Cover image for Inference and prediction in large dimensions
Title:
Inference and prediction in large dimensions
Personal Author:
Series:
Wiley series in probability and statistics
Publication Information:
Chichester : Hoboken, NJ : John Wiley/Dunod, 2007
Physical Description:
x, 316 p. : ill. ; 24 cm.
ISBN:
9780470017616
Added Author:
Added Corporate Author:

Available:*

Library
Item Barcode
Call Number
Material Type
Status
Searching...
30000010185855 QA276.8 B67 2007 Open Access Book
Searching...

On Order

Summary

Summary

This book offers a predominantly theoretical coverage ofstatistical prediction, with some potential applications discussed,when data and/ or parameters belong to a large or infinitedimensional space. It develops the theory of statisticalprediction, non-parametric estimation by adaptive projection? with applications to tests of fit and prediction, andtheory of linear processes in function spaces with applications toprediction of continuous time processes.

This work is in the Wiley-Dunod Series co-published betweenDunod ( www.dunod.com ) and JohnWiley 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 abbreviationsp. ix
Introductionp. 1
Part I Statistical Prediction Theoryp. 5
1 Statistical predictionp. 7
1.1 Filteringp. 7
1.2 Some examplesp. 8
1.3 The prediction modelp. 9
1.4 P-sufficient statisticsp. 11
1.5 Optimal predictorsp. 15
1.6 Efficient predictorsp. 21
1.7 Loss functions and empirical predictorsp. 28
1.7.1 Loss functionp. 29
1.7.2 Location parametersp. 30
1.7.3 Bayesian predictorsp. 31
1.7.4 Linear predictorsp. 32
1.8 Multidimensional predictionp. 33
2 Asymptotic predictionp. 41
2.1 Introductionp. 41
2.2 The basic problemp. 41
2.3 Parametric prediction for stochastic processesp. 43
2.4 Predicting some common processesp. 47
2.5 Equivalent risksp. 54
2.6 Prediction for small time lagsp. 56
2.7 Prediction for large time lagsp. 58
Part II Inference by Projectionp. 61
3 Estimation by adaptive projectionp. 63
3.1 Introductionp. 63
3.2 A class of functional parametersp. 63
3.3 Oraclep. 66
3.4 Parametric ratep. 68
3.5 Nonparametric ratesp. 72
3.6 Rate in uniform normp. 79
3.7 Adaptive projectionp. 81
3.7.1 Behaviour of truncation indexp. 82
3.7.2 Superoptimal ratep. 85
3.7.3 The general casep. 88
3.7.4 Discussion and implementationp. 92
3.8 Adaptive estimation in continuous timep. 92
4 Functional tests of fitp. 97
4.1 Generalized chi-square testsp. 97
4.2 Tests based on linear estimatorsp. 101
4.2.1 Consistency of the testp. 105
4.2.2 Applicationp. 106
4.3 Efficiency of functional tests of fitp. 107
4.3.1 Adjacent hypothesesp. 107
4.3.2 Bahadur efficiencyp. 110
4.4 Tests based on the uniform normp. 111
4.5 Extensions. Testing regressionp. 113
4.6 Functional tests for stochastic processesp. 115
5 Prediction by projectionp. 117
5.1 A class of nonparametric predictorsp. 117
5.2 Guilbart spacesp. 121
5.3 Predicting the conditional distributionp. 122
5.4 Predicting the conditional distribution functionp. 124
Part III Inference by Kernelsp. 131
6 Kernel method in discrete timep. 133
6.1 Presentation of the methodp. 133
6.2 Kernel estimation in the i.i.d. casep. 135
6.3 Density estimation in the dependent casep. 138
6.3.1 Mean-square error and asymptotic normalityp. 138
6.3.2 Almost sure convergencep. 140
6.4 Regression estimation in the dependent casep. 148
6.4.1 Framework and notationsp. 148
6.4.2 Pointwise convergencep. 150
6.4.3 Uniform convergencep. 157
6.5 Nonparametric prediction by kernelp. 157
6.5.1 Prediction for a stationary Markov process of order kp. 157
6.5.2 Prediction for general processesp. 160
7 Kernel method in continuous timep. 163
7.1 Optimal and superoptimal rates for density estimationp. 163
7.1.1 The optimal frameworkp. 164
7.1.2 The superoptimal casep. 167
7.2 From optimal to superoptimal ratesp. 170
7.2.1 Intermediate ratesp. 170
7.2.2 Classes of processes and examplesp. 172
7.2.3 Mean-square convergencep. 173
7.2.4 Almost sure convergencep. 177
7.2.5 An adaptive approachp. 180
7.3 Regression estimationp. 181
7.3.1 Pointwise almost sure convergencep. 182
7.3.2 Uniform almost sure convergencep. 184
7.4 Nonparametric prediction by kernelp. 186
8 Kernel method from sampled datap. 189
8.1 Density estimationp. 190
8.1.1 High rate samplingp. 190
8.1.2 Adequate sampling schemesp. 193
8.2 Regression estimationp. 198
8.3 Numerical studiesp. 201
Part IV Local Timep. 207
9 The empirical densityp. 209
9.1 Introductionp. 209
9.2 Occupation densityp. 209
9.3 The empirical density estimatorp. 212
9.3.1 Recursivityp. 213
9.3.2 Invariancep. 213
9.4 Empirical density estimator consistencyp. 214
9.5 Rates of convergencep. 217
9.6 Approximation of empirical density by common density estimatorsp. 220
Part V Linear Processes in High Dimensionsp. 227
10 Functional linear processesp. 229
10.1 Modelling in large dimensionsp. 229
10.2 Projection over linearly closed spacesp. 230
10.3 Wold decomposition and linear processes in Hilbert spacesp. 235
10.4 Moving average processes in Hilbert spacesp. 239
10.5 Autoregressive processes in Hilbert spacesp. 243
10.6 Autoregressive processes in Banach spacesp. 254
11 Estimation and prediction of functional linear processesp. 261
11.1 Introductionp. 261
11.2 Estimation of the mean of a functional linear processp. 262
11.3 Estimation of autocovariance operatorsp. 263
11.3.1 The space Sp. 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 operatorsp. 268
11.4 Prediction of autoregressive Hilbertian processesp. 269
11.5 Estimation and prediction of ARC processesp. 272
11.5.1 Estimation of autocovariancep. 275
11.5.2 Sampled datap. 276
11.5.3 Estimation of [rho] and predictionp. 277
Appendixp. 281
A.1 Measure and probabilityp. 281
A.2 Random variablesp. 282
A.3 Function spacesp. 284
A.4 Common function spacesp. 285
A.5 Operators on Hilbert spacesp. 286
A.6 Functional random variablesp. 287
A.7 Conditional expectationp. 287
A.8 Conditional expectation in function spacesp. 288
A.9 Stochastic processesp. 289
A.10 Stationary processes and Wold decompositionp. 290
A.11 Stochastic integral and diffusion processesp. 291
A.12 Markov processesp. 293
A.13 Stochastic convergences and limit theoremsp. 294
A.14 Strongly mixing processesp. 295
A.15 Some other mixing coefficientsp. 296
A.16 Inequalities of exponential typep. 297
Bibliographyp. 299
Indexp. 309