Handbook of quantile regression:
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| Language: | English |
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CRC Press
[2020]
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| Series: | Handbooks of modern statistical methods
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| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | xix, 463 Seiten Diagramme |
| ISBN: | 9780367657574 9781498725286 |
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| adam_text | Contents Preface xvii Contributors xix I Introduction 1 1 A Quantile Regression Memoir 3 Gilbert W. Bassett Jr. and Roger Koenker 1.1 Long ago ............................. ...v ....... . 2 Resampling Methods 3 7 Xuming He 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Introduction ՛.............. .............. .. ................................. .. . і...................... . Paired bootstrap . . . .................. Residual-based bootstrap.............................. ... . ............... ... Generalized bootstrap ....................................... Estimating function bootstrap . . . . . . . ... . . . ... ......... Markov chain marginal bootstrap .......................................... і ... . Resampling methods for clustered data ............................... . . Resampling methods for censored quantile regression ....... Bootstrap for post-model selection inference ................................................. 3 Quantile Regression: Penalized 7 8 9 11 11 12 13 14 15 21 Ivan Mizera 3.1 3.2 3.3 Penalized: how? . . . . .......................................... 3.1.1 A probability path . . . ............................................. . . 3.1.2 Regularization of ill-posed problems .................. Penalized: what?............... 3.2.1 The finite differences of Whittaker and others.................. .. 3.2.2 Functions and theirderivatives................... 3.2.3 Quantile regression with smoothing splines .............. 3.2.4 Quantile smoothing splines......................... . . ............................... . 3.2.5 Total-variation splines
..................................... Penalized: what else? . ... . . . ................... ... . . ......... 3.3.1 Tuning . . ................ . . . ........................................ 3.3.2 Multiple covariates .................................................................................. 3.3.3 Additive fits, confidencebandaids, and other phantasmagorias 4 Bayesian Quantile Regression 21 21 22 23 23 24 26 28 29 31 31 32 34 41 4.1 4.2 4.3 Introduction ........................ ... ■ · Asymmetric Laplace likelihood . ... . .... ................ Empirical likelihood........................... . Huiría Judy Wang and Yunwen Yang 41 42^ 45 ix
Contents 4.4 Nonparametric and semiparametric likelihoods . 4.4.1 Mixture-type likelihood................................ 4.4.2 Approximate likelihood via quantile process 4.5 Discussion 5 6 7 Computational Methods for Quantile Regression 55 Roger Koenker 5.1 Introduction ..................................................................... 5.2 Exterior point methods .................................................. 5.3 Interior point methods .................................................. 5.4 Preprocessing ................................................................. 5.5 First-order, proximal methods ...................................... 5.5.1 Proximal operators and the Moreau envelope . 5.5.2 Alternating direction method of multipliers . . 5.5.3 Proximal performance......................................... 55 57 58 60 61 61 64 65 Survival Analysis: A Quantile Perspective Zhiliang Ying and Tony Sit 6.1 Introduction .............................................. 6.1.1 Notation........................................... 6.1.2 Censoring........................................ 6.2 Important models ..................................... 6.2.1 Parametric models......................... 6.2.2 Nonparametric estimators............ 6.2.2.1 Kaplan-Meier estimator.......................................................... 6.2.2.2 Nelson-Aalen estimator........................................................... 6.2.3 Regression models.................................................................................... 6.2.3.1 Cox proportional hazards
model............................................ 6.2.3.2 Accelerated failure time model............................................... 6.2.3.3 Aalen additive hazard model................................................... 6.3 Quantile estimation based on censored data...................................................... 6.3.1 Quantile estimation ................................................................................. 6.3.2 Median and quantile regression.............................................................. 6.3.3 Discussion and miscellanea.............................. 73 75 75 75 76 78 79 79 80 82 Quantile Regression for Survival Analysis 89 Limin Peng 7.1 Introduction ......................................................................................................... 7.2 Quantile regression for randomly censored data................................................ 7.2.1 Random right censoring with C always known ................................... 7.2.2 Covariate-independent random right censoring................................... 7.2.3 Standard random right censoring............................................................ 7.2.3.1 Approaches based on the principle of self-consistency ... 7.2.3.2 Martingale-based approach....................................................... 7.2.3.3 Locally weighted method......................................................... 7.2.4 Variance estimation and other inference................................................ 7.2.4.1 Variance estimation.............................................................
