非参数模型中变点的检测及删失数据中删失指标随机缺失下回归函数的估计
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摘要
变点问题是统计研究中的热门话题之一。变点检测不仅广泛地应用于工业质量控制领域,而且在金融、经济、医学、计算机等领域也得到了越来越广泛的应用。论文运用小波方法研究了方差模型和危险率模型中的变点问题。
首先,在第二章中我们考虑了非参数回归模型中条件方差的变点检测与估计问题。在观测数据是α-混合序列的假设下,我们运用小波方法构造了一个检验量来检验某一个给定的点是否是条件方差的变点,并且建立了检验量的渐近分布。我们构造了变点的个数、位置以及跳跃幅度的估计量,并且给出了这些估计量的渐近性质。进一步,我们给出了有限样本下数值模拟的结果。
在第三章中,我们考虑了危险率函数的变点问题。在观测是右删失数据的假设下,我们运用小波方法构造了一个检验量来检验某一个给定的点是否是危险率函数的变点,并且建立了检验量的渐近分布。我们构造了变点的个数、位置以及跳跃幅度的估计量,并且给出了这些估计量的渐近性质。有限样本下数值模拟的结果表明这些估计量具有良好的表现,同时用所提方法来分析斯坦福心脏移植数据。
在第四章中,我们将所提方法扩展到了观测为α-混合的右删失数据时的情况,并且建立了检验量的渐近分布以及估计量的渐近性质。我们给出了α-混合的有限样本下数值模拟的结果。
另外,第五章研究了在删失数据中删失指标随机缺失的情况下,回归函数的估计问题。我们运用非参数方法以及倒概率思想给出了回归函数的两种估计量,并且给出了估计量的一致收敛速度以及渐近分布。同时,我们给出了有限样本下数值模拟的结果。
The change point problem is one of the hot topics in statistical research. Change point detection is not only widely used in the industrial field of quality control, but also used in the financial, economic, medical, computer and other fields. In this paper, wavelet methods are adopted to study the detection and estimation of change points in variance models and hazard rate models.
Firstly, in chapter II, we study the detection and estimation of change points in volatility under nonparametric regression models with a-mixing observations. Wavelet methods are applied to construct the test statistic to detect change points in volatility. The asymptotic distribution of the test statistic is established. We also utilize the test statistic to construct the estimators for the number, locations, and jump sizes of the change points in volatility. The asymptotic properties of these estimators are derived. Some simulation studies are conducted to assess the finite sample performances of the proposed procedures.
In Chapter III, we study the detection and estimation of change points in hazard rate models with censored data. Wavelet methods are used to construct test statistic. The asymptotic distribution of the test statistic is explored. We also propose estimators for the number, locations, and jump sizes of the change points in hazard rate. The asymptotic properties of these estimators are derived. Some simulation examples are conducted to assess the finite sample performances of our methods and a real data example is provided.
In Chapter IV, we extend our methods to the case that the censored ob-servations are a-mixing. The asymptotic properties of the test statistic and the estimators are established. Some simulation examples are conducted to assess the finite sample performances of our methods in α-mixing censored case.
Finally, In Chapter V, we consider the estimation of regression function when the censoring indicator is missing at random. We use nonpar ametric technique and inverse probability weighted method to define two kernel estimators for the regression function. We establish the strong uniform convergence with rates and the asymptotic normality of our estimators. Some simulations are conducted to assess the finite sample performances of our methods.
首先,在第二章中我们考虑了非参数回归模型中条件方差的变点检测与估计问题。在观测数据是α-混合序列的假设下,我们运用小波方法构造了一个检验量来检验某一个给定的点是否是条件方差的变点,并且建立了检验量的渐近分布。我们构造了变点的个数、位置以及跳跃幅度的估计量,并且给出了这些估计量的渐近性质。进一步,我们给出了有限样本下数值模拟的结果。
在第三章中,我们考虑了危险率函数的变点问题。在观测是右删失数据的假设下,我们运用小波方法构造了一个检验量来检验某一个给定的点是否是危险率函数的变点,并且建立了检验量的渐近分布。我们构造了变点的个数、位置以及跳跃幅度的估计量,并且给出了这些估计量的渐近性质。有限样本下数值模拟的结果表明这些估计量具有良好的表现,同时用所提方法来分析斯坦福心脏移植数据。
在第四章中,我们将所提方法扩展到了观测为α-混合的右删失数据时的情况,并且建立了检验量的渐近分布以及估计量的渐近性质。我们给出了α-混合的有限样本下数值模拟的结果。
另外,第五章研究了在删失数据中删失指标随机缺失的情况下,回归函数的估计问题。我们运用非参数方法以及倒概率思想给出了回归函数的两种估计量,并且给出了估计量的一致收敛速度以及渐近分布。同时,我们给出了有限样本下数值模拟的结果。
The change point problem is one of the hot topics in statistical research. Change point detection is not only widely used in the industrial field of quality control, but also used in the financial, economic, medical, computer and other fields. In this paper, wavelet methods are adopted to study the detection and estimation of change points in variance models and hazard rate models.
