empirical likelihood ratio statistics based on estimating equations for partial linear models under negatively associated random errors. Using the blockwise empirical likelihood with large block and small block, we proved the statistics is asymptotically chi-squared distributed with freedom equal to the dimension of the parameter of interest. We can build the confidence region for the parameter using this theorem, or making hypothesis tests on the parameter. Finally, we conducted a small sample simulation. The simulation results showed that the blockwise empirical likelihood are more attractable than the ordinary empirical likelihood in terms of higher coverage probability of the confidence region.
Secondly, we made large sample statistical inferences on single-index re-gression models with negatively associated random errors. The score function we use is bias corrected. It is because of the bias correction, that the empirical likelihood ratio statistics we constructed converges to a standard chi-squared dis-tributed random variable, instead of the weighted sum of chi-squared distribution form due to the score function without bias correction. By our conclusion, the estimation of the confidence region of the index in the single-index model can be conducted.
Thirdly, we deal with the censored data often appeared in the survival anal-ysis. We made statistical inferences on the partial linear model under random right censoring. We constructed the score function based on the Buckley-James estimator. Using the martingale representation of the score function, we can rewrite the score function as a partial sum of the counting process martingale by means of the relationship between the counting process martingale and the Doob's expectation martingale. Utilizing the common theory of the linear rank statistics and the Rebolledo martingale central limit theorem, we proved the em-pirical likelihood ratio statistics has asymptotic chi-squared distribution. This conclusion can be used to construct the confidence region of the parameter of in-terest. We also conducted a simulation which illustrated that the Buckley-James estimator is superior to the KSV estimator.
引文
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[2]Chen, H. (1988). Convergence rates for parametric components in a partly linear model. The Annals of Statistics,136-146.
[3]Chen, S. X. (1993). On the accuracy of empirical likelihood confidence re-gions for linear regression model. Annals of the Institute of Statistical Math-ematics,45,621-637.
[4]Chen, S. X. (1994). Empirical likelihood confidence intervals for linear re-gression coefficients. Journal of Multivariate Analysis,49,24-40.
[5]Chen, S.X. (1996). Empirical likelihood confidence intervals for nonparamet-ric density estimation. Biometrika,83,329-341.
[6]Chen, S. X., Cui, H. (2003). An extended empirical likelihood for generalized linear models. Statistica Sinica,13,69-82.
[7]Chen, S. X., Cui, H. (2006). On Bartlett correction of empirical likelihood in the presence of nuisance parameters. Biometrika,93,215-220.
[8]Chen, S. X., Cui, H. (2007). On the second-order properties of empirical likelihood with moment restrictions. Journal of Econometrics,141,492-516.
[9]Chen, S. X., Hall, P. (1993). Smoothed empirical likelihood confidence in-tervals for quantiles. The Annals of Statistics,1166-1181.
[10]Chen, S. X., Wong, C. M. (2009). Smoothed block empirical likelihood for quantiles of weakly dependent processes. Statistica Sinica,19,71.
[11]Chen, S. X., Van Keilegom, I. (2009). A review on empirical likelihood meth-ods for regression. Test,18.415-447.
[12]Cox, D. R. (1972). Regression models and life-tables. Journal of the Royal Statistical Society. Series B (Methodological),187-220.
[13]DiCiccio, T., Hall, P., Romano, J. (1991). Empirical likelihood is Bartlett-correctable. The Annals of Statistics,19,1053-1061.
[14]Duan, N., Li, K. C. (1991). Slicing regression:a link-free regression method. The Annals of Statistics,505-530.
[15]Engle, R. F., Granger, C. W., Rice, J., Weiss, A. (1986). Semiparametric estimates of the relation between weather and electricity sales. Journal of the American statistical Association,81,310-320.
[16]Esary, J. D., Proschan, F., Walkup, D. W. (1967). Association of random variables, with applications. The Annals of Mathematical Statistics,38 1466-1474.
[17]Eubank, R. L., Hart, J. D., Speckman, P. (1990). Trigonometric series re-gression estimators with an application to partially linear models. Journal of multivariate analysis,32,70-83.
[18]Fang, K.T, Li, G., Lu, X. Y. Qin, H. (2013). An Empirical Likelihood Method for Semiparametric Linear Regression with Right Censored Data. Computational and Mathematical Methods in Medicine,2013.
[19]Hall, P. (1990). Pseudo-likelihood theory for empirical likelihood. The An-nals of Statistics,121-140.
[20]Hall, P., La Scala, B. (1990). Methodology and algorithms of empirical like-lihood. International Statistical Review/Revue Internationale de Statistique, 109-127.
[21]Hardle, W., Hall, P., Ichimura, H. (1993). Optimal smoothing in single-index models. The Annals of Statistics,21,157-178.
[22]Hardle, W., Liang, H. (2000). J. Gao (2000):Partially linear models. Physica-Verlag, Heidelberg.
[23]Hardle, W., Tsybakov, A. B. (1993). How sensitive are average derivatives?. Journal of Econometrics,58,31-48.
[24]Heckman, N. E. (1986). Spline smoothing in a partly linear model. Journal of the Royal Statistical Society. Series B (Methodological),244-248.
[25]Hong, S. Y. (1991). Estimation theory of a class of semiparametric regression models. Sciences in China Ser. A,12,1258-1272.
[26]Hristache, M., Juditsky, A., Spokoiny, V. (2001). Direct estimation of the index coefficient in a single-index model. The Annals of Statistics,29,593-623.
[27]Hsing, T., Carroll, R. J. (1992). An asymptotic theory for sliced inverse regression. The Annals of Statistics,1040-1061.
[28]Ichimura, H. (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. Journal of Econometrics,58,71-120.
[29]Joag-Dev, K., Proschan, F. (1983). Negative association of random variables with applications. The Annals of Statistics,11,286-295.
[30]Kaplan, E. L., Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American statistical association,53,457-481.
[31]Kim, T., Kim, H. (2007). On the exponential inequality for negative depen-dent sequence. COMMUNICATIONS-KOREAN MATHEMATICAL SOCI-ETY,22,315.
[32]Kitamura, Y. (1997). Empirical likelihood methods with weakly dependent processes. The Annals of Statistics,25,2084-2102.
[33]Kolaczyk, E. D. (1994). Empirical likelihood for generalized linear models. Statistica Sinica,4,199-218.
[34]Koul, H., Susarla, V., Van Ryzin, J. (1981). Regression analysis with ran-domly right-censored data. The Annals of Statistics,1276-1288.
[35]Lai, T. L., Ying, Z. (1991a). Large sample theory of a modified Buckley-James estimator for regression analysis with censored data. The Annals of Statistics,19,1370-1402.
[36]Lai, T. L., Ying, Z. (1991). Rank regression methods for left-truncated and right-censored data. The Annals of Statistics,19,531-556.
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