7.2.4.2 Second-stage inference................................................................ 7.2.4.3 Model checking..................................................................... 7.3 Quantile regression in other survival settings ......................................... [ 7.3.1 Known random left censoring and/or left truncation......................... 89 90 90 91 92 92 93 94 95 95 gg gg 97 97
Contents 7.4 7.3.2 Censored data with a survival cure fraction....................................... An illustration of quantile regression for survival analysis........................... 8 Survival Analysis with Competing Risks and Semi-competing Risks Data Ruosha Li and Limin Peng 8.1 Competing risks data ......................................................................................... 8.1.1 Introduction ...................................................................... 8.1.2 Cumulative incidence quantile regression.............................................. 8.1.3 Data analysis example............................................................................. 8.1.4 Marginal quantile regression.................................................................... 8.2 Semi-competing risks data ................................................................................ 8.2.1 Introduction ................................. 8.2.2 Cumulative incidence quantile regression.............................................. 8.2.3 Marginal quantile regression.................. ... . ........................................ 8.3 Summary and open problems........................... xi 98 98 105 105 105 106 108 110 112 112 113 114 116 9 Instrumental Variable Quantile Regression 119 Victor Ghernozhukov, Christian Hansen, and Kaspar Wiithrich 9.1 Introduction ................................................................................................. 120 9.2 Model overview
.................................................................................................. 121 9.2.1 The instrumental variable quantile regressionmodel................... 121 9.2.2 Conditions for point identification....................................................... 123 9.2.3 Discussion of the IVQR model . . . .................................................... 124 9.2.4 Examples............................................................................... 126 9.2.5 Comparison to other approaches............... 128 9.3 Basic estimation and inference approaches ................................. 129 9.3.1 Generalized methods of moments and relatedapproaches ...... 130 9.3.2 Inverse quantile regression....................................... 132 9.3.2.1 A useful interpretation of IQR as a GMM estimator........... 133 9.3.3 Weak identification robust inference.................................................... 134 9.3.4 Finite-sample inference ......................................................................... 136 9.4 Advanced inference with high-dimensional X................................................. 136 9.4.1 Neyman orthogonal scores ...................................................................... 136 9.4.2 Estimation and inference using orthogonal scores............................... 138 9.5 Conclusion ........................................................................................................... 139 10 Local Quantile Treatment Effects Blaise Melly and Kaspar Wiithrich 10.1 Introduction
........................................................................................................ 10.2 Framework, estimands and identification ....................................................... 10.2.1 Without covariates.................................................................................. 10.2.2 In the presence of covariates: conditional LQTE .................................. 10.2.3 In the presence of covariates: unconditional LQTE............................ 10.3 Estimation and inference .................................................................................. 10.4 Extensions ........................................................................................................... 10.4.1 Regression discontinuity design .............................................................. 10.4.2 Multi-valued and continuous instruments ........................................... 10.4.3 Testing instrument validity................................................................... 10.5 Comparison to the instrumental variablequantile regression model.............. 10.6 Conclusion and open problems.......................................................................... 145 145 148 148 151 152 154 155 155 156 157 158 160
Contents xii 11 Quantile Regression with Measurement Errors and Missing Data Ying Wei 11.1 Introduction .......................................................................................................... 11.2 Quantile regression with measurement errors .................................................. 11.2.1 Linear quantile regression with measurement errors............................ 11.2.1.1 Semiparametric joint estimating equations . ..................... 11.2.1.2 Other methods for linear quantile regression with measure• ment errors................................................................................. 11.2.2 Nonparametric and semiparametric quantile regression model with measurement errors ................................................................................. 11.3 Quantile regression with missing data .............................................................. 11.3.1 Statistical methods handling missing covariates in quantile regression 11.3.1.1 Multiple imputation algorithm............................................... 11.3.1.2 Modified MI algorithms........................................................... 11.3.1.3 EM algorithm.......................................................................... 11.3.1.4 IPW algorithms....................................................................... 11.3.2 Statistical methods handling missing outcomes in quantile regression 11.3.2.1 Imputation approaches for missing outcomes ........ 11.3.2.2 Statistical methods for longitudinal dropout ...................... 12 Multiple-Output Quantile
Regression Marc Hallin and Miroslav Siman 12.1 Multivariate quantiles, and the ordering of Rd, d џ 2 .... 12.2 Directional approaches ............................................................. 12.2.1 Projection methods ........................ .............................. 12.2.1.1 Marginal (coordinatewise) quantiles............ 12.2.1.2 Quantile biplots.............................................. 12.2.1.3 Directional quantile hyperplanes and contours 12.2.1.4 Relation to halfspace depth............................ 12.2.2 Directional Koenker-Bassett methods......................... 12.2.2.1 Location case (p = 0)..................................... 12.2.2.2 (Nonparametric) regression case (p ^ 1) 12.3 Direct approaches ....................................................................... 12.3.1 Spatial (geometric) quantile methods ................ ··■· 12.3.1.1 A spatial check function.................................. 12.3.1.2 Linear spatial quantile regression................... 12.3.1.3 Nonparametric spatial quantile regression . . 12.3.2 Elliptical quantiles ........................................................... 12.3.2.1 Location case..................................................... 12.3.2.2 Linear regression case...................................... 12.3.3 Depth-based quantiles ...................................................... 12.3.3.1 Halfspace depth quantiles................................ 12.3.3.2 Monge-Kantorovich quantiles......................... 12.4 Some other concepts, and applications ...................................... 12.5
Conclusion ............................... 13 Sample Selection in Quantile Regression: A Survey Manuel Arellano and Stéphane Bonhomme 13.1 Introduction . . ......................................... 13.2 Heckman’s parametric selection model 13.2.1 Two-step estimation in Gaussian models 165 165 166 166 166 169 171 172 173 173 175 177 178 178 178 180 185 185 187 187 187 187 188 189 189 189 191 193 195 195 196 197 197 197 198 200 200 201 203 204 209 209 211 212
Contents xiii 13.3 A quantile generalization ............................................. ..................................... 212 13.3.1 A quantile selection model. ............................ ........................ 212 13.3.2 Estimation...................................................... 213 13.4 Identification..................... 216 13.5 Other approaches ....................................... 216 13.5.1 A likelihood approach............................................................. 217 13.5.2 Control function approaches .............................................................. ... . 217 13.5.3 Link to censoring corrections................................................. 217 13.6 Empirical illustration ......................................................................................... 218 13.7 Conclusion ........................ 221 14 Nonparametric Quantile Regression for Banach-Valued Response 225 Joydeep Chowdhury and Probal Chaudhuri 14.1 Introduction ........................ 225 14.2 Regression quantiles in Banachspaces . ........................................................... 229 14.3 Nonparametric estimation ............................................................................... 231 14.4 Data analysis............... 232 14.4.1 Simulation . .............................................. ....................................... .. . 232 14.4.2 Tecator data ............................................................................................ 234 14.4.3 Pediatric airway
