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Summary
Summary
This book offers a predominantly theoretical coverage of statistical prediction, with some potential applications discussed, when data and/ or parameters belong to a large or infinite dimensional space. It develops the theory of statistical prediction, non-parametric estimation by adaptive projection - with applications to tests of fit and prediction, and theory of linear processes in function spaces with applications to prediction of continuous time processes.
This work is in the Wiley-Dunod Series co-published between Dunod ( www.dunod.com ) and John Wiley and Sons, Ltd.
Author Notes
Denis Bosq is a Professor at the Laboratory of Theoretical and Applied Statistics, University of Pierre & Marie Curie - Paris 6. He has over 100 published papers, 5 books, and is chief editor of the journal 'Statistical Inference for Stochastic Processes' as well as associate editor for the 'Journal of Non-Parametric Statistics'. He is a well-known specialist in the field of non-parametric statistical inference.
Table of Contents
List of abbreviations | p. ix |
Introduction | p. 1 |
Part I Statistical Prediction Theory | p. 5 |
1 Statistical prediction | p. 7 |
1.1 Filtering | p. 7 |
1.2 Some examples | p. 8 |
1.3 The prediction model | p. 9 |
1.4 P-sufficient statistics | p. 11 |
1.5 Optimal predictors | p. 15 |
1.6 Efficient predictors | p. 21 |
1.7 Loss functions and empirical predictors | p. 28 |
1.7.1 Loss function | p. 29 |
1.7.2 Location parameters | p. 30 |
1.7.3 Bayesian predictors | p. 31 |
1.7.4 Linear predictors | p. 32 |
1.8 Multidimensional prediction | p. 33 |
2 Asymptotic prediction | p. 41 |
2.1 Introduction | p. 41 |
2.2 The basic problem | p. 41 |
2.3 Parametric prediction for stochastic processes | p. 43 |
2.4 Predicting some common processes | p. 47 |
2.5 Equivalent risks | p. 54 |
2.6 Prediction for small time lags | p. 56 |
2.7 Prediction for large time lags | p. 58 |
Part II Inference by Projection | p. 61 |
3 Estimation by adaptive projection | p. 63 |
3.1 Introduction | p. 63 |
3.2 A class of functional parameters | p. 63 |
3.3 Oracle | p. 66 |
3.4 Parametric rate | p. 68 |
3.5 Nonparametric rates | p. 72 |
3.6 Rate in uniform norm | p. 79 |
3.7 Adaptive projection | p. 81 |
3.7.1 Behaviour of truncation index | p. 82 |
3.7.2 Superoptimal rate | p. 85 |
3.7.3 The general case | p. 88 |
3.7.4 Discussion and implementation | p. 92 |
3.8 Adaptive estimation in continuous time | p. 92 |
4 Functional tests of fit | p. 97 |
4.1 Generalized chi-square tests | p. 97 |
4.2 Tests based on linear estimators | p. 101 |
4.2.1 Consistency of the test | p. 105 |
4.2.2 Application | p. 106 |
4.3 Efficiency of functional tests of fit | p. 107 |
4.3.1 Adjacent hypotheses | p. 107 |
4.3.2 Bahadur efficiency | p. 110 |
4.4 Tests based on the uniform norm | p. 111 |
4.5 Extensions. Testing regression | p. 113 |
4.6 Functional tests for stochastic processes | p. 115 |
5 Prediction by projection | p. 117 |
5.1 A class of nonparametric predictors | p. 117 |
5.2 Guilbart spaces | p. 121 |
5.3 Predicting the conditional distribution | p. 122 |
5.4 Predicting the conditional distribution function | p. 124 |
Part III Inference by Kernels | p. 131 |
6 Kernel method in discrete time | p. 133 |
6.1 Presentation of the method | p. 133 |
6.2 Kernel estimation in the i.i.d. case | p. 135 |
6.3 Density estimation in the dependent case | p. 138 |
6.3.1 Mean-square error and asymptotic normality | p. 138 |
6.3.2 Almost sure convergence | p. 140 |
