book

Mathematical Statistics and Stochastic Processes

Name: Mathematical Statistics and Stochastic Processes
Author: Denis Bosq
ISBN: 9781848213616

by Denis Bosq

May 2012

Intermediate to advanced

304 pages

4h 44m

English

Wiley

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Cover
Title Page
Copyright
Preface
Part 1: Mathematical Statistics
Chapter 1: Introduction to Mathematical Statistics
1.1. Generalities1.2. Examples of statistics problems1.2.1. Quality control1.2.2. Measurement errors1.2.3. Filtering1.2.4. Confidence intervals1.2.5. Homogeneity testing
Chapter 2: Principles of Decision Theory
2.1. Generalities2.2. The problem of choosing a decision function2.3. Principles of Bayesian statistics2.3.1. Generalities2.3.2. Determination of Bayesian decision functions2.3.3. Admissibility of Bayes’ rules2.4. Complete classes2.5. Criticism of decision theory – the asymptotic point of view2.6. Exercises
Chapter 3: Conditional Expectation
3.1. Definition3.2. Properties and extension3.3. Conditional probabilities and conditional distributions3.3.1. Regular version of the conditional probability3.3.2. Conditional distributions3.3.3. Theorem for integration with respect to the conditional distribution3.3.4. Determination of the conditional distributions in the usual cases3.4. Exercises
Chapter 4: Statistics and Sufficiency
4.1. Samples and empirical distributions4.1.1. Properties of the empirical distribution and the associated statistics4.2. Sufficiency4.2.1. The factorization theorem4.3. Examples of sufficient statistics – an exponential model4.4. Use of a sufficient statistic4.5. Exercises
Chapter 5: Point Estimation
5.1. Generalities5.1.1. Definition – examples5.1.2. Choice of a preference relation5.2. Sufficiency and completeness5.2.1. Sufficiency5.2.2. Complete statistics5.3. The maximum-likelihood method5.3.1. Definition5.3.2. Maximum likelihood and sufficiency5.3.3. Calculating maximum-likelihood estimators5.3.3.1. The Newton–Raphson method5.4. Optimal unbiased estimators5.4.1. Unbiased estimation5.4.1.1. Existence of an unbiased estimator5.4.2. Unbiased minimum-dispersion estimator5.4.2.1. Application to an exponential model5.4.2.2. Application to the Gaussian model5.4.2.3. Use of the Lehmann–Scheffé theorem5.4.3. Criticism of unbiased estimators5.5. Efficiency of an estimator5.5.1. The Fréchet-Darmois-Cramer-Rao inequality5.5.1.1. Calculating I(θ)5.5.1.2. Properties of the Fisher information5.5.1.3. The case of a biased estimator5.5.2. Efficiency5.5.3. Extension to Rk5.5.3.1. Properties of the information matrix5.5.3.2. Efficiency5.5.4. The non-regular case5.5.4.1. “Superefficient” estimators5.5.4.2. Cramer–Rao-type inequalities5.6. The linear regression model5.6.1. Generalities5.6.2. Estimation of the parameter – the Gauss–Markov theorem5.7. Exercises