data............................................................. 234 14.4.4 Cigarette data................................................................... 237 14.4.4.1 Regression of price curve on sales curve . ............................ 237 14.5 Consistency .................................... 240 14.5.1 Additional mathematical details . . .................................................... 245 14.6 Concluding remarks . . . ................................................................................... 249 15 High-Dimensional Quantile Regression 253 Alexandre Belloni, Victor Chernozhukov, and Kengo Kato 15.1 Introduction ........................................................................................................ 253 15.2 Estimation of the conditional quantile function..................... 256 15.2.1 Regularity conditions ................................................ 256 15.2.2 fi-penalized quantile regression............................................................. 257 15.2.3 Refitted quantile regression after selection.......................................... 259 15.2.4 Group lasso for quantile regression models.......................................... 260 15.2.5 Estimation of the conditional density.................................... ............... 261 15.3 Confidence bands for the coefficient process.................................................... 261 15.3.1 Construction of an orthogonal score function.................................... 263 15.3.2 Regularity conditions
............................................................................. 265 15.3.3 Score function estimator......................................................................... 266 15.3.4 Double selection estimator...................................................................... 267 15.3.5 Confidence bands..................................................................................... 267 15.3.6 Confidence bands via inverse statistics................................................. 269 16 Nonconvex Penalized Quantile Regression: A Review of Methods, Theory and Algorithms 273 Lan Wang 16.1 Introduction ............................................................................................... 273 16.2 High-dimensional sparse linearquantile regression ......................................... 275 16.2.1 Background on penalized high-dimensional regression and the choice of penalty function.................................................................................. 275 16.2.2 Nonconvex penalized high-dimensional linear quantile regression . . 276
Contents xiv 16.3 16.4 16.5 16.6 16.2.2.1 Overview............................................... ****** *............... 16.2.2.2 Oracle property of the nonconvex penalized quantile regres sion estimator........................................................................... High-dimensional sparse semiparametric quantile regression......................... 16.3.1 Overview.................................. ............... -............... .................. . * 16.3.2 Nonconvex penalized partially linear additive quantile regression . . 16.3.3 Oracle properties...............................................................;..................... Computational aspects of nonconvex penalized quantile regression............. 16.4.1 Linear programming based algorithms (moderately large p)............. 16.4.2 New iterative coordinate descent algorithm (larger p)......................... Other related problems ................................................................................. ... · 16.5.1 Simultaneous estimation and variable selection at multiple quantiles 16.5.2 Two-stage analysis with quantile-adaptive screening......................... 16.5.2.1 Background................................................................................. 16.5.2.2 Quantile-adaptive model-free nonlinear screening................ Discussion ............................................................................................................ 17 QAR and Quantile Time Series Analysis Zhijie Xiao 17.1 Introduction
......................................................................................................... 17.2 Quantile regression estimation of traditional time series models................... 17.2.1 Quantile regression estimation of the traditional AR model............. 17.2.2 Quantile regressions of other time series models with i.i.d. errors . . 17.2.3 Quantile regression estimation of ARMA models............................... 17.2.4 Quantile regressions with serially correlated errors............................ 17.3 Quantile regressions with ARCH/GARCH errors............................................ 17.4 Quantile regressions with heavy-tailed errors .................................................. 17.5 Quantile regression for nonstationary time series............................................ 17.5.1 Quantile regression for trending time series......................................... 17.5.2 Unit-root quantile regressions.................................................................. 17.5.3 Quantile regression on cointegrated time series................................... 17.6 The QAR process ................................................................................................. 17.6.1 The linear QAR process.............................. 17.6.2 Nonlinear QAR models........................................................................... 17.6.3 Quantile autoregression based on transformations................................ 17.7 Other dynamic quantile models ........................................................................ 17.8 Quantile
spectral analysis............................-....·............................................ 17.8.1 Quantile cross-covariances and quantile spectrum . . .......................... 17.8.2 Quantile periodograms..................................... 17.8.3 Relationship to quantile regression on harmonic regressors................ 17.8.4 Estimation of quantile spectral density................................................... 17.9 Quantile regression based forecasting ...................................................... 17.10Conclusion ..................................... 18 Extremal Quantile Regression Victor Chemozhukov, Iván Femández-Val, and Tetsuya Kaji 18.1 Introduction ............................... 18.2 Extreme quantile models ..................................... 18.2.1 Pareto-type and regularly varying tails............. 18.2.2 Extremal quantile regression models................ 18.3 Estimation and inference methods 276 278 279 279 279 280 281 281 282 283 283 284 284 284 285 293 293 294 295 296 297 298 299 306 307 307 308 310 312 313 317 319 320 322 323 324 325 327 32g oor 333 334 336 336 337 338
Contents 18.3.1 Sampling conditions...................................................................... 18.3.2 Univariave case: Marginal quantiles ......................... ............................ 18.3.2.1 Extreme order approximation .................................................. 18.3.2.2 Intermediate order approximation.............................. 18.3.2.3 Estimation of ξ ........................... 18.3.2.4 Estimation of Αχ .............................. 18.3.2.5 Computing quantiles of the limit extreme valuedistributions 18.3.2.6 Median bias correction and confidence intervals................... 18.3.2.7 Extrapolation estimator for very extremes ........................ 18.3.3 Multivariate case: Conditional quantiles.............................................. 18.3.3.1 Extreme order approximation................................................. 18.3.3.2 Intermediate order approximation ........................................ 18.3.3.3 Estimation of ξ and 7 . ..................... 18.3.3.4 Estimation of Αχ .................................................................... 18.3.3.5 Computing quantiles of the limit extreme valuedistributions 18.3.3.6 Median bias correction and confidence intervals. ................ 18.3.3.7 Extrapolation estimator for very extremes ...................... . 18.3.4 Extreme value versus normal inference..................... ........................... 18.4 Empirical applications....................................................................................... 18.4.1 Value-at-risk
prediction.................................... 18.4.2 Contagion of financial risk............... ...................................................... 19 Quantile Regression Methods for Longitudinal Data Antonio F.· Galvao and Kengo Kato 19.1 Introduction ............... 19.2 Panel quantile regression model . . ........................................... 19.3 Fixed effects estimation..................................................................................... 19.3.1 FE-QR estimator............................................................ 19.3.2 FE-SQR estimator ................................................. 19.3.2.1 Bias correction: Analytical method . . . ............................... 19.3.2.2 Bias correction: Jackknife....................................................... 19.3.3 Alternative FE approaches . ................................................................ 19.3.3.1 Shrinkage ................................................... 19.3.3.2 Minimum distance................................................ 19.3.3.3 Two-step estimation of Canay (2011)................................. 19.4 Correlated random effects.................................................................................. 19.5 Extensions ...................................................................... 19.5.1 Endogeneity............................................................... 19.5.2 Censoring.................................................................................. 19.5.3 Group-level treatments.
