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| Preface | |
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| Acknowledgments | |
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| Notions of Probability | |
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| Introduction | |
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| About Sets | |
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| Axiomatic Development of Probability | |
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| The Conditional Probability and Independent Events | |
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| Calculus of Probability | |
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| Bayes's Theorem | |
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| Selected Counting Rules | |
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| Discrete Random Variables | |
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| Probability Mass and Distribution Functions | |
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| Continuous Random Variables | |
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| Probability Density and Distribution Functions | |
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| The Median of a Distribution | |
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| Selected Reviews from Mathematics | |
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| Some Standard Probability Distributions | |
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| Discrete Distributions | |
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| Continuous Distributions | |
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| Exercises and Complements | |
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| Expectations of Functions of Random Variables | |
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| Introduction | |
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| Expectation and Variance | |
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| The Bernoulli Distribution | |
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| The Binomial Distribution | |
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| The Poisson Distribution | |
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| The Uniform Distribution | |
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| The Normal Distribution | |
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| The Laplace Distribution | |
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| The Gamma Distribution | |
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| The Moments and Moment Generating Function | |
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| The Binomial Distribution | |
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| The Poisson Distribution | |
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| The Normal Distribution | |
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| The Gamma Distribution | |
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| Determination of a Distribution via MGF | |
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| The Probability Generating Function | |
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| Exercises and Complements | |
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| Multivariate Random Variables | |
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| Introduction | |
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| Discrete Distributions | |
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| The Joint, Marginal and Conditional Distributions | |
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| The Multinomial Distribution | |
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| Continuous Distributions | |
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| The Joint, Marginal and Conditional Distributions | |
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| Three and Higher Dimensions | |
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| Covariances and Correlation Coefficients | |
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| The Multinomial Case | |
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| Independence of Random Variables | |
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| The Bivariate Normal Distribution | |
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| Correlation Coefficient and Independence | |
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| The Exponential Family of Distributions | |
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| One-parameter Situation | |
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| Multi-parameter Situation | |
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| Some Standard Probability Inequalities | |
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| Markov and Bernstein-Chernoff Inequalities | |
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| Tchebysheff's Inequality | |
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| Cauchy-Schwarz and Covariance Inequalities | |
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| Jensen's and Lyapunov's Inequalities | |
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| Holder's Inequality | |
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| Bonferroni Inequality | |
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| Central Absolute Moment Inequality | |
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| Exercises and Complements | |
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| Functions of Random Variables and Sampling Distribution | |
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| Introduction | |
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| Using Distribution Functions | |
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| Discrete Cases | |
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| Continuous Cases | |
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| The Order Statistics | |
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| The Convolution | |
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| The Sampling Distribution | |
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| Using the Moment Generating Function | |
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| A General Approach with Transformations | |
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| Several Variable Situations | |
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| Special Sampling Distributions | |
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| The Student's t Distribution | |
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| The F Distribution | |
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| The Beta Distribution | |
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| Special Continuous Multivariate Distributions | |
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| The Normal Distribution | |
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| The t Distribution | |
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| The F Distribution | |
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| Importance of Independence in Sampling Distributions | |
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| Reproductivity of Normal Distributions | |
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| Reproductivity of Chi-square Distributions | |
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| The Student's t Distribution | |
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| The F Distribution | |
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| Selected Review in Matrices and Vectors | |
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| Exercises and Complements | |
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| Concepts of Stochastic Convergence | |
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| Introduction | |
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| Convergence in Probability | |
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| Convergence in Distribution | |
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| Combination of the Modes of Convergence | |
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| The Central Limit Theorems | |
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| Convergence of Chi-square, t, and F Distributions | |
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| The Chi-square Distribution | |
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| The Student's t Distribution | |
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| The F Distribution | |
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| Convergence of the PDF and Percentage Points | |
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| Exercises and Complements | |
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| Sufficiency, Completeness, and Ancillarity | |
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| Introduction | |
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| Sufficiency | |
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| The Conditional Distribution Approach | |
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| The Neyman Factorization Theorem | |
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| Minimal Sufficiency | |
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| The Lehmann-Scheffe Approach | |
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| Information | |
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| One-parameter Situation | |
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| Multi-parameter Situation | |
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| Ancillarity | |
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| The Location, Scale, and Location-Scale Families | |
