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研究生:黃名鉞
研究生(外文):Ming-Yueh Huang
論文名稱:針對不同維度縮減模型之半參數化估計方法
論文名稱(外文):Semiparametric Estimation Approaches for Variant Dimension Reduction Models
指導教授:江金倉江金倉引用關係
指導教授(外文):Chin-Tsang Chiang
口試委員:姚怡慶樊采虹周若珍丘政民
口試委員(外文):Yi-Ching YaoTsai-Hung FanRouh-Jane ChouJeng-Min Chiou
口試日期:2014-06-19
學位類別:博士
校院名稱:國立臺灣大學
系所名稱:數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2014
畢業學年度:102
語文別:英文
論文頁數:105
中文關鍵詞:準確性度量巴拿赫空間歐式類泛函中央極限定理高斯過程指標係數單調性擬概似半參數化分配模型均勻一致性變動線性指標近似常態分佈中央子空間交互驗證估計法反迴歸估計法最佳帶寬擬最小積分平方估計法半參數化效率上界半參數化估計法結構維度充分維度縮減
外文關鍵詞:accuracy measureBanach spaceEuclidean classfunctional central limit theoremGaussian processindex coefficientmonotonicitypseudo likelihoodsemiparametric distribution modelsuniform consistencyvarying linear-indexasymptotic normalitycentral subspacecross-validation estimationinverse regression estimationoptimal bandwidthpseudo least integrated squares estimationsemiparametric efficiency boundsemiparametric estimationstructural dimensionsufficient dimension reduction
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為了用更一般化的形式來刻劃條件分配函數,我們考慮兩種不同的半參數化迴歸模型。
在本論文的第一部分中,我們針對分配迴歸介紹一個更具有彈性的半參數化線性指標迴歸模型。
這個模式能夠描述解釋變數在應變數的值域上之變動效果,同時提供另一選擇性的維度縮減觀點,並且能夠涵蓋許多過去廣為使用的參數化與半參數化迴歸模型。
針對同時存在變動效果與不動效果係數的混合情況,我們提出一個更為可行的擬概似估計法來估計未知的係數。
除此之外,此估計式可以有效地由一個容易執行的演算法來得到。
理論上的均勻一致性與漸進高斯過程可藉由驗證巴拿赫空間上之性質來建立。
在分配函數與線性指標的單調性之下,另一個根據變動準確度量的選擇性估計法更進一步地被提供來估計線性指標中的係數。
在這個研究方向之中,重要的成果包括證明迭代演算估計過程的收斂性以及從漸進的遞迴關係式中建立所得到的估計式之大樣本性質。
根據我們所得到的理論結果,我們可以容易地建立關心參數之信賴區域以及針對不同模型結構之假設檢定。
一般來說,我們發展的估計及推論方法在模擬實驗中表現得相當好,並且在兩個重新分析的資料中發現其可用性。

本論文第二個部分考慮的是充分維度縮減模型,為一個廣為人知的探索性條件分配模型。
利用應變數相對應的計數過程,我們發展一個簡單並且容易執行的半參數估計法來估計中央子空間以及背後的迴歸函數。
與現存的充分維度縮減方法不同的是,中央子空間的基底與維度以及迴歸函數估計式的最佳帶寬可以同時藉由一個交互驗證形式的擬積分平方和準則來得到。
此估計技術允許應變數是離散的以及部分的解釋變數可以是離散化或類別的變數。
更進一步的,此交互驗證形式的最佳化準則之均勻一致性以及所得到估計是的一致性均可在較弱的條件被導出。
同時,我們建立了中央子空間之基底估計的近似常態分佈,其中估計式的維度也是估計的而非已知的真實維度。
在我們的模擬實驗裡也驗證出此方法比過去存在的半參數化估計式表現得都要來得好。
除此之外,它的實用性也在過去分析過的資料中被強調。
整體而言,我們的方法在計算中央子空間的估計時非常有效率、能夠包容各種不同形態的變數,並且在實務上能夠得到近似的最佳帶寬估計式。

To characterize the conditional distribution in a more general form, two variant semiparmaetric regressions are considered.
In the first scenario, we present more flexible semiparametric linear-index regression models for distribution regressions.
Such a model formulation captures varying effects of covariates over the support of a response variable in distribution, offers an alternative perspective on dimension reduction, and covers a lot of widely used parametric and semiparameteric regression models.
A more feasible pseudo likelihood approach is reasonably proposed for the mixed case with both varying and invariant coefficients.