................................................................ 19.5.4 Semiparametric QR for longitudinal data ....................................... . 19.6 Conclusion ........................................................................................................... 20 Quantile Regression Applications in Finance Oliver Linton and Zhijie Xiao 20.1 Introduction ........................................................................................................ 20.2 Quantile regression in risk management.............................. 20.2.1 Value-at-risk............................................................................................ 20.2.2 Expected shortfall . ................................................................................ 20.3 Upper quantile information and financial markets ...................................... 20.4 Quantile regression and portfolio allocation.................................................... XV 338 338 339 339 340 341 342 343 344 344 345 345 346 346 347 349 349 350 350 351 354 363 363 365 366 366 368 369 370 371 371 371 372 373 374 374 374 376 377 378 381 381 383 383 388 391 393
Contents xvi 20.4.1 The mean-ES portfolio construction................... 20.4.2 The multi-quantile portfolio construction · ■ · · 20.5 Stochastic dominance and quantile regression................ 20.6 Quantile dependence ........................................................ 20.6.1 Directional predictability via the quantilogram . 20.6.2 Causality in quantiles............................................ 20.7 Concluding remarks........................................................... 21 Quantile Regression for Genetic and Genomic Applications Laurent Briollais and Gilles Durrieu 21.1 Introduction ......................................................................................................... 21.2 Genetic applications............................................................................................. 21.2.1 Background and definitions..................................................................... 21.2.2 Candidate gene association study of child BMI................................... 21.2.3 GWAS of birthweight .............................................................................. 21.2.4 Genetic association with a set of markers............................................ 21.3 Genomic and other -omic applications.............................................................. 21.3.1 Background.................. 21.3.2 Genomic data pre-processing................................................................. 21.3.3 Sample size determination in gene expression studies......................... 21.3.4 Determination of chromosomal region
aberrations............................... 21.3.5 Robust estimation and outlier determination in genomics................ 21.3.6 Genomic analysis of set of genes........................................................... 21.4 Conclusion ............................................................................................................ 394 395 397 399 400 402 403 409 409 410 410 411 412 414 415 415 416 417 419 420 422 423 22 Quantile Regression Applications in Ecology and the Environmental Sci ences 429 Brian S. Cade 22.1 Introduction ................................................................ 22.2 Water quality trends over time........................ .................................................. 22.2.1 A single site within a watershed ............................................................ 22.2.2 Multiple sites within a watershed......................................... 22.2.3 Estimation with below-detection limit values in a single site within a watershed.................................. ................................................................. 22.2.4 Additional extensions possible for water quality and flow trend anal yses ................................................ 22.3 Herbaceous plant species diversity and atmosphericnitrogen deposition . . 22.3.1 Quantile regression estimates................................................................... 22.3.2 Partial effects of nitrogen deposition and pH and critical loads . . . 22.3.3 Additional possible refinements tothe model........................................ 22.4 Discussion
........................................... Index 429 431 432 436 439 442 444 444 449 ΛΛη 455
Preface Quantile regression constitutes a family of statistical techniques intended to estimate and draw inferences about conditional quantile functions. Median regression as introduced in the eighteenth century by Boscovich and Laplace is a (central) special case. In contrast to conventional mean regression that minimizes sums of squared residuals, median regression minimizes sums of absolute residuals; quantile regression simply replaces symmetric absolute loss by asymmetric linear loss. Since its introduction in Koenker and Bassett (1978), quantile regression has gradually been extended to a wide variety of data analytic settings, including time series, survival analysis, and longitudinal data. By focusing attention on local slices of the conditional dis tribution of response variables, it is capable of providing a more complete, more nuanced view of heterogeneous covariate effects. Applications of quantile regression can now be found throughout the sciences, including astrophysics, chemistry, ecology, economics, finance, ge nomics, medicine, and meteorology. Software for quantile regression is now widely available in all the major statistical computing environments. Our objective for this handbook has been to provide a thorough review of recent devel opments of quantile regression methodology, illustrating its applicability in a wide variety of scientific settings. The intended audience for the volume is students and more senior researchers across a diverse set of disciplines. The text consists of 22 chapters by leading scholars in the field, including reviews of
