Inference and Prediction in Large Dimensions

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ISBN: 9780470017616 | 0470017619
Cover: Hardcover
Copyright: 11/28/2007

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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.

List of abbreviations	p. ix
Introduction	p. 1
Statistical Prediction Theory	p. 5
Statistical prediction	p. 7
Filtering	p. 7
Some examples	p. 8
The prediction model	p. 9
P-sufficient statistics	p. 11
Optimal predictors	p. 15
Efficient predictors	p. 21
Loss functions and empirical predictors	p. 28
Loss function	p. 29
Location parameters	p. 30
Bayesian predictors	p. 31
Linear predictors	p. 32
Multidimensional prediction	p. 33
Asymptotic prediction	p. 41
Introduction	p. 41
The basic problem	p. 41
Parametric prediction for stochastic processes	p. 43
Predicting some common processes	p. 47
Equivalent risks	p. 54
Prediction for small time lags	p. 56
Prediction for large time lags	p. 58
Inference by Projection	p. 61
Estimation by adaptive projection	p. 63
Introduction	p. 63
A class of functional parameters	p. 63
Oracle	p. 66
Parametric rate	p. 68
Nonparametric rates	p. 72
Rate in uniform norm	p. 79
Adaptive projection	p. 81
Behaviour of truncation index	p. 82
Superoptimal rate	p. 85
The general case	p. 88
Discussion and implementation	p. 92
Adaptive estimation in continuous time	p. 92
Functional tests of fit	p. 97
Generalized chi-square tests	p. 97
Tests based on linear estimators	p. 101
Consistency of the test	p. 105
Application	p. 106
Efficiency of functional tests of fit	p. 107
Adjacent hypotheses	p. 107
Bahadur efficiency	p. 110
Tests based on the uniform norm	p. 111
Extensions. Testing regression	p. 113
Functional tests for stochastic processes	p. 115
Prediction by projection	p. 117
A class of nonparametric predictors	p. 117
Guilbart spaces	p. 121
Predicting the conditional distribution	p. 122
Predicting the conditional distribution function	p. 124
Inference by Kernels	p. 131
Kernel method in discrete time	p. 133
Presentation of the method	p. 133
Kernel estimation in the i.i.d. case	p. 135
Density estimation in the dependent case	p. 138
Mean-square error and asymptotic normality	p. 138
Almost sure convergence	p. 140
Regression estimation in the dependent case	p. 148
Framework and notations	p. 148
Pointwise convergence	p. 150
Uniform convergence	p. 157
Nonparametric prediction by kernel	p. 157
Prediction for a stationary Markov process of order k	p. 157
Prediction for general processes	p. 160
Kernel method in continuous time	p. 163
Optimal and superoptimal rates for density estimation	p. 163
The optimal framework	p. 164
The superoptimal case	p. 167
From optimal to superoptimal rates	p. 170
Intermediate rates	p. 170
Classes of processes and examples	p. 172
Mean-square convergence	p. 173
Almost sure convergence	p. 177
An adaptive approach	p. 180
Regression estimation	p. 181
Pointwise almost sure convergence	p. 182
Uniform almost sure convergence	p. 184
Nonparametric prediction by kernel	p. 186
Kernel method from sampled data	p. 189
Density estimation	p. 190
High rate sampling	p. 190
Adequate sampling schemes	p. 193
Regression estimation	p. 198
Numerical studies	p. 201
Local Time	p. 207
The empirical density	p. 209
Introduction	p. 209
Occupation density	p. 209
The empirical density estimator	p. 212
Recursivity	p. 213
Invariance	p. 213
Empirical density estimator consistency	p. 214
Rates of convergence	p. 217
Approximation of empirical density by common density estimators	p. 220
Linear Processes in High Dimensions	p. 227
Functional linear processes	p. 229
Modelling in large dimensions	p. 229
Projection over linearly closed spaces	p. 230
Wold decomposition and linear processes in Hilbert spaces	p. 235
Moving average processes in Hilbert spaces	p. 239
Autoregressive processes in Hilbert spaces	p. 243
Autoregressive processes in Banach spaces	p. 254
Estimation and prediction of functional linear processes	p. 261
Introduction	p. 261
Estimation of the mean of a functional linear process	p. 262
Estimation of autocovariance operators	p. 263
The space S	p. 264
Estimation of C[subscript 0]	p. 264
Estimation of the eigenelements of C[subscript 0]	p. 267
Estimation of cross-autocovariance operators	p. 268
Prediction of autoregressive Hilbertian processes	p. 269
Estimation and prediction of ARC processes	p. 272
Estimation of autocovariance	p. 275
Sampled data	p. 276
Estimation of [rho] and prediction	p. 277
Appendix	p. 281
Measure and probability	p. 281
Random variables	p. 282
Function spaces	p. 284
Common function spaces	p. 285
Operators on Hilbert spaces	p. 286
Functional random variables	p. 287
Conditional expectation	p. 287
Conditional expectation in function spaces	p. 288
Stochastic processes	p. 289
Stationary processes and Wold decomposition	p. 290
Stochastic integral and diffusion processes	p. 291
Markov processes	p. 293
Stochastic convergences and limit theorems	p. 294
Strongly mixing processes	p. 295
Some other mixing coefficients	p. 296
Inequalities of exponential type	p. 297
Bibliography	p. 299
Index	p. 309
Table of Contents provided by Ingram. All Rights Reserved.

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Inference and Prediction in Large Dimensions

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