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| List of Figures | |
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| List of Tables | |
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| Preface | |
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| Introduction | |
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| Preamble | |
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| Likelihood | |
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| Sufficiency | |
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| *Minimal sufficiency | |
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| * Completeness | |
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| Exponential family of distributions | |
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| Exercises | |
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| Point Estimation | |
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| Introduction | |
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| Maximum likelihood estimation | |
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| Method of moments | |
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| Method of least squares | |
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| Goodness-of-estimation. Mean squared error | |
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| Unbiased estimation | |
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| Definition and main properties | |
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| Uniformly minimum variance unbiased estimators. The Cramer-Rao lower bound | |
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| *The Cramer-Rao lower bound for multivariate parameters | |
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| Rao-Blackwell theorem | |
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| *Lehmann-Scheff� theorem | |
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| Exercises | |
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| Confidence Intervals, Bounds, and Regions | |
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| Introduction | |
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| Quoting the estimation error | |
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| Confidence intervals | |
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| Confidence bounds | |
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| *Confidence regions | |
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| Exercises | |
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| Hypothesis Testing | |
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| Introduction | |
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| Simple hypotheses | |
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| Type I and Type II errors | |
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| Choice of a critical value | |
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| The p-value | |
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| Maximal power tests. Neyman-Pearson lemma | |
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| Composite hypotheses | |
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| Power function | |
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| Uniformly most powerful tests | |
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| Generalized likelihood ratio tests | |
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| Hypothesis testing and confidence intervals | |
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| Sequential testing | |
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| Exercises | |
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| Asymptotic Analysis | |
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| Introduction | |
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| Convergence and consistency in MSE | |
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| Convergence and consistency in probability | |
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| Convergence in distribution | |
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| The central limit theorem | |
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| Asymptotically normal consistency | |
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| Asymptotic confidence intervals | |
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| Asymptotically normal consistency of the MLE, Wald's confidence intervals, and tests | |
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| *Multiparameter case | |
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| Asymptotic distribution of the GLRT, Wilks' theorem | |
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| Exercises | |
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| Bayesian Inference | |
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| Introduction | |
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| Choice of priors | |
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| Conjugate priors | |
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| Noninformative (objective) priors | |
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| Point estimation | |
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| Interval estimation. Credible sets | |
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| Hypothesis testing | |
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| Simple hypotheses | |
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| Composite hypotheses | |
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| Testing a point null hypothesis | |
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| Exercises | |
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| *Elements of Statistical Decision Theory | |
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| Introduction and notations | |
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| Risk function and admissibility | |
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| Minimax risk and minimax rules | |
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| Bayes risk and Bayes rules | |
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| Posterior expected loss and Bayes actions | |
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| Admissibility and minimaxity of Bayes rules | |
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| Exercises | |
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| *Linear Models | |
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| Introduction | |
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| Definition and examples | |
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| Estimation of regression coefficients | |
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| Residuals. Estimation of the variance | |
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| Examples | |
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| Estimation of a normal mean | |
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| Comparison between the means of two independent normal samples with a common variance | |
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| Simple linear regression | |
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| Goodness-of-fit. Multiple correlation coefficient | |
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| Confidence intervals and regions for the coefficients | |
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| Hypothesis testing in linear models | |
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| Testing significance of a single predictor | |
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| Testing significance of a group of predictors | |
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| Testing a general linear hypothesis | |
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| Predictions | |
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| Analysis of variance | |
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| One-way ANOVA | |
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| Two-way ANOVA and beyond | |
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| Probabilistic Review | |
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| Introduction | |
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| Basic probabilistic laws | |
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| Random variables | |
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| Expected value and the variance | |
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| Chebyshev's and Markov's inequalities | |
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| Expectation of functions and the Jensen's inequality | |
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| Joint distribution | |
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| Covariance, correlation, and the Cauchy-Schwarz inequality | |
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| Expectation and variance of a sum of random variables | |
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| Conditional distribution and Bayes Theorem | |
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| Distributions of functions of random variables | |
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| Random vectors | |
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| Special families of distributions | |
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| Bernoulli and binomial distributions | |
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| Geometric and negative binomial distributions | |
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| Hypergeometric distribution | |
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| Poisson distribution | |
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| Uniform distribution | |
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| Exponential distribution | |
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| Weibull distribution | |
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| Gamma-distribution | |
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| Beta-distribution | |
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| Cauchy distribution | |
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| Normal distribution | |
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| Log-normal distribution | |
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| �<sup>2</sup> distribution | |
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| t-distribution | |
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| F-distribution | |
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| Multinormal distribution | |
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| Definition and main properties | |
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| Projections of normal vectors | |
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| Solutions of Selected Exercises | |
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| Chapter 1 | |
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| Chapter 2 | |
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| Chapter 3 | |
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| Chapter 4 | |
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| Chapter 5 | |
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| Chapter 6 | |
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| Chapter 7 | |
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| Index | |