recent work on resampling methods for inference, nonparametric penalty methods for func tion estimation, computational methods for large data applications, and Bayesian methods. Survival analysis has been a fertile growth area for quantile regression methods, and there are three chapters devoted to these developments. Causal inference has been another very active research area, and there are two chapters devoted to this topic. High-dimensional models and model selection have become obsessions in the recent statistical literature; two chapters confront these issues. Time series, longitudinal data, errors in variables, missing data, and sample selection all pose unique challenges for quantile regression modeling and are treated in four distinct chapters. Extension of quantile regression methods to multivari ate and functional response are surveyed in two chapters. And finally, three chapters are devoted to surveys of applications in ecology, genomics, and finance. We would like to offer our profound thanks to all the contributors to the handbook for their dedicated work on this project and for their prior contributions to the literature on quantile regression. We would also like to express our appreciation to Rob Calver, our editor, for his encouragement throughout the course of the project, and to Richard Leigh, our copy-editor, for his careful treatment of the manuscript. Finally, to the many others who have contributed to the development of quantile regression methods through theory and applications, we say thanks and express the hope that this volume will help to foster
further developments. Roger Koenker, Victor Chernozhukov, Xuming He, and Limin Peng April 2017 XVII
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Contents Preface xvii Contributors xix I Introduction 1 1 A Quantile Regression Memoir 3 Gilbert W. Bassett Jr. and Roger Koenker 1.1 Long ago . .v . . 2 Resampling Methods 3 7 Xuming He 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Introduction ՛. . . . . . і. . Paired bootstrap . . . . Residual-based bootstrap. . . . . Generalized bootstrap . Estimating function bootstrap . . . . . . . . . . . . . Markov chain marginal bootstrap . і . . Resampling methods for clustered data . . . Resampling methods for censored quantile regression . Bootstrap for post-model selection inference . 3 Quantile Regression: Penalized 7 8 9 11 11 12 13 14 15 21 Ivan Mizera 3.1 3.2 3.3 Penalized: how? . . . . . 3.1.1 A probability path . . . . . . 3.1.2 Regularization of ill-posed problems . Penalized: what?. 3.2.1 The finite differences of Whittaker and others. . 3.2.2 Functions and theirderivatives. 3.2.3 Quantile regression with smoothing splines . 3.2.4 Quantile smoothing splines. . . . . 3.2.5 Total-variation splines
. Penalized: what else? . . . . . . . . . . 3.3.1 Tuning . . . . . . . 3.3.2 Multiple covariates . 3.3.3 Additive fits, confidencebandaids, and other phantasmagorias 4 Bayesian Quantile Regression 21 21 22 23 23 24 26 28 29 31 31 32 34 41 4.1 4.2 4.3 Introduction . . ■ · Asymmetric Laplace likelihood . . . . . Empirical likelihood. . Huiría Judy Wang and Yunwen Yang 41 42^ 45 ix
Contents 4.4 Nonparametric and semiparametric likelihoods . 4.4.1 Mixture-type likelihood. 4.4.2 Approximate likelihood via quantile process 4.5 Discussion 5 6 7 Computational Methods for Quantile Regression 55 Roger Koenker 5.1 Introduction . 5.2 Exterior point methods . 5.3 Interior point methods . 5.4 Preprocessing . 5.5 First-order, proximal methods . 5.5.1 Proximal operators and the Moreau envelope . 5.5.2 Alternating direction method of multipliers . . 5.5.3 Proximal performance. 55 57 58 60 61 61 64 65 Survival Analysis: A Quantile Perspective Zhiliang Ying and Tony Sit 6.1 Introduction . 6.1.1 Notation. 6.1.2 Censoring. 6.2 Important models . 6.2.1 Parametric models. 6.2.2 Nonparametric estimators. 6.2.2.1 Kaplan-Meier estimator. 6.2.2.2 Nelson-Aalen estimator. 6.2.3 Regression models. 6.2.3.1 Cox proportional hazards
model. 6.2.3.2 Accelerated failure time model. 6.2.3.3 Aalen additive hazard model. 6.3 Quantile estimation based on censored data. 6.3.1 Quantile estimation . 6.3.2 Median and quantile regression. 6.3.3 Discussion and miscellanea. 73 75 75 75 76 78 79 79 80 82 Quantile Regression for Survival Analysis 89 Limin Peng 7.1 Introduction . 7.2 Quantile regression for randomly censored data. 7.2.1 Random right censoring with C always known . 7.2.2 Covariate-independent random right censoring. 7.2.3 Standard random right censoring. 7.2.3.1 Approaches based on the principle of self-consistency . 7.2.3.2 Martingale-based approach. 7.2.3.3 Locally weighted method. 7.2.4 Variance estimation and other inference. 7.2.4.1 Variance estimation.