In addition, the estimator are effectively computed through a simple and easily implemented algorithm.
Theoretically, its uniform consistency and asymptotic Gaussian process are established by justifying some properties on Banach spaces.
Under the monotonicity of distribution in linear-index, an alternative estimation procedure is further developed based on a varying accuracy measure.
In this research direction, other important achievements include showing the convergence of an iterative computation procedure and establishing the large sample properties of the resulting estimator from the asymptotic recursion relation for the estimators.
As a consequence of our general theoretical frameworks, it is convenient to construct confidence bands for the parameters of interest and tests for the hypotheses of various qualitative structures in distribution.
Generally, the developed estimation and inference procedures perform quite well in the conducted simulations and are demonstrated to be useful in reanalyzing data from the studies of house-price in Boston and World Values Survey.

In the second scenario of this thesis, we consider the sufficient dimension reduction model, which is a well-known exploratory model for describing the conditional distribution of interest.
With the associated counting process of a response, a simple and easily implemented semiparametric approach is developed to estimate the central subspace and underlying regression function.
Different from the existing sufficient dimension reduction approaches, two essential elements, basis and structural dimension, of the central subspace and the optimal bandwidth of a kernel distribution estimator can be simultaneously estimated through a cross-validation version of the pseudo sum of integrated squares.
One attractive merit of this estimation technique is that it allows a response to be discrete and some of covariates to be discrete or categorical.
Further, the uniform consistency of the cross-validation optimization function and the consistency of the resulting estimators are derived under very mild conditions.
Meanwhile, we establish the asymptomatic normality of the central subspace estimator with an estimated rather than exact structural dimension.
It is also demonstrated by our extensive numerical experiments that the developed approach dramatically outperforms the semiparametric competitors.
In addition, the applicability and practicality of the proposal are highlighted through data from previous studies.
Overall speaking, our methodology is computationally efficient in estimating the central subspace and the conditional distribution, highly flexible in adapting diverse types of a response and covariates, and practically feasible to obtain an asymptotically optimal bandwidth estimator.

Contents
Acknowledgements ................................ i
Abstract (in Chinese) .............................. ii
Abstract (in English) ............................... iv
Contents ................................................ vi
List of Tables ......................................... ix
List of Figures ........................................ xi
1 Introduction 1
1.1 Semiparametric Single-Index Distribution Models . . . . . . . . . . . . . 2
1.2 AnillustratedExample-BostonHouse-PriceData . . . . . . . . . . . . 4
1.3 VaryingLinear-IndexDistributionModel . . . . . . . . . . . . . . . . . 4
1.4 SufficientDimensionReduction....................... 7
1.5 Overview ................................... 10
2 Review of Existing Approaches 12
2.1 Pseudo Estimation in Single-Index Distribution Models . . . . . . . . . 12
2.2 ReviewofInverseRegression ........................ 14
2.3 Semiparametric Approaches for Sufficient Dimension Reduction . . . . . 17
3 Estimation and Inferences for Varying Linear-Index Distribution Models 20
3.1 EstimationofIndexCoefficients....................... 20
3.1.1 Pseudo Likelihood Estimation . . . . . . . . . . . . . . . . . . . . 21
3.1.2 Estimation under the Monotonic Structure . . . . . . . . . . . . 24
3.2 AsymptoticProperties............................ 26
3.2.1 AsymptoticPropertiesofthePMLE ................ 26
3.2.2 Asymptotic Properties of the AUC-based Estimator . . . . . . . 33
3.3 InferenceProcedures............................. 39
3.3.1 InferencesforIndexCoefficients................... 40
3.3.2 ModelChecking ........................... 42
3.3.3 TestfortheMonotonicStructure.................. 44
4 Effective Estimation for Sufficient Dimension Reduction 49
4.1 EstimationfortheCentralSubspace .................... 49
4.1.1 Background.............................. 49
4.1.2 Cross-Validation Estimation Criterion . . . . . . . . . . . . . . . 53
4.2 AsymptoticProperties............................ 56
4.2.1 NotationsandAssumptions..................... 56
4.2.2 TechnicalLemmas .......................... 57
4.2.3 ConsistencyandAsymptoticNormality . . . . . . . . . . . . . . 59
5 Monte Carlo Simulations 69
5.1 VaryingCoefficientsEstimation....................... 69
5.2 TestsforModelStructures.......................... 71
5.3 EstimationfortheCentralSubspace .................... 73
5.4 AComparisonwiththeSRApproach ................... 75
6 Empirical Examples 83
6.1 BostonHouse-PriceStudy.......................... 83
6.2 WorldValuesSurveyData.......................... 87
7 Concluding Remarks and Discussion 90
7.1 Summary ................................... 90
7.2 QuantileRegressions............................. 91
7.3 Multi-PhaseDistributionModels ...................... 92
7.4 VaryingMultiple-IndicesModels ...................... 94
7.5 VariableSelection............................... 94
7.6 CensoredSurvivalData ........................... 95
Bibliography 97
Vita 103

Arcones, M. A. and Gine, E. (1993). Limit theorems for U-processes. Ann. Probab. 21 1494–1542.
