Mathematical Statistics and Stochastic Processes

Denis Bosq is Professor emeritus Université Pierre et Marie Curie (Paris 6) France.

• Preface xiiiPART 1. MATHEMATICAL STATISTICS 1Chapter 1. Introduction to Mathematical Statistics 31.1. Generalities 31.2. Examples of statistics problems 4Chapter 2. Principles of Decision Theory 92.1. Generalities 92.2. The problem of choosing a decision function 112.3. Principles of Bayesian statistics 132.4. Complete classes 172.5. Criticism of decision theory – the asymptotic point of view 182.6. Exercises 18Chapter 3. Conditional Expectation 213.1. Definition 213.2. Properties and extension 223.3. Conditional probabilities and conditional distributions 243.4. Exercises 26Chapter 4. Statistics and Sufficiency 294.1. Samples and empirical distributions 294.2. Sufficiency 314.3. Examples of sufficient statistics – an exponential model 334.4. Use of a sufficient statistic 354.5. Exercises 36Chapter 5. Point Estimation 395.1. Generalities 395.2. Sufficiency and completeness 425.3. The maximum-likelihood method 455.4. Optimal unbiased estimators 495.5. Efficiency of an estimator 565.6. The linear regression model 655.7. Exercises 68Chapter 6. Hypothesis Testing and Confidence Regions 736.1. Generalities 736.2. The Neyman–Pearson (NP) lemma 756.3. Multiple hypothesis tests (general methods) 806.4. Case where the ratio of the likelihoods is monotonic 846.5. Tests relating to the normal distribution 866.6. Application to estimation: confidence regions 866.7. Exercises 90Chapter 7. Asymptotic Statistics 1017.1. Generalities 1017.2. Consistency of the maximum likelihood estimator 1037.3. The limiting distribution of the maximum likelihood estimator 1047.4. The likelihood ratio test 1067.5. Exercises 108Chapter 8. Non-Parametric Methods and Robustness 1138.1. Generalities 1138.2. Non-parametric estimation 1148.3. Non-parametric tests 1178.4. Robustness 1218.5. Exercises 124PART 2. STATISTICS FOR STOCHASTIC PROCESSES 131Chapter 9. Introduction to Statistics for Stochastic Processes 1339.1. Modeling a family of observations 1339.2. Processes 1349.3. Statistics for stochastic processes 1379.4. Exercises 138Chapter 10. Weakly Stationary Discrete-Time Processes 14110.1. Autocovariance and spectral density 14110.2. Linear prediction and Wold decomposition 14410.3. Linear processes and the ARMA model 14610.4. Estimating the mean of a weakly stationary process 14910.5. Estimating the autocovariance 15110.6. Estimating the spectral density 15110.7. Exercises 155Chapter 11. Poisson Processes – A Probabilistic and Statistical Study 16311.1. Introduction 16311.2. The axioms of Poisson processes 16411.3. Interarrival time 16611.4. Properties of the Poisson process 16811.5. Notions on generalized Poisson processes 17011.6. Statistics of Poisson processes 17211.7. Exercises 177Chapter 12. Square-Integrable Continuous-Time Processes 18312.1. Definitions 18312.2. Mean-square continuity 18312.3. Mean-square integration 18412.4. Mean-square differentiation 18712.5. The Karhunen–Loeve theorem 18812.6. Wiener processes 18912.7. Notions on weakly stationary continuous-time processes 19512.8. Exercises 197Chapter 13. Stochastic Integration and Diffusion Processes 20313.1. Itô integral 20313.2. Diffusion processes 20613.3. Processes defined by stochastic differential equations and stochastic integrals 21213.4. Notions on statistics for diffusion processes 21513.5. Exercises 216Chapter 14. ARMA Processes 21914.1. Autoregressive processes 21914.2. Moving average processes 22314.3. General ARMA processes 22414.4. Non-stationary models 22614.5. Statistics of ARMA processes 22814.6. Multidimensional processes 23214.7. Exercises 233Chapter 15. Prediction 23915.1. Generalities 23915.2. Empirical methods of prediction 24015.3. Prediction in the ARIMA model 24215.4. Prediction in continuous time 24415.5. Exercises 245PART 3. SUPPLEMENT 249Chapter 16. Elements of Probability Theory 25116.1. Measure spaces: probability spaces 25116.2. Measurable functions: real random variables 25316.3. Integrating real random variables 25516.4. Random vectors 25916.5. Independence 26116.6. Gaussian vectors 26216.7. Stochastic convergence 26416.8. Limit theorems 265Appendix. Statistical Tables 267A1.1. Random numbers 267A1.2. Distribution function of the standard normal distribution 268A1.3. Density of the standard normal distribution 269A1.4. Percentiles (tp) of Student’s distribution 270A1.5. Ninety-fifth percentiles of Fisher–Snedecor distributions 271A1.6. Ninety-ninth percentiles of Fisher–Snedecor distributions 272A1.7. Percentiles (χ2 p) of the χ2 distribution with n degrees of freedom 273A1.8. Individual probabilities of the Poisson distribution 274A1.9. Cumulative probabilities of the Poisson distribution 275A1.10. Binomial coefficients Ck n for n ≤ 30 and 0 ≤ k ≤ 7 276A1.11. Binomial coefficients Ck n for n ≤ 30 and 8 ≤ k ≤ 15 277Bibliography 279Index 281

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