Intermediate Mathematical Statistics

1 Sufficiency. - 1. 1 Introduction. - 1. 2 Factorization criterion. - 1. 3 Distribution of statistics conditional on a sufficient statistic. - 1. 4 Joint sufficiency. - 1. 5 Minimal sufficiency. - 2 Unbiased point estimators. - 2. 1 Introduction. - 2. 2 Rao-Blackwell theorem. - 2. 3 The role of sufficient statistics. - 2. 4 Completeness. - 2. 5 Joint completeness. - 2. 6 Sufficiency completeness and independence. - 2. 7 Minimum-variance bounds. - 2. 8 Computation of a minimum-variance bound. - 2. 9 Minimum attainable variance. - 2. 10 Mean square error. - 2. 11 Two parameters. - 3 Elementary decision theory and Bayesian methods. - 3. 1 Comments on classical techniques. - 3. 2 Loss functions. - 3. 3 Decision theory. - 3. 4 Bayes decisions. - 3. 5 Using data. - 3. 6 Computing posterior distributions. - 3. 7 Conjugate distributions. - 3. 8 Distribution of the next observation. - 3. 9 More than one parameter. - 3. 10 Decision functions. - 3. 11 Bayes estimators. - 3. 12 Admissibility. - 4 Methods of estimation. - 4. 1 Introduction. - 4. 2 Maximum likelihood estimation. - 4. 3 Locating the maximum likelihood estimator. - 4. 4 Estimation of a function of a parameter. - 4. 5 Truncation and censoring. - 4. 6 Estimation of several parameters. - 4. 7 Approximation techniques. - 4. 8 Large-sample properties. - 4. 9 Method of least squares. - 4. 10 Normal equations. - 4. 11 Solution of the normal equations (non-singular case). - 4. 12 Use of matrices. - 4. 13 Best unbiased linear estimation. - 4. 14 Co variance matrix. - 4. 15 Relaxation of assumptions. - 5 Hypothesis testing I. - 5. 1 Introduction. - 5. 2 Statistical hypothesis. - 5. 3 Simple null hypothesis against simple alternative. - 5. 4 Applications of the Neyman-Pearson theorem. - 5. 5 Uniformly most powerful tests for a single parameter. - 5. 6 Most powerful randomized tests. - 5. 7 Hypothesis testing as a decision process. - 5. 8 Minimax and Bayes tests. - 6 Hypothesis testing II. - 6. 1 Two-sided tests for a single parameter. - 6. 2 Neyman-Pearson theorem extension (nonrandomized version). - 6. 3 Regular exponential family of distributions. - 6. 4 Uniformly most powerful unbiased test of ? = ?0 against ? ? ?0. - 6. 5 Nuisance parameters. - 6. 6 Similar tests. - 6. 7 Composite hypotheses-several parameters. - 6. 8 Likelihood ratio tests. - 6. 9 Bayes methods. - 6. 10 Loss function for one-sided hypotheses. - 6. 11 Testing ? = ?0 against ? ? ?0. - 7 Interval estimation. - 7. 1 One parameter Bayesian confidence intervals. - 7. 2 Two parameters Bayesian confidence regions. - 7. 3 Confidence intervals (classical). - 7. 4 Most selective limits. - 7. 5 Relationship to best tests. - 7. 6 Unbiased confidence intervals. - 7. 7 Nuisance parameters. - 7. 8 Discrete distributions. - 7. 9 Relationship between classical and Bayesian intervals. - 7. 10 Large-sample confidence intervals. - Appendix 1 Functions of random variables. - A 1. 1 Introduction. - A 1. 2 Transformations: discrete distributions. - A1. 3 Continuous distributions. - A 1. 4 The order statistics. - Appendix 2 The regular exponential family of distributions. - A2. 1 Single parameter. - A2. 2 Several parameters. - A2. 3 The regular exponential family of bivariate distributions. - Further exercises. - Brief solutions to further exercises. - Further reading. - Author index. Language: English