Barlow, R. E., Bartholomew, D. J., Bremner, J. M., and Brunk, H. D. (1972). Statistical inference under order restrictions: the theory and application of isotonic regression. Wiley, New York.
Bickel, P. J., Klaassen, C. A. J., Ritov, Y., and Wellner, J. A. (1998). Efficient and adaptive estimation for semiparametric models. Springer, New York.
Borisenko, A. A. and Nikolaevski&;#301;, Y. A. (1991). Grassmann manifolds and Grassmann image of submanifolds. Uspekhi Mat. Nauk 46 41–83, 240.
Breiman, L. and Friedman, J. H. (1985). Estimating optimal transformations for mul- tiple regression and correlation. J. Amer. Statist. Assoc. 80 580–619.
Buckley, J. and James, I. (1979). Linear regression with censored data. Biometrika 66 429–436.
Bura, E. and Cook, R. D. (2001). Extending sliced inverse regression: the weighted chi-squared test. J. Amer. Statist. Assoc. 96 996–1003.
Cavanagh, C. and Sherman, R. P. (1998). Rank estimators for monotonic index models. J. Econometrics 84 351–381.
Chiang, C. T. and Huang, M. Y. (2012). New estimation and inference procedures for a single-index conditional distribution model. J. Multivariate Anal. 111 271–285.
Cook, R. D. (1998). Regression graphics. John Wiley, New York.
Cook, R. D. and Weisberg, S. (1991). Comment on “sliced inverse regression for dimen-
sion reduction” by K.-C. Li. J. Amer. Statist. Assoc. 86 328–332.
Cosslett, S. R. (1983). Distribution-free maximum likelihood estimator of the binary
choice model. Econometrica 51 765–782.
Cosslett, S. R. (1987). Efficiency bounds for distribution-free estimators of the binary
choice and the censored regression models. Econometrica 55 559–585.
Delecroix, M., Hardle, W., and Hristache, M. (2003). Efficient estimation in conditional
single-index regression. J. Multivariate Anal. 86 213–226.
Fahrmeir, L. and Tutz, G. (2001). Multivariate statistical modelling based on generalized
linear models. Springer, New York.
Fan, J. and Huang, T. (2005). Profile likelihood inferences on semiparametric varying-
coefficient partially linear models. Bernoulli 11 1031–1057.
Fox, J. and Anderson, R. (2006). Effect displays for multinomial and proportional-odds
logit models. Sociological Methodology 36 225–255.
Hall, P. and Yao, Q. (2005). Approximating conditional distribution functions using
dimension reduction. Ann. Statist. 33 1404–1421.
Hamilton, R. S. (1982). The inverse function theorem of Nash and Moser. Bull. Amer.
Math. Soc. 7 65–222.
Han, A. K. (1987). Nonparametric analysis of a generalized regression model: The
maximum rank correlation estimator. J. Econometrics 35 303–316.
Hardle, W., Hall, P., and Ichimura, H. (1993). Optimal smoothing in single-index models. Ann. Statist. 21 157–178.
Hardle, W., Hall, P., and Marron, J. S. (1988). How far are automatically chosen regression smoothing parameters from their optimum? J. Amer. Statist. Assoc. 83 86–101.
Hardle, W. and Marron, J. S. (1985). Optimal bandwidth selection in nonparametric regression function estimation. Ann. Statist. 13 1465–1481.
Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Statist. 19 293–325.
Ichimura, H. (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J. Econometrics 58 71–120.
Klein, R. W. and Spady, R. H. (1993). An efficient semiparametric estimator for binary response models. Econometrica 61 387–421.
Kong, E. and Xia, Y. (2012). A single-index quantile regression model and its estimation. Econometric Theory 28 730–768.
Kosorok, M. R. (2008). Introduction to empirical processes and semiparametric infer- ence. Springer, New York.
