Mathematical Statistics and Stochastic Processes

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ISBN: 9781848213616 | 1848213611
Cover: Hardcover
Copyright: 5/14/2012

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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.

Preface	p. xiii
Mathematical Statistics	p. 1
Introduction to Mathematical Statistics	p. 3
Generalities	p. 3
Examples of statistics problems	p. 4
Quality control	p. 4
Measurement errors	p. 5
Filtering	p. 5
Confidence intervals	p. 6
Homogeneity testing	p. 7
Principles of Decision Theory	p. 9
Generalities	p. 9
The problem of choosing a decision function	p. 11
Principles of Bayesian statistics	p. 13
Generalities	p. 13
Determination of Bayesian decision functions	p. 14
Admissibility of Bayes' rules	p. 16
Complete classes	p. 17
Criticism of decision theory - the asymptotic point of view	p. 18
Exercises	p. 18
Conditional Expectation	p. 21
Definition	p. 21
Properties and extension	p. 22
Conditional probabilities and conditional distributions	p. 24
Regular version of the conditional probability	p. 24
Conditional distributions	p. 24
Theorem for integration with respect to the conditional distribution	p. 25
Determination of the conditional distributions in the usual cases	p. 25
Exercises	p. 26
Statistics and Sufficiency	p. 29
Samples and empirical distributions	p. 29
Properties of the empirical distribution and the associated statistics	p. 30
Sufficiency	p. 31
The factorization theorem	p. 31
Examples of sufficient statistics - an exponential model	p. 33
Use of a sufficient statistic	p. 35
Exercises	p. 36
Point Estimation	p. 39
Generalities	p. 39
Definition - examples	p. 39
Choice of a preference relation	p. 40
Sufficiency and completeness	p. 42
Sufficiency	p. 42
Complete statistics	p. 43
The maximum-likelihood method	p. 45
Definition	p. 45
Maximum likelihood and sufficiency	p. 47
Calculating maximum-likelihood estimators	p. 47
The Newton-Raphson method	p. 48
Optimal unbiased estimators	p. 49
Unbiased estimation	p. 49
Existence of an unbiased estimator	p. 50
Unbiased minimum-dispersion estimator	p. 51
Application to an exponential model	p. 52
Application to the Gaussian model	p. 54
Use of the Lehmann-Scheffé theorem	p. 55
Criticism of unbiased estimators	p. 55
Efficiency of an estimator	p. 56
The Fréchet-Darmois-Cramer-Rao inequality	p. 56
Calculating I(¿)	p. 57
Properties of the Fisher information	p. 58
The case of a biased estimator	p. 59
Efficiency	p. 60
Extension to R^k	p. 60
Properties of the information matrix	p. 62
Efficiency	p. 62
The non-regular case	p. 64
"Superefficient" estimators	p. 64
Cramer-Rao-type inequalities	p. 64
The linear regression model	p. 65
Generalities	p. 65
Estimation of the parameter - the Gauss-Markov theorem	p. 66
Exercises	p. 68
Hypothesis Testing and Confidence Regions	p. 73
Generalities	p. 73
The problem	p. 73
Use of decision theory	p. 73
Preference relation	p. 74
Generalization	p. 74
Preference relation	p. 75
Sufficiency	p. 75
The Neyman-Pearson (NP) lemma	p. 75
Multiple hypothesis tests (general methods)	p. 80
Testing a simple hypothesis against a composite one	p. 80
The ¿ test	p. 81
The ¿ test	p. 81
General case - unbiased tests	p. 82
Relation between unbiased tests and unbiased decision functions	p. 83
Case where the ratio of the likelihoods is monotonic	p. 84
Generalities	p. 84
Unilateral tests	p. 84
Bilateral tests	p. 85
Tests relating to the normal distribution	p. 86
Application to estimation: confidence regions	p. 86
First preference relation on confidence regions	p. 88
Relation to tests	p. 88
Exercises	p. 90
Asymptotic Statistics	p. 101
Generalities	p. 101
Consistency of the maximum likelihood estimator	p. 103
The limiting distribution of the maximum likelihood estimator	p. 104
The likelihood ratio test	p. 106
Exercises	p. 108
Non-Parametric Methods and Robustness	p. 113
Generalities	p. 113
Non-parametric estimation	p. 114
Empirical estimators	p. 114
Distribution and density estimation	p. 114
Convergence of the estimator	p. 115
Regression estimation	p. 116
Non-parametric tests	p. 117
The ¿² test	p. 117
