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[UserCP=5]<p>Quantile Regression <br/></p><p>Roger Koenker <br/>University of Illinois <br/></p><p>Contents <br/>viii Contents <br/> Inference for Quantile Regression 68 <br/> 3.1 The Finite-Sample Distribution of Regression Quantiles 68 <br/> 3.2 Heuristic Introduction to Quantile Regression <br/> Asymptotics 71 <br/> 3.2.1 Confidence Intervals for the Sample Quantiles 72 <br/> 3.2.2 Quantile Regression Asymptotics with IID Errors 73 <br/> 3.2.3 Quantile Regression Asymptotics in Non-IID <br/> Settings 74 <br/> 3.3 Wald Tests 75 <br/> 3.3.1 Two-Sample Tests of Location Shift 75 <br/> 3.3.2 General Linear Hypotheses 76 <br/> 3.4 Estimation of Asymptotic Covariance Matrices 77 <br/> 3.4.1 Scalar Sparsity Estimation 77 <br/> 3.4.2 Covariance Matrix Estimation in Non-IID Settings 79 <br/> 3.5 Rank-Based Inference 81 <br/> 3.5.1 Rank Tests for Two-Sample Location Shift 81 <br/> 3.5.2 Linear Rank Statistics 84 <br/> 3.5.3 Asymptotics of Linear Rank Statistics 85 <br/> 3.5.4 Rank Tests Based on Regression Rankscores 87 <br/> 3.5.5 Confidence Intervals Based on Regression <br/> Rankscores 91 <br/> 3.6 Quantile Likelihood Ratio Tests 92 <br/> 3.7 Inference on the Quantile Regression Process 95 <br/> 3.7.1 Wald Processes 97 <br/> 3.7.2 Quantile Likelihood Ratio Processes 98 <br/> 3.7.3 The Regression Rankscore Process Revisited 98 <br/> 3.8 Tests of the Location-Scale Hypothesis 98 <br/> 3.9 Resampling Methods and the Bootstrap 105 <br/> 3.9.1 Bootstrap Refinements, Smoothing, and <br/> Subsampling 107 <br/> 3.9.2 Resampling on the Subgradient Condition 108 <br/> 3.10 Monte Carlo Comparison of Methods 110 <br/> 3.10.1 Model 1: Location-Shift Model 111 <br/> 3.10.2 Model 2: Location–Scale-Shift Model 112 <br/> 3.11 Problems 113 <br/> Asymptotic Theory of Quantile Regression 116 <br/> 4.1 Consistency 117 <br/>4.1.1 Univariate Sample Quantiles 117 <br/>4.1.2 Linear Quantile Regression 118 <br/>4.2 Rates of Convergence 120 <br/>4.3 Bahadur Representation 122 <br/>4.4 Nonlinear Quantile Regression 123 <br/>4.5 The Quantile Regression Rankscore Process 124 <br/>4.6 Quantile Regression Asymptotics under Dependent <br/>Conditions 126 <br/> Contents <br/> 6.4.3 Interior vs. Exterior: Computational <br/> Comparison 202 <br/> 6.4.4 Computational Complexity 204 <br/> 6.5 Preprocessing for Quantile Regression 206 <br/> 6.5.1 “Selecting” Univariate Quantiles 207 <br/> 6.5.2 Implementation 207 <br/> 6.5.3 Confidence Bands 208 <br/> 6.5.4 Choosing 209 <br/> 6.6 Nonlinear Quantile Regression 211 <br/> 6.7 Inequality Constraints 213 <br/> 6.8 Weighted Sums of ρτ -Functions 214 <br/> 6.9 Sparsity 216 <br/> 6.10 Conclusion 220 <br/> 6.11 Problems 220 <br/> Nonparametric Quantile Regression 222 <br/> 7.1 Locally Polynomial Quantile Regression 222 <br/> 7.1.1 Average Derivative Estimation 226 <br/> 7.1.2 Additive Models 228 <br/> 7.2 Penalty Methods for Univariate Smoothing 229 <br/> 7.2.1 Univariate Roughness Penalties 229 <br/> 7.2.2 Total Variation Roughness Penalties 230 <br/> 7.3 Penalty Methods for Bivariate Smoothing 235 <br/> 7.3.1 Bivariate Total Variation Roughness Penalties 235 <br/> 7.3.2 Total Variation Penalties for Triograms 236 <br/> 7.3.3 Penalized Triogram Estimation as Linear <br/> Program 240 <br/> 7.3.4 On Triangulation 241 <br/> 7.3.5 On Sparsity 242 <br/> 7.3.6 Automatic Selection 242 <br/> 7.3.7 Boundary and Qualitative Constraints 243 <br/> 7.3.8 Model of Chicago Land Values 243 <br/> 7.3.9 Taut Strings and Edge Detection 246 <br/> 7.4 Additive Models and the Role of Sparsity 248 <br/> Twilight Zone of Quantile Regression 250 <br/> 8.1 Quantile Regression for Survival Data 250 <br/>8.1.1 Quantile Functions or Hazard Functions? 252 <br/>8.1.2 Censoring 253 <br/>8.2 Discrete Response Models 255 <br/>8.2.1 Binary Response 255 <br/>8.2.2 Count Data 259 <br/>8.3 Quantile Autoregression 260 <br/>8.3.1 Quantile Autoregression and Comonotonicity 261 <br/>8.4 Copula Functions and Nonlinear Quantile Regression 265 <br/>8.4.1 Copula Functions 265 <br/></p><p>Contents xi <br/>8.5 High-Breakdown Alternatives to Quantile Regression 268 <br/>8.6 Multivariate Quantiles 272 <br/>8.6.1 The Oja Median and Its Extensions 273 <br/>8.6.2 Half-Space Depth and Directional Quantile <br/>Regression 275 <br/>8.7 Penalty Methods for Longitudinal Data 276 <br/>8.7.1 Classical Random Effects as Penalized <br/>Least Squares 276 <br/>8.7.2 Quantile Regression with Penalized Fixed Effects 278 <br/>8.8 Causal Effects and Structural Models 281 <br/>8.8.1 Structural Equation Models 281 <br/>8.8.2 Chesher’s Causal Chain Model 283 <br/>8.8.3 Interpretation of Structural Quantile Effects 284 <br/>8.8.4 Estimation and Inference 285 <br/>8.9 Choquet Utility, Risk, and Pessimistic Portfolios 287 <br/> 8.9.1 Choquet Expected Utility 287 <br/> 8.9.2 Choquet Risk Assessment 289 <br/> 8.9.3 Pessimistic Portfolios 291 <br/> Conclusion 293 <br/> Quantile Regression in R: Vignette 295 <br/> A.Introduction 295 <br/> A.What Is Vignette? 296 <br/> A.Getting Started 296 <br/> A.Object Orientation 298 <br/> A.Formal Inference 299 <br/> A.More on Testing 305 <br/> A.Inference on the Quantile Regression Process 307 <br/> A.Nonlinear Quantile Regression 308 <br/> A.Nonparametric Quantile Regression 310 <br/> A.10 Conclusion 316 <br/> Asymptotic Critical Values 317 <br/> References 319 <br/> Name Index 337 <br/> Subject Index 342 <br/></p><br/>[/UserCP]
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