## Book description

Generally, books on mathematical statistics are restricted to the case of independent identically distributed random variables. In this book however, both this case AND the case of dependent variables, i.e. statistics for discrete and continuous time processes, are studied. This second case is very important for today's practitioners.

Mathematical Statistics and Stochastic Processes is based on decision theory and asymptotic statistics and contains up-to-date information on the relevant topics of theory of probability, estimation, confidence intervals, non-parametric statistics and robustness, second-order processes in discrete and continuous time and diffusion processes, statistics for discrete and continuous time processes, statistical prediction, and complements in probability.

This book is aimed at students studying courses on probability with an emphasis on measure theory and for all practitioners who apply and use statistics and probability on a daily basis.

1. Cover
2. Title Page
4. Preface
5. Part 1: Mathematical Statistics
1. Chapter 1: Introduction to Mathematical Statistics
1. 1.1. Generalities
2. 1.2. Examples of statistics problems
2. Chapter 2: Principles of Decision Theory
1. 2.1. Generalities
2. 2.2. The problem of choosing a decision function
3. 2.3. Principles of Bayesian statistics
4. 2.4. Complete classes
5. 2.5. Criticism of decision theory – the asymptotic point of view
6. 2.6. Exercises
3. Chapter 3: Conditional Expectation
1. 3.1. Definition
2. 3.2. Properties and extension
3. 3.3. Conditional probabilities and conditional distributions
4. 3.4. Exercises
4. Chapter 4: Statistics and Sufficiency
1. 4.1. Samples and empirical distributions
2. 4.2. Sufficiency
3. 4.3. Examples of sufficient statistics – an exponential model
4. 4.4. Use of a sufficient statistic
5. 4.5. Exercises
5. Chapter 5: Point Estimation
1. 5.1. Generalities
2. 5.2. Sufficiency and completeness
3. 5.3. The maximum-likelihood method
1. 5.3.1. Definition
2. 5.3.2. Maximum likelihood and sufficiency
3. 5.3.3. Calculating maximum-likelihood estimators
4. 5.4. Optimal unbiased estimators
1. 5.4.1. Unbiased estimation
2. 5.4.2. Unbiased minimum-dispersion estimator
3. 5.4.3. Criticism of unbiased estimators
5. 5.5. Efficiency of an estimator
1. 5.5.1. The Fréchet-Darmois-Cramer-Rao inequality
2. 5.5.2. Efficiency
3. 5.5.3. Extension to Rk
4. 5.5.4. The non-regular case
6. 5.6. The linear regression model
7. 5.7. Exercises
6. Chapter 6: Hypothesis Testing and Confidence Regions
1. 6.1. Generalities
1. 6.1.1. The problem
2. 6.1.2. Use of decision theory
3. 6.1.3. Generalization
4. 6.1.4. Sufficiency
2. 6.2. The Neyman-Pearson (NP) lemma
3. 6.3. Multiple hypothesis tests (general methods)
1. 6.3.1. Testing a simple hypothesis against a composite one
2. 6.3.2. General case – unbiased tests
4. 6.4. Case where the ratio of the likelihoods is monotonic
5. 6.5. Tests relating to the normal distribution
6. 6.6. Application to estimation: confidence regions
7. 6.7. Exercises
7. Chapter 7: Asymptotic Statistics
8. Chapter 8: Non-Parametric Methods and Robustness
1. 8.1. Generalities
2. 8.2. Non-parametric estimation
1. 8.2.1. Empirical estimators
2. 8.2.2. Distribution and density estimation
3. 8.2.3. Regression estimation
3. 8.3. Non-parametric tests
4. 8.4. Robustness
5. 8.5. Exercises
6. Part 2: Statistics for Stochastic Processes
1. Chapter 9: Introduction to Statistics for Stochastic Processes
1. 9.1. Modeling a family of observations
2. 9.2. Processes
3. 9.3. Statistics for stochastic processes
4. 9.4. Exercises
2. Chapter 10: Weakly Stationary Discrete-Time Processes
1. 10.1. Autocovariance and spectral density
2. 10.2. Linear prediction and Wold decomposition
3. 10.3. Linear processes and the ARMA model
4. 10.4. Estimating the mean of a weakly stationary process
5. 10.5. Estimating the autocovariance
6. 10.6. Estimating the spectral density
7. 10.7. Exercises
3. Chapter 11: Poisson Processes – A Probabilistic and Statistical Study
4. Chapter 12: Square-Integrable Continuous-Time Processes
1. 12.1. Definitions
2. 12.2. Mean-square continuity
3. 12.3. Mean-square integration
4. 12.4. Mean-square differentiation
5. 12.5. The Karhunen–Loeve theorem
6. 12.6. Wiener processes
7. 12.7. Notions on weakly stationary continuous-time processes
8. 12.8. Exercises
5. Chapter 13: Stochastic Integration and Diffusion Processes
6. Chapter 14: ARMA Processes
1. 14.1. Autoregressive processes
2. 14.2. Moving average processes
3. 14.3. General ARMA processes
4. 14.4. Non-stationary models
5. 14.5. Statistics of ARMA processes
6. 14.6. Multidimensional processes
7. 14.7. Exercises
7. Chapter 15: Prediction
1. 15.1. Generalities
2. 15.2. Empirical methods of prediction
3. 15.3. Prediction in the ARIMA model
4. 15.4. Prediction in continuous time
5. 15.5. Exercises
7. Part 3: Supplement
1. Chapter 16: Elements of Probability Theory
8. Appendix: Statistical Tables
9. Bibliography
10. Index

## Product information

• Title: Mathematical Statistics and Stochastic Processes
• Author(s):
• Release date: May 2012
• Publisher(s): Wiley
• ISBN: 9781848213616