6.4 Regression estimation in the dependent case | p. 148 |
6.4.1 Framework and notations | p. 148 |
6.4.2 Pointwise convergence | p. 150 |
6.4.3 Uniform convergence | p. 157 |
6.5 Nonparametric prediction by kernel | p. 157 |
6.5.1 Prediction for a stationary Markov process of order k | p. 157 |
6.5.2 Prediction for general processes | p. 160 |
7 Kernel method in continuous time | p. 163 |
7.1 Optimal and superoptimal rates for density estimation | p. 163 |
7.1.1 The optimal framework | p. 164 |
7.1.2 The superoptimal case | p. 167 |
7.2 From optimal to superoptimal rates | p. 170 |
7.2.1 Intermediate rates | p. 170 |
7.2.2 Classes of processes and examples | p. 172 |
7.2.3 Mean-square convergence | p. 173 |
7.2.4 Almost sure convergence | p. 177 |
7.2.5 An adaptive approach | p. 180 |
7.3 Regression estimation | p. 181 |
7.3.1 Pointwise almost sure convergence | p. 182 |
7.3.2 Uniform almost sure convergence | p. 184 |
7.4 Nonparametric prediction by kernel | p. 186 |
8 Kernel method from sampled data | p. 189 |
8.1 Density estimation | p. 190 |
8.1.1 High rate sampling | p. 190 |
8.1.2 Adequate sampling schemes | p. 193 |
8.2 Regression estimation | p. 198 |
8.3 Numerical studies | p. 201 |
Part IV Local Time | p. 207 |
9 The empirical density | p. 209 |
9.1 Introduction | p. 209 |
9.2 Occupation density | p. 209 |
9.3 The empirical density estimator | p. 212 |
9.3.1 Recursivity | p. 213 |
9.3.2 Invariance | p. 213 |
9.4 Empirical density estimator consistency | p. 214 |
9.5 Rates of convergence | p. 217 |
9.6 Approximation of empirical density by common density estimators | p. 220 |
Part V Linear Processes in High Dimensions | p. 227 |
10 Functional linear processes | p. 229 |
10.1 Modelling in large dimensions | p. 229 |
10.2 Projection over linearly closed spaces | p. 230 |
10.3 Wold decomposition and linear processes in Hilbert spaces | p. 235 |
10.4 Moving average processes in Hilbert spaces | p. 239 |
10.5 Autoregressive processes in Hilbert spaces | p. 243 |
10.6 Autoregressive processes in Banach spaces | p. 254 |
11 Estimation and prediction of functional linear processes | p. 261 |
11.1 Introduction | p. 261 |
11.2 Estimation of the mean of a functional linear process | p. 262 |
11.3 Estimation of autocovariance operators | p. 263 |
11.3.1 The space S | p. 264 |
11.3.2 Estimation of C[subscript 0] | p. 264 |
11.3.3 Estimation of the eigenelements of C[subscript 0] | p. 267 |
11.3.4 Estimation of cross-autocovariance operators | p. 268 |
11.4 Prediction of autoregressive Hilbertian processes | p. 269 |
11.5 Estimation and prediction of ARC processes | p. 272 |
11.5.1 Estimation of autocovariance | p. 275 |
11.5.2 Sampled data | p. 276 |
11.5.3 Estimation of [rho] and prediction | p. 277 |
Appendix | p. 281 |
A.1 Measure and probability | p. 281 |
A.2 Random variables | p. 282 |
A.3 Function spaces | p. 284 |
A.4 Common function spaces | p. 285 |
A.5 Operators on Hilbert spaces | p. 286 |
A.6 Functional random variables | p. 287 |
A.7 Conditional expectation | p. 287 |
A.8 Conditional expectation in function spaces | p. 288 |
A.9 Stochastic processes | p. 289 |
A.10 Stationary processes and Wold decomposition | p. 290 |
A.11 Stochastic integral and diffusion processes | p. 291 |
A.12 Markov processes | p. 293 |
A.13 Stochastic convergences and limit theorems | p. 294 |
A.14 Strongly mixing processes | p. 295 |
A.15 Some other mixing coefficients | p. 296 |
A.16 Inequalities of exponential type | p. 297 |
Bibliography | p. 299 |
Index | p. 309 |