Chapter 6: Hypothesis Testing and Confidence Regions
6.1. Generalities6.1.1. The problem6.1.2. Use of decision theory6.1.2.1. Preference relation6.1.3. Generalization6.1.3.1. Preference relation6.1.4. Sufficiency6.2. The Neyman-Pearson (NP) lemma6.3. Multiple hypothesis tests (general methods)6.3.1. Testing a simple hypothesis against a composite one6.3.1.1. The γ test6.3.1.2. The λ test6.3.2. General case – unbiased tests6.3.2.1. Relation beween unbiased tests and unbiased decision functions6.4. Case where the ratio of the likelihoods is monotonic6.4.1. Generalities6.4.2. Unilateral tests6.4.3. Bilateral tests6.5. Tests relating to the normal distribution6.6. Application to estimation: confidence regions6.6.1. First preference relation on confidence regions6.6.2. Relation to tests6.7. Exercises
Chapter 7: Asymptotic Statistics
7.1. Generalities7.2. Consistency of the maximum likelihood estimator7.3. The limiting distribution of the maximum likelihood estimator7.4. The likelihood ratio test7.5. Exercises
Chapter 8: Non-Parametric Methods and Robustness
8.1. Generalities8.2. Non-parametric estimation8.2.1. Empirical estimators8.2.2. Distribution and density estimation8.2.2.1. Convergence of the estimator8.2.3. Regression estimation8.3. Non-parametric tests8.3.1. The χ2 test8.3.2. The Kolmogorov–Smirnov test8.3.3. The Cramer–von Mises test8.3.4. Rank test8.4. Robustness8.4.1. An example of a robust test8.4.2. An example of a robust estimator8.4.3. A general definition of a robust estimator8.5. Exercises
Part 2: Statistics for Stochastic Processes
Chapter 9: Introduction to Statistics for Stochastic Processes
9.1. Modeling a family of observations9.2. Processes9.2.1. The distribution of a process9.2.2. Gaussian processes9.2.3. Stationary processes9.2.4. Markov processes9.3. Statistics for stochastic processes9.4. Exercises
Chapter 10: Weakly Stationary Discrete-Time Processes
10.1. Autocovariance and spectral density10.2. Linear prediction and Wold decomposition10.3. Linear processes and the ARMA model10.3.1. Spectral density of a linear process10.4. Estimating the mean of a weakly stationary process10.5. Estimating the autocovariance10.6. Estimating the spectral density10.6.1. The periodogram10.6.2. Convergent estimators of the spectral density10.7. Exercises
Chapter 11: Poisson Processes – A Probabilistic and Statistical Study
11.1. Introduction11.2. The axioms of Poisson processes11.3. Interarrivai time11.4. Properties of the Poisson process11.5. Notions on generalized Poisson processes11.6. Statistics of Poisson processes11.7. Exercises
Chapter 12: Square-Integrable Continuous-Time Processes
12.1. Definitions12.2. Mean-square continuity12.3. Mean-square integration12.4. Mean-square differentiation12.5. The Karhunen–Loeve theorem12.6. Wiener processes12.6.1. Karhunen-Loeve decomposition12.6.2. Statistics of Wiener processes12.7. Notions on weakly stationary continuous-time processes12.7.1. Estimating the mean12.7.2. Estimating the autocovariance12.7.3. The case of a process observed at discrete instants12.8. Exercises
Chapter 13: Stochastic Integration and Diffusion Processes
13.1. Itô integral13.2. Diffusion processes13.3. Processes defined by stochastic differential equations and stochastic integrals13.4. Notions on statistics for diffusion processes13.5. Exercises
Chapter 14: ARMA Processes
14.1. Autoregressive processes14.2. Moving average processes14.3. General ARMA processes14.4. Non-stationary models14.4.1. The Box-Cox transformation14.4.2. Eliminating the trend by differentiation14.4.3. Eliminating the seasonality14.4.4. Introducing exogenous variables14.5. Statistics of ARMA processes14.5.1. Identification14.5.2. Estimation14.5.3. Verification14.6. Multidimensional processes14.7. Exercises
Chapter 15: Prediction
15.1. Generalities15.2. Empirical methods of prediction15.2.1. The empirical mean15.2.2. Exponential smoothing15.2.3. Naive predictors15.2.4. Trend adjustment15.3. Prediction in the ARIMA model15.4. Prediction in continuous time15.5. Exercises
Part 3: Supplement
Chapter 16: Elements of Probability Theory
16.1. Measure spaces: probability spaces16.2. Measurable functions: real random variables16.3. Integrating real random variables16.4. Random vectors16.5. Independence16.6. Gaussian vectors16.7. Stochastic convergence16.8. Limit theorems
Appendix: Statistical Tables
A1.1. Random numbersA1.2. Distribution function of the standard normal distributionA1.3. Density of the standard normal distributionA1.4. Percentiles (tp) of Student’s distributionA1.5. Ninety-fifth percentiles of Fisher–Snedecor distributionsA1.6. Ninety-ninth percentiles of Fisher–Snedecor distributionsA1.7. Percentiles (χ2p) of the χ2 distribution with n degrees of freedomA1.8. Individual probabilities of the Poisson distributionA1.9. Cumulative probabilities of the Poisson distributionA1.10. Binomial coefficients Ckn for n ≤ 30 and 0 ≤ k ≤ 7A1.11. Binomial coefficients Ckn for n ≤ 30 and 8 ≤ k ≤ 15
Bibliography
Index

Content preview from Mathematical Statistics and Stochastic Processes

Chapter 2 Principles of Decision Theory

2.1. Generalities

The examples given in Chapter 1 show that a statistical problem may be characterized by the following elements:

– a probability distribution that is not entirely known;

– observations;

– a decision to be made.

Formalization: Wald’s decision theory provided a common framework for statistics problems. We take:

1) A triplet where P is a family of probabilities on the measurable space is called a statistical model. We often set P = {Pθ, θ ∈ Θ} where we suppose that is injective, and Θ is called the parameter space.

2) A measurable space is called a decision (or action) space.

3) A set of measurable mappings is called a set of decision functions (d.f.) (or decision rules).

Description: From an observation that follows an unknown distribution P ∈ P, a statistician chooses an element a ∈ D using an element d from .

Preference relation ...

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Mathematical Statistics and Stochastic Processes

by Denis Bosq

Chapter 2

Principles of Decision Theory

2.1. Generalities

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