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E-raamat: Model-Free Prediction and Regression: A Transformation-Based Approach to Inference

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Model-Free Prediction and Regression: A Transformation-Based Approach to InferenceSuurem pilt
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The Model-Free Prediction Principle expounded upon in this monograph is based on the simple notion of transforming a complex dataset to one that is easier to work with, e.g., i.i.d. or Gaussian. As such, it restores the emphasis on observable quantities, i.e., current and future data, as opposed to unobservable model parameters and estimates thereof, and yields optimal predictors in diverse settings such as regression and time series. Furthermore, the Model-Free Bootstrap takes us beyond point prediction in order to construct frequentist prediction intervals without resort to unrealistic assumptions such as normality.

Prediction has been traditionally approached via a model-based paradigm, i.e., (a) fit a model to the data at hand, and (b) use the fitted model to extrapolate/predict future data. Due to both mathematical and computational constraints, 20th century statistical practice focused mostly on parametric models. Fortunately, with the advent of widely accessible powerful computing in the late 1970s, computer-intensive methods such as the bootstrap and cross-validation freed practitioners from the limitations of parametric models, and paved the way towards the `big data' era of the 21st century. Nonetheless, there is a further step one may take, i.e., going beyond even nonparametric models; this is where the Model-Free Prediction Principle is useful.

Interestingly, being able to predict a response variable Y associated with a regressor variable X taking on any possible value seems to inadvertently also achieve the main goal of modeling, i.e., trying to describe how Y depends on X. Hence, as prediction can be treated as a by-product of model-fitting, key estimation problems can be addressed as a by-product of being able to perform prediction. In other words, a practitioner can use Model-Free Prediction ideas in order to additionally obtain point estimates and confidence intervals for relevant parameters leading to an alternative, transformation-based approach to statistical inference.

Arvustused

The monograph restores the emphasis on observable quantities. Considering model-free and model-based prediction, the monograph emphasises on model-free approach but also shows the close relation between these two approaches. The book is of interest for both academics and practitioners in the field of data analysis. (Pavel Stoynov, zbMATH 1397.62008, 2018)

This self-contained and fascinating book, intended for advanced graduate and Ph.D. students, teachers, researchers and practitioners, is useful, well written, and directly oriented toward real data applications. (Gilles Teyssière, Mathematical Reviews, February, 2017)