Firstly, in chapter II, we study the detection and estimation of change points in volatility under nonparametric regression models with a-mixing observations. Wavelet methods are applied to construct the test statistic to detect change points in volatility. The asymptotic distribution of the test statistic is established. We also utilize the test statistic to construct the estimators for the number, locations, and jump sizes of the change points in volatility. The asymptotic properties of these estimators are derived. Some simulation studies are conducted to assess the finite sample performances of the proposed procedures.
In Chapter III, we study the detection and estimation of change points in hazard rate models with censored data. Wavelet methods are used to construct test statistic. The asymptotic distribution of the test statistic is explored. We also propose estimators for the number, locations, and jump sizes of the change points in hazard rate. The asymptotic properties of these estimators are derived. Some simulation examples are conducted to assess the finite sample performances of our methods and a real data example is provided.
In Chapter IV, we extend our methods to the case that the censored ob-servations are a-mixing. The asymptotic properties of the test statistic and the estimators are established. Some simulation examples are conducted to assess the finite sample performances of our methods in α-mixing censored case.
Finally, In Chapter V, we consider the estimation of regression function when the censoring indicator is missing at random. We use nonpar ametric technique and inverse probability weighted method to define two kernel estimators for the regression function. We establish the strong uniform convergence with rates and the asymptotic normality of our estimators. Some simulations are conducted to assess the finite sample performances of our methods.
引文
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[3]Bouezmarni, T. and Rombouts, J.V.K. (2008). Density and hazard rate estima-tion for censored and a-mixing data using gamma kernels [J]. Journal of Non-parametric Statistics.
[4]Cai, Z.W. (1998). Asymptotic properties of Kaplan-Meier estimator for censored dependent data [J]. Statistics and Probability Letters,37,381-389.
[5]Carrasco, M., Chernovb, M., Florensc, J. and Ghyselsd, E. (2007). Efficient esti-mation of general dynamic models with a continuum of moment conditions [J]. Journal of Econometrics,140,529-573.
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[10]Chen, G., Choi, Y. and Zhou, Y. (2008). Detections of changes in return by a wavelet smoother with conditional heteroscedastic volatility [J]. Journal of Econo-metrics,143,227-262.
[11]Chen, S., Dahl, G. and Khan, S. (2005). Nonparametric Identification and Estima-tion of a Censored Location-Scale Regression Model [J]. Journal of the American Statistical Association,100,212-221.
[12]Davydov, Y. (1968). Convergence of distributions generated by stationary stochastic process [J]. Theory of Probability and Its Applications,13,691-696.
[13]Dikta, D. (1998). On semiparametric random censorship models [J]. Journal of Statistical Planning and Inference,66,253-279.
[14]Doukhan, P. (1994). Mixing:Properties and Examples [M]. vol.85. Springer-Verlag, New York.
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[17]Fan, J. and Gijbels, I. (1996) Local Polynomial Modeling and its Applications, Chapman and Hall, London.
[18]Gijbels, I. and Gurler, U. (2003). Estimation of a change point in a hazard function based on censored data [J]. Lifetime Data Analysis,9,395-411.
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[22]Jiang, J.C. and Doksum, K. (2003). On local polynomial estimation of hazard rates and their derivatives under random censoring [J]. Lecture Notes-Monograph Series,42,463-481.
[23]Jorion, P. (1988). On jump processes in the foreign exchange and stock markets [J]. Review of Financial Studies,1,427-445.
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[25]Lamoreux, C.G. and Lastrapes, W.D. (1990). Persistence in variance, structural change and the GARCH model [J]. Journal of Business and Economic Statistics, 48,225-234.
[26]Li, Y. and Xie, Z. (1999). The wavelet identification of thresholds and time delay of threshold autoregressive models [J]. Statistica Sinica,9,153-166.