The Kolmogorov-Smirnov test	p. 120
The Cramer-von Mises test	p. 120
Rank test	p. 121
Robustness	p. 121
An example of a robust test	p. 122
An example of a robust estimator	p. 122
A general definition of a robust estimator	p. 123
Exercises	p. 124
Statistics for Stochastic Processes	p. 131
Introduction to Statistics for Stochastic Processes	p. 133
Modeling a family of observations	p. 133
Processes	p. 134
The distribution of a process	p. 134
Gaussian processes	p. 135
Stationary processes	p. 135
Markov processes	p. 136
Statistics for stochastic processes	p. 137
Exercises	p. 138
Weakly Stationary Discrete-Time Processes	p. 141
Autocovariance and spectral density	p. 141
Linear prediction and Wold decomposition	p. 144
Linear processes and the ARMA model	p. 146
Spectral density of a linear process	p. 147
Estimating the mean of a weakly stationary process	p. 149
Estimating the autocovariance	p. 151
Estimating the spectral density	p. 151
The periodogram	p. 151
Convergent estimators of the spectral density	p. 154
Exercises	p. 155
Poisson Processes - A Probabilistic and Statistical Study	p. 163
Introduction	p. 163
The axioms of Poisson processes	p. 164
Interarrival time	p. 166
Properties of the Poisson process	p. 168
Notions on generalized Poisson processes	p. 170
Statistics of Poisson processes	p. 172
Exercises	p. 177
Square-Integrable Continuous-Time Processes	p. 183
Definitions	p. 183
Mean-square continuity	p. 183
Mean-square integration	p. 184
Mean-square differentiation	p. 187
The Karhunen-Loeve theorem	p. 188
Wiener processes	p. 189
Karhunen-Loeve decomposition	p. 192
Statistics of Wiener processes	p. 193
Notions on weakly stationary continuous-time processes	p. 195
Estimating the mean	p. 196
Estimating the autocovariance	p. 196
The case of a process observed at discrete instants	p. 197
Exercises	p. 197
Stochastic Integration and Diffusion Processes	p. 203
Itô integral	p. 203
Diffusion processes	p. 206
Processes defined by stochastic differential equations and stochastic integrals	p. 212
Notions on statistics for diffusion processes	p. 215
Exercises	p. 216
ARMA Processes	p. 219
Autoregressive processes	p. 219
Moving average processes	p. 223
General ARMA processes	p. 224
Non-stationary models	p. 226
The Box-Cox transformation	p. 226
Eliminating the trend by differentiation	p. 227
Eliminating the seasonality	p. 227
Introducing exogenous variables	p. 228
Statistics of ARMA processes	p. 228
Identification	p. 228
Estimation	p. 230
Verification	p. 232
Multidimensional processes	p. 232
Exercises	p. 233
Prediction	p. 239
Generalities	p. 239
Empirical methods of prediction	p. 240
The empirical mean	p. 240
Exponential smoothing	p. 240
Native predictors	p. 241
Trend adjustment	p. 241
Prediction in the ARIMA model	p. 242
Prediction in continuous time	p. 244
Exercises	p. 245
Supplement	p. 249
Elements of Probability Theory	p. 251
Measure spaces: probability spaces	p. 251
Measurable functions: real random variables	p. 253
Integrating real random variables	p. 255
Random vectors	p. 259
Independence	p. 261
Gaussian vectors	p. 262
Stochastic convergence	p. 264
Limit theorems	p. 265
Statistical Tables	p. 267
Random numbers	p. 267
Distribution function of the standard normal distribution	p. 268
Density of the standard normal distribution	p. 269
Percentiles (t_p) of Student's distribution	p. 270
Ninety-fifth percentiles of Fisher-Snedecor distributions	p. 271
Ninety-ninth percentiles of Fisher-Snedecor distributions	p. 272
Percentiles (¿²_p) of the & Chi;² distribution with n degrees of freedom	p. 273
Individual probabilities of the Poisson distribution	p. 274
Cumulative probabilities of the Poisson distribution	p. 275
Binomial coefficients C^k_n for n ≤ 30 and 0 ≤ k ≤ 7	p. 276
Binomial coefficients C^k_n for n ≤ 30 and 8 ≤ k ≤ 15	p. 277
Bibliography	p. 279
Index	p. 281
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Mathematical Statistics and Stochastic Processes

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Mathematical Statistics and Stochastic Processes

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