Part I The Model-Free Prediction Principle
1 Prediction: Some Heuristic Notions
3(10)
1.1 To Explain or to Predict?
3(3)
1.2 Model-Based Prediction
6(3)
1.3 Model-Free Prediction
9(4)
2 The Model-Free Prediction Principle
13(20)
2.1 Introduction
13(1)
2.2 Model-Free Approach to Prediction
14(3)
2.2.1 Motivation: The i.i.d. Case
14(1)
2.2.2 The Model-Free Prediction Principle
14(3)
2.3 Tools for Identifying a Transformation Towards i.i.d.--Ness
17(5)
2.3.1 Model-Free Prediction as an Optimization Problem
17(1)
2.3.2 Transformation into Gaussianity as a Stepping Stone
17(2)
2.3.3 Existence of a Transformation Towards i.i.d.--Ness
19(1)
2.3.4 A Simple Check of the Model-Free Prediction Principle
20(1)
2.3.5 Model-Free Model-Fitting in Practice
21(1)
2.4 Model-Free Predictive Distributions
22(11)
2.4.1 Prediction Intervals and Asymptotic Validity
22(1)
2.4.2 Predictive Roots and Model-Free Bootstrap
23(3)
2.4.3 Limit Model-Free Resampling Algorithm
26(2)
2.4.4 Prediction of Discrete Variables
28(5)
Part II Independent Data: Regression
3 Model-Based Prediction in Regression
33(24)
3.1 Model-Based Regression
33(3)
3.2 Model-Based Prediction in Regression
36(1)
3.3 A First Application of the Model-Free Prediction Principle
37(1)
3.4 Model-Free/Model-Based Prediction
38(2)
3.5 Model-Free/Model-Based Prediction Intervals
40(3)
3.6 Pertinent Prediction Intervals
43(4)
3.6.1 The i.i.d. Case
43(2)
3.6.2 Asymptotic Pertinence of Bootstrap Prediction Intervals
45(2)
3.7 Application to Linear Regression
47(10)
3.7.1 Better Prediction Intervals in Linear Regression
47(2)
3.7.2 Simulation: Prediction Intervals in Linear Regression
49(1)
3.7.3 Model-Free vs. Least Squares: A Reconciliation
50(3)
Appendix 1 The Solution of Eq. (3.9)
53(1)
Appendix 2 L1 vs. L2 Cross-Validation
54(3)
4 Model-Free Prediction in Regression
57(24)
4.1 Introduction
57(1)
4.2 Constructing the Transformation Towards i.i.d.--Ness
58(5)
4.3 Model-Free Optimal Predictors
63(5)
4.3.1 Model-Free and Limit Model-Free Optimal Predictors
63(1)
4.3.2 Asymptotic Equivalence of Point Predictors
64(3)
4.3.3 Cross-Validation for Model-Free Prediction
67(1)
4.4 Model-Free Bootstrap
68(2)
4.5 Predictive Model-Free Bootstrap
70(1)
4.6 Model-Free Diagnostics
71(1)
4.7 Simulations
72(9)
4.7.1 When a Nonparametric Regression Model Is True
72(4)
4.7.2 When a Nonparametric Regression Model Is Not True
76(2)
Acknowledgements
78(1)
Appendix 1 High-Dimensional and/or Functional Regressors
78(3)
5 Model-Free vs. Model-Based Confidence Intervals
81(16)
5.1 Introduction
81(1)
5.2 Model-Based Confidence Intervals in Regression
82(3)
5.3 Model-Free Confidence Intervals Without an Additive Model
85(4)
5.4 Simulations
89(8)
5.4.1 When a Nonparametric Regression Model Is True
89(3)
5.4.2 When a Nonparametric Regression Model Is Not True
92(1)
Acknowledgements
93(4)
Part III Dependent Data: Time Series
6 Linear Time Series and Optimal Linear Prediction
97(16)
6.1 Introduction
97(2)
6.2 Optimal Linear Prediction
99(1)
6.3 Linear Prediction Using the Complete Process History
100(3)
6.3.1 Autocovariance Matrix Estimation
101(1)
6.3.2 Data-Based Choice of the Banding Parameter
102(1)
6.4 Correcting a Matrix Towards Positive Definiteness
103(3)
6.4.1 Eigenvalue Thresholding
103(1)
6.4.2 Shrinkage of Problematic Eigenvalues
104(1)
6.4.3 Shrinkage Towards White Noise
105(1)
6.4.4 Shrinkage Towards a Second Order Estimate
106(1)
6.5 Estimating the Length n Vector γ(n)
106(1)
6.6 Linear Prediction Based on the Model-Free Prediction Principle
107(6)
6.6.1 A First Idea: The Discrete Fourier Transform
107(1)
6.6.2 Whitening and the Model-Free Linear Predictor
108(3)
6.6.3 From Point Predictors to Prediction Intervals
111(1)
Acknowledgements
112(1)
7 Model-Based Prediction in Autoregression
113(28)
7.1 Introduction
113(1)
7.2 Prediction Intervals in AR Models: Laying the Foundation
114(5)
7.2.1 Forward and Backward Bootstrap for Prediction
114(2)
7.2.2 Prediction Intervals for Autoregressive Processes
116(1)
7.2.3 Pertinent Prediction Intervals in Model-Based Autoregression
117(2)
7.3 Bootstrap Prediction Intervals for Linear Autoregressions
119(8)
7.3.1 Forward Bootstrap with Fitted Residuals
120(1)
7.3.2 Forward Bootstrap with Predictive Residuals
121(1)
7.3.3 Forward Bootstrap Based on Studentized Roots
122(1)