[27]Liang, H., Mammitzsch, V. and Steinebach, J. (2005). Nonlinear wavelet density and hazard rate estimation for censored data under dependent observations [J]. Statistics and Decisions,23,161-180.
[28]Liebscher, E. (2002). Kernel density and hazard rate estimation for censored data under α-mixing condition [J]. Annals of The Institute of Statistical Mathematics, 54,19-28.
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[32]Lu, W. and Liang, Y. (2008). Analysis of competing risks data with missing cause of failure under additive hazards model [J]. Statistica Sinica,18,219-234.
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[35]Miiller, H.G. and Wang, J.L. (1990). Nonparametric analysis of changes in hazard rates for censored survival data:an alternative to change-point problems [J]. Biometrika,77,305-314.
[36]Miiller, T.G. (1992). Change points in nonparametric regression analysis [J]. The Annals of Statistics,20,737-761.
[37]Muller, H.G. and Wang, J.L. (1994). Hazard rate estimation under random cen-soring with varying kernels and bandwidths [J]. Biometrics,50,61-76.
Nason, G.P. (2008). Wavelet methods in statistics with R. Springer, New York.
Nadaraya, E. A. (1964) On estimating regression [J]. Theory of Probability and its Applications,9,141-142.
[40]Nguyen, H.T., Rogers, G.S. and Walker, E.A. (1984). Estimation in change-point hazard rate model [J]. Biometrika,71,299-304.
[41]Ogden, T. (1996). Essential Wavelets for Statistical Applications and Data Anal-ysis. Birkhauser.
[42]Pagan, A.R. and Schwert, G.W. (1990). Alternative models for conditional stock volatility [J]. Journal of Econometrics,45,267-290.
[43]Patil, P. (1997). Nonparametric hazard rate estimation by orthogonal wavelet methods [J]. Journal of Statistical Planning and Inference,60,153-168.
[44]Patil, P.N. and Wood, A.T. (2004). Counting process intensity estimation by orthogonal wavelet methods [J]. Bernoulli,10,1-24.
[45]Perron, B. (2001). Jumps in the volatility of financial markets. Working Paper, University of Montreal.
[46]Raimondo, M. and Tajvidi, N. (2004). A peaks over threshold model for change-point detection by wavelets [J]. Statistica Sinica,14,395-412.
[47]Ramlau-Hansen, H. (1983). Smoothing counting process intensities by means of kernel functions [J]. The Annals of Statistics,11,453-466.
Roussas, G.G. (1990). Asymptotic normality of the kernel estimate under depen-dence conditions:application to hazard rate [J]. Journal of Statistical Planning and Inference,25,81-104.
[49]Roussas, G.G. and Ioannides (1987). Moment inequalities for mixing sequences of random variables [J]. Stochastic Analysis and Applications,5,61-120.
[50]Stuart, L. and Joseph, G.I. (2000). Estimation with correlated censored survival data with missing covariates [J]. Biostatistics,1,315-327.
[51]Stute, W. (1982). Law of logarithm for kernel density estimators [J]. Annals of Probability,10,414-422.
[52]Tanner, M. and Wong, W.H. (1983). The estimation of the hazard function from randomly censored data by the kernel method [J]. The Annals of Statistics,11, 989-993.
[53]Vlaar, P. and Palm, F. (1993). The message in weekly exchange rates in the European monetary system:mean revision, conditional heteroscedasticity, and jumps [J]. Journal of Business and Economic Statistics,11,351-360.
[54]Volkonskii, V. A. and Rozanov, Y. A. (1959). Some limit theorems for random functions, I [J]. Theory of Probability and Its Applications,4,178-197.
[55]Van der Laan, M.J. and McKeague, I.W. (1998). Efficient estimation from right-censored data when failure indicators are missing at random [J].Annals of Statis-tics,26,164-182.
[56]Wang, Q.H. and Dinse, G.E. (2011). Linear regression analysis of survival data with missing censoring indicators [J]. Lifetime Data Analysis,17,256-279.
[57]Wang, Q.H. and Ng, K.W. (2008). Asymptotically efficient product-limit esti-mators with censoring indicators missing at random [J]. Statistica Sinica,18, 749-768.
[58]Wang, Y. (1995). Jump and Sharp cusp detection by wavelets [J]. Biometrika, 82,385-397.
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[67]Zhou, Y., Wan, A.T.K, Xie, S. and Wang, X. (2010). Wavelet analysis of change-points in a non-parametric regression with heteroscedastic variance [J]. Journal of Econometrics,159,183-201.
