Handbook of Quantile Regression 1st Edition by Roger Koenker ISBN 1351646567 9781351646567

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ISBN 10: 1351646567
ISBN 13: 9781351646567
Author: Roger Koenker

Quantile regression constitutes an ensemble of statistical techniques intended to estimate and draw inferences about conditional quantile functions. Median regression, as introduced in the 18th century by Boscovich and Laplace, is a 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 the 1970’s by Koenker and Bassett, quantile regression has been gradually 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 distribution 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, genomics, medicine, and meteorology. Software for quantile regression is now widely available in all the major statistical computing environments. The objective of this volume is to provide a comprehensive review of recent developments of quantile regression methodology illustrating its applicability in a wide range of scientific settings. The intended audience of the volume is researchers and graduate students across a diverse set of disciplines.

Handbook of Quantile Regression 1st Table of contents:

1. A Quantile Regression Memoir

1.1 Long ago …

2. Resampling Methods

2.1 Introduction

2.2 Paired bootstrap

2.3 Residual-based bootstrap

2.4 Generalized bootstrap

2.5 Estimating function bootstrap

2.6 Markov chain marginal bootstrap

2.7 Resampling methods for clustered data

2.8 Resampling methods for censored quantile regression

2.9 Bootstrap for post-model selection inference

3. Quantile Regression: Penalized

3.1 Penalized: how?

3.1.1 A probability path

3.1.2 Regularization of ill-posed problems

3.2 Penalized: what?

3.2.1 The finite differences of Whittaker and others

3.2.2 Functions and their derivatives

3.2.3 Quantile regression with smoothing splines

3.2.4 Quantile smoothing splines

3.2.5 Total-variation splines

3.3 Penalized: what else?

3.3.1 Tuning

3.3.2 Multiple covariates

3.3.3 Additive fits, confidence bandaids, and other phantasmagorias

4. Bayesian Quantile Regression

4.1 Introduction

4.2 Asymmetric Laplace likelihood

4.3 Empirical likelihood

4.4 Nonparametric and semiparametric likelihoods

4.4.1 Mixture-type likelihood

4.4.2 Approximate likelihood via quantile process

4.5 Discussion

5. Computational Methods for Quantile Regression

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

6. Survival Analysis: A Quantile Perspective

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

7. Quantile Regression for Survival Analysis

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

7.3.2 Censored data with a survival cure fraction

7.4 An illustration of quantile regression for survival analysis

8. Survival Analysis with Competing Risks and Semi-competing Risks Data

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

9. Instrumental Variable Quantile Regression

9.1 Introduction

9.2 Model overview

9.2.1 The instrumental variable quantile regression model

9.2.2 Conditions for point identification

9.2.3 Discussion of the IVQR model

9.2.4 Examples

9.2.5 Comparison to other approaches

9.3 Basic estimation and inference approaches

9.3.1 Generalized methods of moments and related approaches

9.3.2 Inverse quantile regression

9.3.2.1 A useful interpretation of IQR as a GMM estimator.

9.3.3 Weak identification robust inference

9.3.4 Finite-sample inference

9.4 Advanced inference with high-dimensional X

9.4.1 Neyman orthogonal scores

9.4.2 Estimation and inference using orthogonal scores

9.5 Conclusion

10. Local Quantile Treatment Effects

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 variable quantile regression model

10.6 Conclusion and open problems

11. Quantile Regression with Measurement Errors and Missing Data

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 measurement 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

12.1 Multivariate quantiles, and the ordering of

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 ()

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

13.1 Introduction

13.2 Heckman’s parametric selection model

13.2.1 Two-step estimation in Gaussian models

13.3 A quantile generalization

13.3.1 A quantile selection model

13.3.2 Estimation

13.4 Identification

13.5 Other approaches

13.5.1 A likelihood approach

13.5.2 Control function approaches

13.5.3 Link to censoring corrections

13.6 Empirical illustration

13.7 Conclusion

14. Nonparametric Quantile Regression for Banach-Valued Response

14.1 Introduction

14.2 Regression quantiles in Banach spaces

14.3 Nonparametric estimation

14.4 Data analysis

14.4.1 Simulation

14.4.2 Tecator data

14.4.3 Pediatric airway data

14.4.4 Cigarette data

14.4.4.1 Regression of price curve on sales curve

14.5 Consistency

14.5.1 Additional mathematical details

14.6 Concluding remarks

15. High-Dimensional Quantile Regression

15.1 Introduction

15.2 Estimation of the conditional quantile function

15.2.1 Regularity conditions

15.2.2 -penalized quantile regression

15.2.3 Refitted quantile regression after selection

15.2.4 Group lasso for quantile regression models

15.2.5 Estimation of the conditional density

15.3 Confidence bands for the coefficient process

15.3.1 Construction of an orthogonal score function

15.3.2 Regularity conditions

15.3.3 Score function estimator

15.3.4 Double selection estimator

15.3.5 Confidence bands

15.3.6 Confidence bands via inverse statistics

16. Nonconvex Penalized Quantile Regression: A Review of Methods, Theory and Algorithms

16.1 Introduction

16.2 High-dimensional sparse linear quantile regression

16.2.1 Background on penalized high-dimensional regression and the choice of penalty function

16.2.2 Nonconvex penalized high-dimensional linear quantile regression

16.2.2.1 Overview

16.2.2.2 Oracle property of the nonconvex penalized quantile regression estimator

16.3 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

16.4 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)

16.5 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

16.6 Discussion

17. QAR and Quantile Time Series Analysis

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

17.10 Conclusion

18. Extremal Quantile Regression

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

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

18.3.2.4 Estimation of

18.3.2.5 Computing quantiles of the limit extreme value distributions

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 and

18.3.3.4 Estimation of

18.3.3.5 Computing quantiles of the limit extreme value distributions

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

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

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

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

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

22. Quantile Regression Applications in Ecology and the Environmental Sciences

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 analyses

22.3 Herbaceous plant species diversity and atmospheric nitrogen 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 to the model

22.4 Discussion

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