Inference and Prediction in Large Dimensions

Introduction.

Part I Statistical Prediction Theory.

1 Statistical prediction.

1.1 Filtering.

1.2 Some examples.

1.3 The prediction model.

1.4 P-sufficient statistics.

1.5 Optimal predictors.

1.6 Efficient predictors.

1.7 Loss functions and empirical predictors.

1.7.1 Loss function.

1.7.2 Location parameters.

1.7.3 Bayesian predictors.

1.7.4 Linear predictors.

1.8 Multidimensional prediction.

2 Asymptotic prediction.

2.1 Introduction.

2.2 The basic problem.

2.3 Parametric prediction for stochastic processes.

2.4 Predicting some common processes.

2.5 Equivalent risks.

2.6 Prediction for small time lags.

2.7 Prediction for large time lags.

Part II Inference by Projection.

3 Estimation by adaptive projection.

3.1 Introduction.

3.2 A class of functional parameters.

3.3 Oracle.

3.4 Parametric rate.

3.5 Nonparametric rates.

3.6 Rate in uniform norm.

3.7 Adaptive projection.

3.7.1 Behaviour of truncation index.

3.7.2 Superoptimal rate.

3.7.3 The general case.

3.7.4 Discussion and implementation.

3.8 Adaptive estimation in continuous time.

4 Functional tests of fit.

4.1 Generalized chi-square tests.

4.2 Tests based on linear estimators.

4.2.1 Consistency of the test.

4.2.2 Application.

4.3 Efficiency of functional tests of fit.

4.3.1 Adjacent hypotheses.

4.3.2 Bahadur efficiency.

4.4 Tests based on the uniform norm.

4.5 Extensions. Testing regression.

4.6 Functional tests for stochastic processes.

5 Prediction by projection.

5.1 A class of nonparametric predictors.

5.2 Guilbart spaces.

5.3 Predicting the conditional distribution.

5.4 Predicting the conditional distribution function.

Part III Inference by Kernels.

6 Kernel method in discrete time.

6.1 Presentation of the method.

6.2 Kernel estimation in the i.i.d. case.

6.3 Density estimation in the dependent case.

6.3.1 Mean-square error and asymptotic normality.

6.3.2 Almost sure convergence.

6.4 Regression estimation in the dependent case.

6.4.1 Framework and notations.

6.4.2 Pointwise convergence.

6.4.3 Uniform convergence.

6.5 Nonparametric prediction by kernel.

6.5.1 Prediction for a stationary Markov process of order k.

6.5.2 Prediction for general processes.

7 Kernel method in continuous time.

7.1 Optimal and superoptimal rates for density estimation.

7.1.1 The optimal framework.

7.1.2 The superoptimal case.

7.2 From optimal to superoptimal rates.

7.2.1 Intermediate rates.

7.2.2 Classes of processes and examples.

7.2.3 Mean-square convergence.

7.2.4 Almost sure convergence.

7.2.5 An adaptive approach.

7.3 Regression estimation.

7.3.1 Pointwise almost sure convergence.

7.3.2 Uniform almost sure convergence.

7.4 Nonparametric prediction by kernel.

8 Kernel method from sampled data.

8.1 Density estimation.

8.1.1 High rate sampling.

8.1.2 Adequate sampling schemes.

8.2 Regression estimation.

8.3 Numerical studies.

Part IV Local Time.

9 The empirical density.

9.1 Introduction.

9.2 Occupation density.

9.3 The empirical density estimator.

9.3.1 Recursivity.

9.3.2 Invariance.

9.4 Empirical density estimator consistency.

9.5 Rates of convergence.

9.6 Approximation of empirical density by common density estimators.

Part V Linear Processes in High Dimensions.

10 Functional linear processes.

10.1 Modelling in large dimensions.

10.2 Projection over linearly closed spaces.

10.3 Wold decomposition and linear processes in Hilbert spaces.

10.4 Moving average processes in Hilbert spaces.

10.5 Autoregressive processes in Hilbert spaces.

10.6 Autoregressive processes in Banach spaces.

11 Estimation and prediction of functional linear processes.

11.1 Introduction.

11.2 Estimation of the mean of a functional linear process.

11.3 Estimation of autocovariance operators.

11.3.1 The space S.

11.3.2 Estimation of C₀.

11.3.3 Estimation of the eigenelements of C₀.

11.3.4 Estimation of cross-autocovariance operators.

11.4 Prediction of autoregressive Hilbertian processes.

11.5 Estimation and prediction of ARC processes.

11.5.1 Estimation of autocovariance.

11.5.2 Sampled data.

11.5.3 Estimation of p  and prediction.

Appendix.

A.1 Measure and probability.

A.2 Random variables.

A.3 Function spaces.

A.4 Common function spaces.

A.5 Operators on Hilbert spaces.

A.6 Functional random variables.

A.7 Conditional expectation.

A.8 Conditional expectation in function spaces.

A.9 Stochastic processes.

A.10 Stationary processes and Wold decomposition.

A.11 Stochastic integral and diffusion processes.

A.12 Markov processes.

A.13 Stochastic convergences and limit theorems.

A.14 Strongly mixing processes.

A.15 Some other mixing coefficients.

A.16 Inequalities of exponential type.

Bibliography.

Index.