[2]Bijwaard, G. (2004). Dymamic economoc aspect of migration [J]. Medium econometrische toepassingen,1,26-30.
[3]Bouezmarni, T. and Rombouts, J.V.K. (2008). Density and hazard rate estima-tion for censored and a-mixing data using gamma kernels [J]. Journal of Non-parametric Statistics.
[4]Cai, Z.W. (1998). Asymptotic properties of Kaplan-Meier estimator for censored dependent data [J]. Statistics and Probability Letters,37,381-389.
[5]Carrasco, M., Chernovb, M., Florensc, J. and Ghyselsd, E. (2007). Efficient esti-mation of general dynamic models with a continuum of moment conditions [J]. Journal of Econometrics,140,529-573.
[6]Chang, I.S., Chen, C.H. and Hsiung, C.A. (1994). Estimation in change-point hazard rate models with random censorship [J]. Lecture Notes-Monograph Series, 23,78-92.
[7]Chen, X. (2009). Change point analysis of survival data with application in clinical trials. Ph.D. Dissertation. The university of Texas at Dallas.
[8]Chen, F. (2011). Maximum local partial likelihood estimators for the counting process intensity function and its derivatives [J]. Statistica Sinica,21,107-128.
[9]Chen, G., Choi, Y. and Zhou, Y. (2005). Nonparametric estimation of structural change points in volatility models for time series [J]. Journal of Econometrics, 126,79-114.
[10]Chen, G., Choi, Y. and Zhou, Y. (2008). Detections of changes in return by a wavelet smoother with conditional heteroscedastic volatility [J]. Journal of Econo-metrics,143,227-262.
[11]Chen, S., Dahl, G. and Khan, S. (2005). Nonparametric Identification and Estima-tion of a Censored Location-Scale Regression Model [J]. Journal of the American Statistical Association,100,212-221.
[12]Davydov, Y. (1968). Convergence of distributions generated by stationary stochastic process [J]. Theory of Probability and Its Applications,13,691-696.
[13]Dikta, D. (1998). On semiparametric random censorship models [J]. Journal of Statistical Planning and Inference,66,253-279.
[14]Doukhan, P. (1994). Mixing:Properties and Examples [M]. vol.85. Springer-Verlag, New York.
[15]Drost, F., Nijman, T. and Werker, B. (1998). Estimation and testing in models containing both jumps and conditional heteroscedasticity [J]. Journal of Business and Economic Statistics,16,237-243.
[16]Fan, J. and Gijbels, I. (1994). Censored regression:local linear approximations and their applications [J]. Journal of the American Statistical Association,89, 560-570.
[17]Fan, J. and Gijbels, I. (1996) Local Polynomial Modeling and its Applications, Chapman and Hall, London.
[18]Gijbels, I. and Gurler, U. (2003). Estimation of a change point in a hazard function based on censored data [J]. Lifetime Data Analysis,9,395-411.
[19]Guessoum, Z. and Ould-Said, E. (2008). On nonparametric estimation of the regression function under random censorship model [J]. Statistics and Decisions, 26,159-177.
[20]Ibragimov, I. A. and Linnik, Yu. A. (1971). Independent and Stationary Sequences of Random Variables, Wolters-Noordohoff, Groningen, Germany
[21]Inclan, C. and Tiao, G.C. (1994). Use of cumulative sums of squares for ret-rospective detection of change of variance, Journal of the American Statistical Association [J].89,913-923.
[22]Jiang, J.C. and Doksum, K. (2003). On local polynomial estimation of hazard rates and their derivatives under random censoring [J]. Lecture Notes-Monograph Series,42,463-481.
[23]Jorion, P. (1988). On jump processes in the foreign exchange and stock markets [J]. Review of Financial Studies,1,427-445.
[24]Kohler, M. (1997). On the universal consistency of a least squares spline regression estimator [J]. Mathematical Methods of Statistics,6,349-364.
[25]Lamoreux, C.G. and Lastrapes, W.D. (1990). Persistence in variance, structural change and the GARCH model [J]. Journal of Business and Economic Statistics, 48,225-234.
[26]Li, Y. and Xie, Z. (1999). The wavelet identification of thresholds and time delay of threshold autoregressive models [J]. Statistica Sinica,9,153-166.
[27]Liang, H., Mammitzsch, V. and Steinebach, J. (2005). Nonlinear wavelet density and hazard rate estimation for censored data under dependent observations [J]. Statistics and Decisions,23,161-180.
