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Part 1: Mathematical Statistics
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Part 1: Mathematical Statistics
by Denis Bosq
Mathematical Statistics and Stochastic Processes
Cover
Title Page
Copyright
Preface
Part 1: Mathematical Statistics
Chapter 1: Introduction to Mathematical Statistics
1.1. Generalities
1.2. Examples of statistics problems
1.2.1. Quality control
1.2.2. Measurement errors
1.2.3. Filtering
1.2.4. Confidence intervals
1.2.5. Homogeneity testing
Chapter 2: Principles of Decision Theory
2.1. Generalities
2.2. The problem of choosing a decision function
2.3. Principles of Bayesian statistics
2.3.1. Generalities
2.3.2. Determination of Bayesian decision functions
2.3.3. Admissibility of Bayes’ rules
2.4. Complete classes
2.5. Criticism of decision theory – the asymptotic point of view
2.6. Exercises
Chapter 3: Conditional Expectation
3.1. Definition
3.2. Properties and extension
3.3. Conditional probabilities and conditional distributions
3.3.1. Regular version of the conditional probability
3.3.2. Conditional distributions
3.3.3. Theorem for integration with respect to the conditional distribution
3.3.4. Determination of the conditional distributions in the usual cases
3.4. Exercises
Chapter 4: Statistics and Sufficiency
4.1. Samples and empirical distributions
4.1.1. Properties of the empirical distribution and the associated statistics
4.2. Sufficiency
4.2.1. The factorization theorem
4.3. Examples of sufficient statistics – an exponential model
4.4. Use of a sufficient statistic
4.5. Exercises
Chapter 5: Point Estimation
5.1. Generalities
5.1.1. Definition – examples
5.1.2. Choice of a preference relation
5.2. Sufficiency and completeness
5.2.1. Sufficiency
5.2.2. Complete statistics
5.3. The maximum-likelihood method
5.3.1. Definition
5.3.2. Maximum likelihood and sufficiency
5.3.3. Calculating maximum-likelihood estimators
5.3.3.1. The Newton–Raphson method
5.4. Optimal unbiased estimators
5.4.1. Unbiased estimation
5.4.1.1. Existence of an unbiased estimator
5.4.2. Unbiased minimum-dispersion estimator
5.4.2.1. Application to an exponential model
5.4.2.2. Application to the Gaussian model
5.4.2.3. Use of the Lehmann–Scheffé theorem
5.4.3. Criticism of unbiased estimators
5.5. Efficiency of an estimator
5.5.1. The Fréchet-Darmois-Cramer-Rao inequality
5.5.1.1. Calculating I(θ)
5.5.1.2. Properties of the Fisher information
5.5.1.3. The case of a biased estimator
5.5.2. Efficiency
5.5.3. Extension to Rk
5.5.3.1. Properties of the information matrix
5.5.3.2. Efficiency
5.5.4. The non-regular case
5.5.4.1. “Superefficient” estimators
5.5.4.2. Cramer–Rao-type inequalities
5.6. The linear regression model
5.6.1. Generalities
5.6.2. Estimation of the parameter – the Gauss–Markov theorem
5.7. Exercises
Chapter 6: Hypothesis Testing and Confidence Regions
6.1. Generalities
6.1.1. The problem
6.1.2. Use of decision theory
6.1.2.1. Preference relation
6.1.3. Generalization
6.1.3.1. Preference relation
6.1.4. Sufficiency
6.2. The Neyman-Pearson (NP) lemma
6.3. Multiple hypothesis tests (general methods)
6.3.1. Testing a simple hypothesis against a composite one
6.3.1.1. The γ test
6.3.1.2. The λ test
6.3.2. General case – unbiased tests
6.3.2.1. Relation beween unbiased tests and unbiased decision functions
6.4. Case where the ratio of the likelihoods is monotonic
6.4.1. Generalities
6.4.2. Unilateral tests
6.4.3. Bilateral tests
6.5. Tests relating to the normal distribution
6.6. Application to estimation: confidence regions
6.6.1. First preference relation on confidence regions
6.6.2. Relation to tests
6.7. Exercises
Chapter 7: Asymptotic Statistics
7.1. Generalities
7.2. Consistency of the maximum likelihood estimator
7.3. The limiting distribution of the maximum likelihood estimator
7.4. The likelihood ratio test
7.5. Exercises
Chapter 8: Non-Parametric Methods and Robustness
8.1. Generalities
8.2. Non-parametric estimation
8.2.1. Empirical estimators