7.2.4.2 Second-stage inference. 7.2.4.3 Model checking. 7.3 Quantile regression in other survival settings . ' [ 7.3.1 Known random left censoring and/or left truncation. 89 90 90 91 92 92 93 94 95 95 gg gg 97 97
Contents 7.4 7.3.2 Censored data with a survival cure fraction. An illustration of quantile regression for survival analysis. 8 Survival Analysis with Competing Risks and Semi-competing Risks Data Ruosha Li and Limin Peng 8.1 Competing risks data . 8.1.1 Introduction . 8.1.2 Cumulative incidence quantile regression. 8.1.3 Data analysis example. 8.1.4 Marginal quantile regression. 8.2 Semi-competing risks data . 8.2.1 Introduction . 8.2.2 Cumulative incidence quantile regression. 8.2.3 Marginal quantile regression. . . . 8.3 Summary and open problems. xi 98 98 105 105 105 106 108 110 112 112 113 114 116 9 Instrumental Variable Quantile Regression 119 Victor Ghernozhukov, Christian Hansen, and Kaspar Wiithrich 9.1 Introduction . 120 9.2 Model overview
. 121 9.2.1 The instrumental variable quantile regressionmodel. 121 9.2.2 Conditions for point identification. 123 9.2.3 Discussion of the IVQR model . . . . 124 9.2.4 Examples. 126 9.2.5 Comparison to other approaches. 128 9.3 Basic estimation and inference approaches . 129 9.3.1 Generalized methods of moments and relatedapproaches . 130 9.3.2 Inverse quantile regression. 132 9.3.2.1 A useful interpretation of IQR as a GMM estimator. 133 9.3.3 Weak identification robust inference. 134 9.3.4 Finite-sample inference . 136 9.4 Advanced inference with high-dimensional X. 136 9.4.1 Neyman orthogonal scores . 136 9.4.2 Estimation and inference using orthogonal scores. 138 9.5 Conclusion . 139 10 Local Quantile Treatment Effects Blaise Melly and Kaspar Wiithrich 10.1 Introduction
. 10.2 Framework, estimands and identification . 10.2.1 Without covariates. 10.2.2 In the presence of covariates: conditional LQTE . 10.2.3 In the presence of covariates: unconditional LQTE. 10.3 Estimation and inference . 10.4 Extensions . 10.4.1 Regression discontinuity design . 10.4.2 Multi-valued and continuous instruments . 10.4.3 Testing instrument validity. 10.5 Comparison to the instrumental variablequantile regression model. 10.6 Conclusion and open problems. 145 145 148 148 151 152 154 155 155 156 157 158 160
Contents xii 11 Quantile Regression with Measurement Errors and Missing Data Ying Wei 11.1 Introduction . 11.2 Quantile regression with measurement errors . 11.2.1 Linear quantile regression with measurement errors. 11.2.1.1 Semiparametric joint estimating equations . . 11.2.1.2 Other methods for linear quantile regression with measure• ment errors. 11.2.2 Nonparametric and semiparametric quantile regression model with measurement errors . 11.3 Quantile regression with missing data . 11.3.1 Statistical methods handling missing covariates in quantile regression 11.3.1.1 Multiple imputation algorithm. 11.3.1.2 Modified MI algorithms. 11.3.1.3 EM algorithm. 11.3.1.4 IPW algorithms. 11.3.2 Statistical methods handling missing outcomes in quantile regression 11.3.2.1 Imputation approaches for missing outcomes . 11.3.2.2 Statistical methods for longitudinal dropout . 12 Multiple-Output Quantile
Regression Marc Hallin and Miroslav Siman 12.1 Multivariate quantiles, and the ordering of Rd, d џ 2 . 12.2 Directional approaches . 12.2.1 Projection methods . . 12.2.1.1 Marginal (coordinatewise) quantiles. 12.2.1.2 Quantile biplots. 12.2.1.3 Directional quantile hyperplanes and contours 12.2.1.4 Relation to halfspace depth. 12.2.2 Directional Koenker-Bassett methods. 12.2.2.1 Location case (p = 0). 12.2.2.2 (Nonparametric) regression case (p ^ 1) 12.3 Direct approaches . 12.3.1 Spatial (geometric) quantile methods . ··■· 12.3.1.1 A spatial check function. 12.3.1.2 Linear spatial quantile regression. 12.3.1.3 Nonparametric spatial quantile regression . . 12.3.2 Elliptical quantiles . 12.3.2.1 Location case. 12.3.2.2 Linear regression case. 12.3.3 Depth-based quantiles . 12.3.3.1 Halfspace depth quantiles. 12.3.3.2 Monge-Kantorovich quantiles. 12.4 Some other concepts, and applications . 12.5
Conclusion . 13 Sample Selection in Quantile Regression: A Survey Manuel Arellano and Stéphane Bonhomme 13.1 Introduction . . . 13.2 Heckman’s parametric selection model 13.2.1 Two-step estimation in Gaussian models 165 165 166 166 166 169 171 172 173 173 175 177 178 178 178 180 185 185 187 187 187 187 188 189 189 189 191 193 195 195 196 197 197 197 198 200 200 201 203 204 209 209 211 212
Contents xiii 13.3 A quantile generalization . . 212 13.3.1 A quantile selection model. . . 212 13.3.2 Estimation. 213 13.4 Identification. 216 13.5 Other approaches . 216 13.5.1 A likelihood approach. 217 13.5.2 Control function approaches . . . 217 13.5.3 Link to censoring corrections. 217 13.6 Empirical illustration . 218 13.7 Conclusion . 221 14 Nonparametric Quantile Regression for Banach-Valued Response 225 Joydeep Chowdhury and Probal Chaudhuri 14.1 Introduction . 225 14.2 Regression quantiles in Banachspaces . . 229 14.3 Nonparametric estimation . 231 14.4 Data analysis. 232 14.4.1 Simulation . . . . . 232 14.4.2 Tecator data . 234 14.4.3 Pediatric airway