[28]Liebscher, E. (2002). Kernel density and hazard rate estimation for censored data under α-mixing condition [J]. Annals of The Institute of Statistical Mathematics, 54,19-28.
Loeve, M. (1991). Probability Theory,4th ed. Springer, New York.
[30]Loader, C.R. (1991). Inference for a hazard rate change point [J]. Biometrika,78, 749-757.
[31]Luan. Y. and Xie, Z. (2001). The wavelet identification for jump points of deriva-tive in regression model [J]. Statistics and Probability Letters,53,167-180.
[32]Lu, W. and Liang, Y. (2008). Analysis of competing risks data with missing cause of failure under additive hazards model [J]. Statistica Sinica,18,219-234.
[33]Matthews, D.E. and Farewell, V.T. (1982). On testing for a constant against a change-point alternative [J]. Biometrics,38,463-468.
[34]Miller, R. and Halpern, J. (1982). Regression with censored data [J]. Biometrika, 69,521-531.
[35]Miiller, H.G. and Wang, J.L. (1990). Nonparametric analysis of changes in hazard rates for censored survival data:an alternative to change-point problems [J]. Biometrika,77,305-314.
[36]Miiller, T.G. (1992). Change points in nonparametric regression analysis [J]. The Annals of Statistics,20,737-761.
[37]Muller, H.G. and Wang, J.L. (1994). Hazard rate estimation under random cen-soring with varying kernels and bandwidths [J]. Biometrics,50,61-76.
Nason, G.P. (2008). Wavelet methods in statistics with R. Springer, New York.
Nadaraya, E. A. (1964) On estimating regression [J]. Theory of Probability and its Applications,9,141-142.
[40]Nguyen, H.T., Rogers, G.S. and Walker, E.A. (1984). Estimation in change-point hazard rate model [J]. Biometrika,71,299-304.
[41]Ogden, T. (1996). Essential Wavelets for Statistical Applications and Data Anal-ysis. Birkhauser.
[42]Pagan, A.R. and Schwert, G.W. (1990). Alternative models for conditional stock volatility [J]. Journal of Econometrics,45,267-290.
[43]Patil, P. (1997). Nonparametric hazard rate estimation by orthogonal wavelet methods [J]. Journal of Statistical Planning and Inference,60,153-168.
[44]Patil, P.N. and Wood, A.T. (2004). Counting process intensity estimation by orthogonal wavelet methods [J]. Bernoulli,10,1-24.
[45]Perron, B. (2001). Jumps in the volatility of financial markets. Working Paper, University of Montreal.
[46]Raimondo, M. and Tajvidi, N. (2004). A peaks over threshold model for change-point detection by wavelets [J]. Statistica Sinica,14,395-412.
[47]Ramlau-Hansen, H. (1983). Smoothing counting process intensities by means of kernel functions [J]. The Annals of Statistics,11,453-466.
Roussas, G.G. (1990). Asymptotic normality of the kernel estimate under depen-dence conditions:application to hazard rate [J]. Journal of Statistical Planning and Inference,25,81-104.
[49]Roussas, G.G. and Ioannides (1987). Moment inequalities for mixing sequences of random variables [J]. Stochastic Analysis and Applications,5,61-120.
[50]Stuart, L. and Joseph, G.I. (2000). Estimation with correlated censored survival data with missing covariates [J]. Biostatistics,1,315-327.
[51]Stute, W. (1982). Law of logarithm for kernel density estimators [J]. Annals of Probability,10,414-422.
[52]Tanner, M. and Wong, W.H. (1983). The estimation of the hazard function from randomly censored data by the kernel method [J]. The Annals of Statistics,11, 989-993.
[53]Vlaar, P. and Palm, F. (1993). The message in weekly exchange rates in the European monetary system:mean revision, conditional heteroscedasticity, and jumps [J]. Journal of Business and Economic Statistics,11,351-360.
[54]Volkonskii, V. A. and Rozanov, Y. A. (1959). Some limit theorems for random functions, I [J]. Theory of Probability and Its Applications,4,178-197.
[55]Van der Laan, M.J. and McKeague, I.W. (1998). Efficient estimation from right-censored data when failure indicators are missing at random [J].Annals of Statis-tics,26,164-182.
[56]Wang, Q.H. and Dinse, G.E. (2011). Linear regression analysis of survival data with missing censoring indicators [J]. Lifetime Data Analysis,17,256-279.
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