7.3.4 Backward Bootstrap
123(3)
7.3.5 Generalized Bootstrap Prediction Intervals
126(1)
7.4 Alternative Approaches to Bootstrap Prediction Intervals for Linear Autoregressions
127(1)
7.5 Simulations: Linear AR Models
128(6)
7.5.1 Unconditional Coverage Level
128(4)
7.5.2 Conditional Coverage Level
132(2)
7.6 Bootstrap Prediction Intervals for Nonparametric Autoregression
134(7)
7.6.1 Nonparametric Autoregression with i.i.d. Errors
134(3)
7.6.2 Nonparametric Autoregression with Heteroscedastic Errors
137(2)
Acknowledgements
139(2)
8 Model-Free Inference for Markov Processes
141(36)
8.1 Introduction
141(1)
8.2 Prediction and Bootstrap for Markov Processes
142(3)
8.2.1 Notation and Definitions
142(2)
8.2.2 Forward vs. Backward Bootstrap for Prediction Intervals
144(1)
8.3 Bootstrap Based on Estimates of Transition Density
145(3)
8.4 The Local Bootstrap for Markov Processes
148(3)
8.5 Hybrid Backward Markov Bootstrap for Nonparametric Autoregression
151(2)
8.6 Prediction Intervals for Markov Processes Based on the Model-Free Prediction Principle
153(6)
8.6.1 Theoretical Transformation
154(1)
8.6.2 Estimating the Transformation from Data
155(4)
8.7 Finite-Sample Performance of Model-Free Prediction Intervals
159(4)
8.8 Model-Free Confidence Intervals in Markov Processes
163(7)
8.8.1 Finite-Sample Performance of Confidence Intervals
167(3)
8.9 Discrete-Valued Markov Processes
170(7)
8.9.1 Transition Densities and Local Bootstrap
171(3)
8.9.2 Model-Free Bootstrap
174(2)
Acknowledgements
176(1)
9 Predictive Inference for Locally Stationary Time Series
177(22)
9.1 Introduction
177(2)
9.2 Model-Based Inference
179(6)
9.2.1 Theoretical Optimal Point Prediction
179(1)
9.2.2 Trend Estimation and Practical Prediction
180(3)
9.2.3 Model-Based Predictors and Prediction Intervals
183(2)
9.3 Model-Free Inference
185(14)
9.3.1 Constructing the Theoretical Transformation
186(1)
9.3.2 Kernel Estimation of the "Uniformizing" Transformation
187(1)
9.3.3 Local Linear Estimation of the "Uniformizing" Transformation
188(1)
9.3.4 Estimation of the Whitening Transformation
189(1)
9.3.5 Model-Free Point Predictors and Prediction Intervals
190(3)
9.3.6 Special Case: Strictly Stationary Data
193(1)
9.3.7 Local Stationarity in a Higher-Dimensional Marginal
194(1)
Acknowledgements
195(4)
Part IV Case Study: Model-Free Volatility Prediction for Financial Time Series
10 Model-Free vs. Model-Based Volatility Prediction
199(38)
10.1 Introduction
199(3)
10.2 Three Illustrative Datasets
202(3)
10.3 Normalization and Variance-Stabilization
205(10)
10.3.1 Definition of the NoVaS Transformation
205(1)
10.3.2 Choosing the Parameters of NoVaS
206(1)
10.3.3 Simple NoVaS Algorithm
207(6)
10.3.4 Exponential NoVaS Algorithm
213(2)
10.4 Model-Based Volatility Prediction
215(6)
10.4.1 Some Basic Notions: L1 vs. L2
215(4)
10.4.2 Do Financial Returns Have a Finite Fourth Moment?
219(2)
10.5 Model-Free Volatility Prediction
221(9)
10.5.1 Transformation Towards i.i.d.--Ness
222(2)
10.5.2 Volatility Prediction Using NoVaS
224(2)
10.5.3 Optimizing NoVaS for Volatility Prediction
226(4)
10.5.4 Summary of Data-Analytic Findings on Volatility Prediction
230(1)
10.6 Model-Free Prediction Intervals for Financial Returns
230(2)
10.7 Time-Varying NoVaS: Robustness Against Structural Breaks
232(5)
Acknowledgements
235(2)
References 237
Dimitris N. Politis is Professor of Mathematics and Adjunct Professor of Economics at the University of California, San Diego. His research interests include Time Series Analysis, Resampling and Subsampling, Nonparametric Function Estimation, and Model-free Prediction. He has served as Editor of the IMS Bulletin (2010-2013), Co-Editor of the Journal of Time Series Analysis (2013-present), Co-Editor of the Journal of Nonparametric Statistics (2008-2011), and as Associate Editor for several journals including Bernoulli, the Journal of the American Statistical Association, and the Journal of the Royal Statistical Society, Series B. He is a fellow of the Institute of Mathematical Statistics (IMS) and the American Statistical Association, former fellow of the John Simon Guggenheim Memorial Foundation, and co-founder (with M. Akritas and S.N. Lahiri) of the International Society for NonParametric Statistics.
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