Li, B. and Wang, S. (2007). On directional regression for dimension reduction. J. Amer. Statist. Assoc. 102 997–1008.
Li, K.-C. (1991). Sliced inverse regression for dimension reduction. J. Amer. Statist. Assoc. 86 316–342.
Li, K.-C. (1992). On principal Hessian directions for data visualization and dimension reduction: another application of Stein’s lemma. J. Amer. Statist. Assoc. 87 1025– 1039.
Ma, Y. and Zhu, L. (2012). A semiparametric approach to dimension reduction. J. Amer. Statist. Assoc. 107 168–179.
Ma, Y. and Zhu, L. (2013). Efficient estimation in sufficient dimension reduction. Ann. Statist. 41 250–268.
McIntosh, M. W. and Pepe, M. S. (2002). Combining several screening tests: optimality of the risk score. Biometrics 58 657–664.
Mokkadem, A., Pelletier, M., and Thiam, B. (2008). Large and moderate deviations principles for kernel estimators of the multivariate regression. Math. Methods Statist. 17 146–172.
Nelder, J. A. and Wedderburn, R. W. M. (1972). Generalized linear models. J. Roy. Statist. Soc. A135 370–384.
Nolan, D. and Pollard, D. (1987). U-processes: rates of convergence. Ann. Statist. 15 780–799.
Nolan, D. and Pollard, D. (1988). Functional limit theorems for U-processes. Ann. Probab. 16 1291–1298.
Pakes, A. and Pollard, D. (1989). Simulation and the asymptotics of optimization estimators. Econometrica 57 1027–1057.
Peng, L. and Huang, Y. (2007). Survival analysis with temporal covariate effects. Biometrika 94 719–733.
Pollard, D. (1984). Convergence of stochastic processes. Springer, New York.
Powell, J. L., Stock, J. H., and Stoker, T. M. (1989). Semiparametric estimation of
index coefficients. Econometrica 57 1403–1430.
Rudin, W. (1976). Principles of mathematical analysis. McGraw-Hill, New York.
Sherman, R. P. (1993). The limiting distribution of the maximum rank correlation estimator. Econometrica 61 123–137.
Sherman, R. P. (1994). Maximal inequalities for degenerate U-processes with applica- tions to optimization estimators. Ann. Statist. 22 439–459.
Tsiatis, A. A. (2006). Semiparametric theory and missing data. Springer, New York. Wang, H. and Xia, Y. (2008). Sliced regression for dimension reduction. J. Amer.
Statist. Assoc. 103 811–821.
Wu, T. Z., Yu, K., and Yu, Y. (2010). Single-index quantile regression. J. Multivariate
Anal. 101 1607–1621.
Xia, Y. (2007). A constructive approach to the estimation of dimension reduction
directions. Ann. Statist. 35 2654–2690.
Xia, Y. (2009). Model checking in regression via dimension reduction. Biometrika 96
133–148.
Xia, Y., Tong, H., Li, W. K., and Zhu, L.-X. (2002). An adaptive estimation of dimension
reduction space. J. Roy. Statist. Soc. B64 363–410.
Ye, Z. and Weiss, R. E. (2003). Using the bootstrap to select one of a new class of
dimension reduction methods. J. Amer. Statist. Assoc. 98 968–979.
Zeng, P. and Zhu, Y. (2010). An integral transform method for estimating the central
mean and central subspaces. J. Multivariate Anal. 101 271–290.
Zheng, Y., Cai, T., and Feng, Z. (2006). Application of the time-dependent ROC curves
for prognostic accuracy with multiple biomarkers. Biometrics 62 279–287.
Zhu, L., Miao, B., and Peng, H. (2006). On sliced inverse regression with high-
dimensional covariates. J. Amer. Statist. Assoc. 101 630–643.
Zhu, L., Wang, T., Zhu, L., and Ferre, L. (2010a). Sufficient dimension reduction
through discretization-expectation estimation. Biometrika 97 295–304.
Zhu, L. P., Zhu, L. X., and Feng, Z. H. (2010b). Dimension reduction in regressions
through cumulative slicing estimation. J. Amer. Statist. Assoc. 105 1455–1466. Zhu, Y. and Zeng, P. (2006). Fourier methods for estimating the central subspace and
the central mean subspace in regression. J. Amer. Statist. Assoc. 101 1638–1651.
Zou, H. (2006). The adaptive lasso and its oracle properties. J. Amer. Statist. Assoc. 101 1418–1429.

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