data. 234 14.4.4 Cigarette data. 237 14.4.4.1 Regression of price curve on sales curve . . 237 14.5 Consistency . 240 14.5.1 Additional mathematical details . . . 245 14.6 Concluding remarks . . . . 249 15 High-Dimensional Quantile Regression 253 Alexandre Belloni, Victor Chernozhukov, and Kengo Kato 15.1 Introduction . 253 15.2 Estimation of the conditional quantile function. 256 15.2.1 Regularity conditions . 256 15.2.2 fi-penalized quantile regression. 257 15.2.3 Refitted quantile regression after selection. 259 15.2.4 Group lasso for quantile regression models. 260 15.2.5 Estimation of the conditional density. . 261 15.3 Confidence bands for the coefficient process. 261 15.3.1 Construction of an orthogonal score function. 263 15.3.2 Regularity conditions
. 265 15.3.3 Score function estimator. 266 15.3.4 Double selection estimator. 267 15.3.5 Confidence bands. 267 15.3.6 Confidence bands via inverse statistics. 269 16 Nonconvex Penalized Quantile Regression: A Review of Methods, Theory and Algorithms 273 Lan Wang 16.1 Introduction . 273 16.2 High-dimensional sparse linearquantile regression . 275 16.2.1 Background on penalized high-dimensional regression and the choice of penalty function. 275 16.2.2 Nonconvex penalized high-dimensional linear quantile regression . . 276
Contents xiv 16.3 16.4 16.5 16.6 16.2.2.1 Overview. ****** *. 16.2.2.2 Oracle property of the nonconvex penalized quantile regres sion estimator. High-dimensional sparse semiparametric quantile regression. 16.3.1 Overview. . -. . . * ' 16.3.2 Nonconvex penalized partially linear additive quantile regression . . 16.3.3 Oracle properties.;. Computational aspects of nonconvex penalized quantile regression. 16.4.1 Linear programming based algorithms (moderately large p). 16.4.2 New iterative coordinate descent algorithm (larger p). Other related problems . . · 16.5.1 Simultaneous estimation and variable selection at multiple quantiles 16.5.2 Two-stage analysis with quantile-adaptive screening. 16.5.2.1 Background. 16.5.2.2 Quantile-adaptive model-free nonlinear screening. Discussion . 17 QAR and Quantile Time Series Analysis Zhijie Xiao 17.1 Introduction
. 17.2 Quantile regression estimation of traditional time series models. 17.2.1 Quantile regression estimation of the traditional AR model. 17.2.2 Quantile regressions of other time series models with i.i.d. errors . . 17.2.3 Quantile regression estimation of ARMA models. 17.2.4 Quantile regressions with serially correlated errors. 17.3 Quantile regressions with ARCH/GARCH errors. 17.4 Quantile regressions with heavy-tailed errors . 17.5 Quantile regression for nonstationary time series. 17.5.1 Quantile regression for trending time series. 17.5.2 Unit-root quantile regressions. 17.5.3 Quantile regression on cointegrated time series. 17.6 The QAR process . 17.6.1 The linear QAR process. 17.6.2 Nonlinear QAR models. 17.6.3 Quantile autoregression based on transformations. 17.7 Other dynamic quantile models . 17.8 Quantile
spectral analysis.-.·. 17.8.1 Quantile cross-covariances and quantile spectrum . . . 17.8.2 Quantile periodograms. 17.8.3 Relationship to quantile regression on harmonic regressors. 17.8.4 Estimation of quantile spectral density. 17.9 Quantile regression based forecasting . 17.10Conclusion . 18 Extremal Quantile Regression Victor Chemozhukov, Iván Femández-Val, and Tetsuya Kaji 18.1 Introduction . 18.2 Extreme quantile models . 18.2.1 Pareto-type and regularly varying tails. 18.2.2 Extremal quantile regression models. 18.3 Estimation and inference methods 276 278 279 279 279 280 281 281 282 283 283 284 284 284 285 293 293 294 295 296 297 298 299 306 307 307 308 310 312 313 317 319 320 322 323 324 325 327 32g oor 333 334 336 336 337 338
Contents 18.3.1 Sampling conditions. 18.3.2 Univariave case: Marginal quantiles . . 18.3.2.1 Extreme order approximation . 18.3.2.2 Intermediate order approximation. 18.3.2.3 Estimation of ξ . 18.3.2.4 Estimation of Αχ . 18.3.2.5 Computing quantiles of the limit extreme valuedistributions 18.3.2.6 Median bias correction and confidence intervals. 18.3.2.7 Extrapolation estimator for very extremes . 18.3.3 Multivariate case: Conditional quantiles. 18.3.3.1 Extreme order approximation. 18.3.3.2 Intermediate order approximation . 18.3.3.3 Estimation of ξ and 7 . . 18.3.3.4 Estimation of Αχ . 18.3.3.5 Computing quantiles of the limit extreme valuedistributions 18.3.3.6 Median bias correction and confidence intervals. . 18.3.3.7 Extrapolation estimator for very extremes . . 18.3.4 Extreme value versus normal inference. . 18.4 Empirical applications. 18.4.1 Value-at-risk
prediction. 18.4.2 Contagion of financial risk. . 19 Quantile Regression Methods for Longitudinal Data Antonio F.· Galvao and Kengo Kato 19.1 Introduction . 19.2 Panel quantile regression model . . . 19.3 Fixed effects estimation. 19.3.1 FE-QR estimator. 19.3.2 FE-SQR estimator . 19.3.2.1 Bias correction: Analytical method . . . . 19.3.2.2 Bias correction: Jackknife. 19.3.3 Alternative FE approaches . . 19.3.3.1 Shrinkage . 19.3.3.2 Minimum distance. 19.3.3.3 Two-step estimation of Canay (2011). 19.4 Correlated random effects. 19.5 Extensions . 19.5.1 Endogeneity. 19.5.2 Censoring. 19.5.3 Group-level treatments.
