指導教授：江金倉臺灣大學：數學研究所黃名鉞Huang, Ming-YuehMing-YuehHuang2014-11-302018-06-282014-11-302018-06-282014http://ntur.lib.ntu.edu.tw//handle/246246/264036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 1031028448 bytesapplication/pdf論文公開時間：2019/07/31論文使用權限：同意有償授權(權利金給回饋學校)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thesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/264036/1/ntu-103-D99221001-1.pdf