Inference and prediction in large dimensions

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.

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

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