. 19.5.4 Semiparametric QR for longitudinal data . . 19.6 Conclusion . 20 Quantile Regression Applications in Finance Oliver Linton and Zhijie Xiao 20.1 Introduction . 20.2 Quantile regression in risk management. 20.2.1 Value-at-risk. 20.2.2 Expected shortfall . . 20.3 Upper quantile information and financial markets . 20.4 Quantile regression and portfolio allocation. XV 338 338 339 339 340 341 342 343 344 344 345 345 346 346 347 349 349 350 350 351 354 363 363 365 366 366 368 369 370 371 371 371 372 373 374 374 374 376 377 378 381 381 383 383 388 391 393
Contents xvi 20.4.1 The mean-ES portfolio construction. 20.4.2 The multi-quantile portfolio construction · ■ · · 20.5 Stochastic dominance and quantile regression. 20.6 Quantile dependence . 20.6.1 Directional predictability via the quantilogram . 20.6.2 Causality in quantiles. 20.7 Concluding remarks. 21 Quantile Regression for Genetic and Genomic Applications Laurent Briollais and Gilles Durrieu 21.1 Introduction . 21.2 Genetic applications. 21.2.1 Background and definitions. 21.2.2 Candidate gene association study of child BMI. 21.2.3 GWAS of birthweight . 21.2.4 Genetic association with a set of markers. 21.3 Genomic and other -omic applications. 21.3.1 Background. 21.3.2 Genomic data pre-processing. 21.3.3 Sample size determination in gene expression studies. 21.3.4 Determination of chromosomal region
aberrations. 21.3.5 Robust estimation and outlier determination in genomics. 21.3.6 Genomic analysis of set of genes. 21.4 Conclusion . 394 395 397 399 400 402 403 409 409 410 410 411 412 414 415 415 416 417 419 420 422 423 22 Quantile Regression Applications in Ecology and the Environmental Sci ences 429 Brian S. Cade 22.1 Introduction . 22.2 Water quality trends over time. . 22.2.1 A single site within a watershed . 22.2.2 Multiple sites within a watershed. 22.2.3 Estimation with below-detection limit values in a single site within a watershed. . 22.2.4 Additional extensions possible for water quality and flow trend anal yses . 22.3 Herbaceous plant species diversity and atmosphericnitrogen deposition . . 22.3.1 Quantile regression estimates. 22.3.2 Partial effects of nitrogen deposition and pH and critical loads . . . 22.3.3 Additional possible refinements tothe model. 22.4 Discussion
. Index 429 431 432 436 439 442 444 444 449 ΛΛη 455
Preface Quantile regression constitutes a family of statistical techniques intended to estimate and draw inferences about conditional quantile functions. Median regression as introduced in the eighteenth century by Boscovich and Laplace is a (central) special case. In contrast to conventional mean regression that minimizes sums of squared residuals, median regression minimizes sums of absolute residuals; quantile regression simply replaces symmetric absolute loss by asymmetric linear loss. Since its introduction in Koenker and Bassett (1978), quantile regression has gradually been extended to a wide variety of data analytic settings, including time series, survival analysis, and longitudinal data. By focusing attention on local slices of the conditional dis tribution of response variables, it is capable of providing a more complete, more nuanced view of heterogeneous covariate effects. Applications of quantile regression can now be found throughout the sciences, including astrophysics, chemistry, ecology, economics, finance, ge nomics, medicine, and meteorology. Software for quantile regression is now widely available in all the major statistical computing environments. Our objective for this handbook has been to provide a thorough review of recent devel opments of quantile regression methodology, illustrating its applicability in a wide variety of scientific settings. The intended audience for the volume is students and more senior researchers across a diverse set of disciplines. The text consists of 22 chapters by leading scholars in the field, including reviews of
recent work on resampling methods for inference, nonparametric penalty methods for func tion estimation, computational methods for large data applications, and Bayesian methods. Survival analysis has been a fertile growth area for quantile regression methods, and there are three chapters devoted to these developments. Causal inference has been another very active research area, and there are two chapters devoted to this topic. High-dimensional models and model selection have become obsessions in the recent statistical literature; two chapters confront these issues. Time series, longitudinal data, errors in variables, missing data, and sample selection all pose unique challenges for quantile regression modeling and are treated in four distinct chapters. Extension of quantile regression methods to multivari ate and functional response are surveyed in two chapters. And finally, three chapters are devoted to surveys of applications in ecology, genomics, and finance. We would like to offer our profound thanks to all the contributors to the handbook for their dedicated work on this project and for their prior contributions to the literature on quantile regression. We would also like to express our appreciation to Rob Calver, our editor, for his encouragement throughout the course of the project, and to Richard Leigh, our copy-editor, for his careful treatment of the manuscript. Finally, to the many others who have contributed to the development of quantile regression methods through theory and applications, we say thanks and express the hope that this volume will help to foster
further developments. Roger Koenker, Victor Chernozhukov, Xuming He, and Limin Peng April 2017 XVII |
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| spelling | Handbook of quantile regression edited by Roger Koenker, Victor Chernozhukov, Xuming He, Limin Peng Boca Raton CRC Press [2020] © 2018 xix, 463 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Handbooks of modern statistical methods Includes bibliographical references and index Quantile regression Regression analysis Quantil (DE-588)4224812-7 gnd rswk-swf Regressionsanalyse (DE-588)4129903-6 gnd rswk-swf (DE-588)4143413-4 Aufsatzsammlung gnd-content Regressionsanalyse (DE-588)4129903-6 s Quantil (DE-588)4224812-7 s b DE-604 Koenker, Roger 1947- (DE-588)131874632 edt Chernozhukov, Victor (DE-588)129354759 edt He, Xuming ca. 20/21. Jahrhundert (DE-588)170895238 edt Peng, Limin (DE-588)136267823 edt Digitalisierung UB Bamberg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032516567&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| title | Handbook of quantile regression |
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| title_exact_search | Handbook of quantile regression |
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| title_full | Handbook of quantile regression edited by Roger Koenker, Victor Chernozhukov, Xuming He, Limin Peng |
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| topic | Quantile regression Regression analysis Quantil (DE-588)4224812-7 gnd Regressionsanalyse (DE-588)4129903-6 gnd |
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