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Quantile Regression  

Roger Koenker  
University of Illinois  


Contents  
viii  Contents  
 Inference for Quantile Regression  68  
 3.1 The Finite-Sample Distribution of Regression Quantiles  68  
 3.2 Heuristic Introduction to Quantile Regression   
 Asymptotics  71  
 3.2.1  Confidence Intervals for the Sample Quantiles  72  
 3.2.2  Quantile Regression Asymptotics with IID Errors  73  
 3.2.3  Quantile Regression Asymptotics in Non-IID   
 Settings  74  
 3.3 Wald Tests  75  
 3.3.1  Two-Sample Tests of Location Shift  75  
 3.3.2  General Linear Hypotheses  76  
 3.4 Estimation of Asymptotic Covariance Matrices  77  
 3.4.1  Scalar Sparsity Estimation  77  
 3.4.2  Covariance Matrix Estimation in Non-IID Settings  79  
 3.5 Rank-Based Inference  81  
 3.5.1  Rank Tests for Two-Sample Location Shift  81  
 3.5.2  Linear Rank Statistics  84  
 3.5.3  Asymptotics of Linear Rank Statistics  85  
 3.5.4  Rank Tests Based on Regression Rankscores  87  
 3.5.5  Confidence Intervals Based on Regression   
  Rankscores  91  
 3.6 Quantile Likelihood Ratio Tests  92  
 3.7 Inference on the Quantile Regression Process  95  
 3.7.1  Wald Processes  97  
 3.7.2  Quantile Likelihood Ratio Processes  98  
 3.7.3  The Regression Rankscore Process Revisited  98  
 3.8 Tests of the Location-Scale Hypothesis  98  
 3.9 Resampling Methods and the Bootstrap  105  
 3.9.1  Bootstrap Refinements, Smoothing, and   
 Subsampling  107  
 3.9.2  Resampling on the Subgradient Condition  108  
 3.10 Monte Carlo Comparison of Methods  110  
 3.10.1 Model 1: Location-Shift Model  111  
 3.10.2 Model 2: Location–Scale-Shift Model  112  
 3.11 Problems  113  
 Asymptotic Theory of Quantile Regression  116  
 4.1 Consistency  117  
4.1.1  Univariate Sample Quantiles  117  
4.1.2  Linear Quantile Regression  118  
4.2 Rates of Convergence  120  
4.3 Bahadur Representation  122  
4.4 Nonlinear Quantile Regression  123  
4.5 The Quantile Regression Rankscore Process  124  
4.6 Quantile Regression Asymptotics under Dependent   
Conditions  126  
 Contents   
 6.4.3  Interior vs. Exterior: Computational   
 Comparison  202  
 6.4.4  Computational Complexity  204  
 6.5 Preprocessing for Quantile Regression  206  
 6.5.1  “Selecting” Univariate Quantiles  207  
 6.5.2  Implementation  207  
 6.5.3  Confidence Bands  208  
 6.5.4  Choosing  209  
 6.6 Nonlinear Quantile Regression  211  
 6.7 Inequality Constraints  213  
 6.8 Weighted Sums of ρτ -Functions  214  
 6.9 Sparsity  216  
 6.10 Conclusion  220  
 6.11 Problems  220  
 Nonparametric Quantile Regression  222  
 7.1 Locally Polynomial Quantile Regression  222  
 7.1.1  Average Derivative Estimation  226  
 7.1.2  Additive Models  228  
 7.2 Penalty Methods for Univariate Smoothing  229  
 7.2.1  Univariate Roughness Penalties  229  
 7.2.2  Total Variation Roughness Penalties  230  
 7.3 Penalty Methods for Bivariate Smoothing  235  
 7.3.1  Bivariate Total Variation Roughness Penalties  235  
 7.3.2  Total Variation Penalties for Triograms  236  
 7.3.3  Penalized Triogram Estimation as Linear   
  Program  240  
 7.3.4  On Triangulation  241  
 7.3.5  On Sparsity  242  
 7.3.6  Automatic Selection  242  
 7.3.7  Boundary and Qualitative Constraints  243  
 7.3.8  Model of Chicago Land Values  243  
 7.3.9  Taut Strings and Edge Detection  246  
 7.4 Additive Models and the Role of Sparsity   248  
 Twilight Zone of Quantile Regression   250  
 8.1 Quantile Regression for Survival Data   250  
8.1.1  Quantile Functions or Hazard Functions?   252  
8.1.2  Censoring   253  
8.2 Discrete Response Models   255  
8.2.1  Binary Response  255  
8.2.2  Count Data  259  
8.3 Quantile Autoregression  260  
8.3.1  Quantile Autoregression and Comonotonicity  261  
8.4 Copula Functions and Nonlinear Quantile Regression  265  
8.4.1  Copula Functions  265  

Contents  xi  
8.5 High-Breakdown Alternatives to Quantile Regression  268  
8.6 Multivariate Quantiles  272  
8.6.1  The Oja Median and Its Extensions  273  
8.6.2  Half-Space Depth and Directional Quantile   
Regression  275  
8.7 Penalty Methods for Longitudinal Data  276  
8.7.1  Classical Random Effects as Penalized   
Least Squares  276  
8.7.2  Quantile Regression with Penalized Fixed Effects  278  
8.8 Causal Effects and Structural Models  281  
8.8.1  Structural Equation Models  281  
8.8.2  Chesher’s Causal Chain Model  283  
8.8.3  Interpretation of Structural Quantile Effects  284  
8.8.4  Estimation and Inference  285  
8.9 Choquet Utility, Risk, and Pessimistic Portfolios  287  
 8.9.1  Choquet Expected Utility  287  
 8.9.2  Choquet Risk Assessment  289  
 8.9.3  Pessimistic Portfolios  291  
 Conclusion   293  
 Quantile Regression in R: Vignette  295  
 A.Introduction  295  
 A.What Is Vignette?  296  
 A.Getting Started  296  
 A.Object Orientation  298  
 A.Formal Inference  299  
 A.More on Testing  305  
 A.Inference on the Quantile Regression Process  307  
 A.Nonlinear Quantile Regression  308  
 A.Nonparametric Quantile Regression  310  
 A.10 Conclusion  316  
 Asymptotic Critical Values  317  
 References  319  
 Name Index  337  
 Subject Index  342  

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