8.2.2. Distribution and density estimation
8.2.2.1. Convergence of the estimator
8.2.3. Regression estimation
8.3. Non-parametric tests
8.3.1. The χ2 test
8.3.2. The Kolmogorov–Smirnov test
8.3.3. The Cramer–von Mises test
8.3.4. Rank test
8.4. Robustness
8.4.1. An example of a robust test
8.4.2. An example of a robust estimator
8.4.3. A general definition of a robust estimator
8.5. Exercises
Part 2: Statistics for Stochastic Processes
Chapter 9: Introduction to Statistics for Stochastic Processes
9.1. Modeling a family of observations
9.2. Processes
9.2.1. The distribution of a process
9.2.2. Gaussian processes
9.2.3. Stationary processes
9.2.4. Markov processes
9.3. Statistics for stochastic processes
9.4. Exercises
Chapter 10: Weakly Stationary Discrete-Time Processes
10.1. Autocovariance and spectral density
10.2. Linear prediction and Wold decomposition
10.3. Linear processes and the ARMA model
10.3.1. Spectral density of a linear process
10.4. Estimating the mean of a weakly stationary process
10.5. Estimating the autocovariance
10.6. Estimating the spectral density
10.6.1. The periodogram
10.6.2. Convergent estimators of the spectral density
10.7. Exercises
Chapter 11: Poisson Processes – A Probabilistic and Statistical Study
11.1. Introduction
11.2. The axioms of Poisson processes
11.3. Interarrivai time
11.4. Properties of the Poisson process
11.5. Notions on generalized Poisson processes
11.6. Statistics of Poisson processes
11.7. Exercises
Chapter 12: Square-Integrable Continuous-Time Processes
12.1. Definitions
12.2. Mean-square continuity
12.3. Mean-square integration
12.4. Mean-square differentiation
12.5. The Karhunen–Loeve theorem
12.6. Wiener processes
12.6.1. Karhunen-Loeve decomposition
12.6.2. Statistics of Wiener processes
12.7. Notions on weakly stationary continuous-time processes
12.7.1. Estimating the mean
12.7.2. Estimating the autocovariance
12.7.3. The case of a process observed at discrete instants
12.8. Exercises
Chapter 13: Stochastic Integration and Diffusion Processes
13.1. Itô integral
13.2. Diffusion processes
13.3. Processes defined by stochastic differential equations and stochastic integrals
13.4. Notions on statistics for diffusion processes
13.5. Exercises
Chapter 14: ARMA Processes
14.1. Autoregressive processes
14.2. Moving average processes
14.3. General ARMA processes
14.4. Non-stationary models
14.4.1. The Box-Cox transformation
14.4.2. Eliminating the trend by differentiation
14.4.3. Eliminating the seasonality
14.4.4. Introducing exogenous variables
14.5. Statistics of ARMA processes
14.5.1. Identification
14.5.2. Estimation
14.5.3. Verification
14.6. Multidimensional processes
14.7. Exercises
Chapter 15: Prediction
15.1. Generalities
15.2. Empirical methods of prediction
15.2.1. The empirical mean
15.2.2. Exponential smoothing
15.2.3. Naive predictors
15.2.4. Trend adjustment
15.3. Prediction in the ARIMA model
15.4. Prediction in continuous time
15.5. Exercises
Part 3: Supplement
Chapter 16: Elements of Probability Theory
16.1. Measure spaces: probability spaces
16.2. Measurable functions: real random variables
16.3. Integrating real random variables
16.4. Random vectors
16.5. Independence
16.6. Gaussian vectors
16.7. Stochastic convergence
16.8. Limit theorems
Appendix: Statistical Tables
A1.1. Random numbers
A1.2. Distribution function of the standard normal distribution
A1.3. Density of the standard normal distribution
A1.4. Percentiles (tp) of Student’s distribution
A1.5. Ninety-fifth percentiles of Fisher–Snedecor distributions
A1.6. Ninety-ninth percentiles of Fisher–Snedecor distributions
A1.7. Percentiles (χ2p) of the χ2 distribution with n degrees of freedom
A1.8. Individual probabilities of the Poisson distribution
A1.9. Cumulative probabilities of the Poisson distribution
A1.10. Binomial coefficients Ckn for n ≤ 30 and 0 ≤ k ≤ 7
A1.11. Binomial coefficients Ckn for n ≤ 30 and 8 ≤ k ≤ 15
Bibliography
Index
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