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| Its Role in the Recovery of Information | |
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| Completeness | |
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| Complete Sufficient Statistics | |
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| Basu's Theorem | |
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| Exercises and Complements | |
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| Point Estimation | |
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| Introduction | |
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| Finding Estimators | |
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| The Method of Moments | |
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| The Method of Maximum Likelihood | |
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| Criteria to Compare Estimators | |
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| Unbiasedness, Variance and Mean Squared Error | |
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| Best Unbiased and Linear Unbiased Estimators | |
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| Improved Unbiased Estimator via Sufficiency | |
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| The Rao-Blackwell Theorem | |
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| Uniformly Minimum Variance Unbiased Estimator | |
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| The Cramer-Rao Inequality and UMVUE | |
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| The Lehmann-Scheffe Theorems and UMVUE | |
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| A Generalization of the Cramer-Rao Inequality | |
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| Evaluation of Conditional Expectations | |
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| Unbiased Estimation Under Incompleteness | |
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| Does the Rao-Blackwell Theorem Lead to UMVUE? | |
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| Consistent Estimators | |
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| Exercises and Complements | |
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| Tests of Hypotheses | |
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| Introduction | |
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| Error Probabilities and the Power Function | |
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| The Concept of a Best Test | |
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| Simple Null Versus Simple Alternative Hypotheses | |
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| Most Powerful Test via the Neyman-Pearson Lemma | |
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| Applications: No Parameters Are Involved | |
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| Applications: Observations Are Non-IID | |
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| One-Sided Composite Alternative Hypothesis | |
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| UMP Test via the Neyman-Pearson Lemma | |
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| Monotone Likelihood Ratio Property | |
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| UMP Test via MLR Property | |
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| Simple Null Versus Two-Sided Alternative Hypotheses | |
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| An Example Where UMP Test Does Not Exist | |
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| An Example Where UMP Test Exists | |
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| Unbiased and UMP Unbiased Tests | |
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| Exercises and Complements | |
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| Confidence Interval Estimation | |
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| Introduction | |
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| One-Sample Problems | |
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| Inversion of a Test Procedure | |
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| The Pivotal Approach | |
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| The Interpretation of a Confidence Coefficient | |
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| Ideas of Accuracy Measures | |
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| Using Confidence Intervals in the Tests of Hypothesis | |
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| Two-Sample Problems | |
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| Comparing the Location Parameters | |
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| Comparing the Scale Parameters | |
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| Multiple Comparisons | |
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| Estimating a Multivariate Normal Mean Vector | |
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| Comparing the Means | |
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| Comparing the Variances | |
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| Exercises and Complements | |
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| Bayesian Methods | |
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| Introduction | |
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| Prior and Posterior Distributions | |
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| The Conjugate Priors | |
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| Point Estimation | |
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| Credible Intervals | |
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| Highest Posterior Density | |
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| Contrasting with the Confidence Intervals | |
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| Tests of Hypotheses | |
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| Examples with Non-Conjugate Priors | |
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| Exercises and Complements | |
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| Likelihood Ratio and Other Tests | |
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| Introduction | |
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| One-Sample Problems | |
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| LR Test for the Mean | |
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| LR Test for the Variance | |
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| Two-Sample Problems | |
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| Comparing the Means | |
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| Comparing the Variances | |
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| Bivariate Normal Observations | |
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| Comparing the Means: The Paired Difference t Method | |
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| LR Test for the Correlation Coefficient | |
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| Tests for the Variances | |
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| Exercises and Complements | |
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| Large-Sample Inference | |
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| Introduction | |
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| The Maximum Likelihood Estimation | |
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| Confidence Intervals and Tests of Hypothesis | |
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| The Distribution-Free Population Mean | |
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| The Binomial Proportion | |
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| The Poisson Mean | |
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| The Variance Stabilizing Transformations | |
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| The Binomial Proportion | |
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| The Poisson Mean | |
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| The Correlation Coefficient | |
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| Exercises and Complements | |
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| Sample Size Determination: Two-Stage Procedures | |
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| Introduction | |
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| The Fixed-Width Confidence Interval | |
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| Stein's Sampling Methodology | |
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| Some Interesting Properties | |
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| The Bounded Risk Point Estimation | |
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| The Sampling Methodology | |
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| Some Interesting Properties | |
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| Exercises and Complements | |
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| Appendix | |
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| Abbreviations and Notation | |
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| A Celebration of Statistics: Selected Biographical Notes | |
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| Selected Statistical Tables | |
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| The Standard Normal Distribution Function | |
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| Percentage Points of the Chi-Square Distribution | |
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| Percentage Points of the Student's t Distribution | |
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| Percentage Points of the F Distribution | |
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| References | |
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| Index | |