| |

| |

Preface | |

| |

| |

Acknowledgments | |

| |

| |

| |

Notions of Probability | |

| |

| |

| |

Introduction | |

| |

| |

| |

About Sets | |

| |

| |

| |

Axiomatic Development of Probability | |

| |

| |

| |

The Conditional Probability and Independent Events | |

| |

| |

| |

Calculus of Probability | |

| |

| |

| |

Bayes's Theorem | |

| |

| |

| |

Selected Counting Rules | |

| |

| |

| |

Discrete Random Variables | |

| |

| |

| |

Probability Mass and Distribution Functions | |

| |

| |

| |

Continuous Random Variables | |

| |

| |

| |

Probability Density and Distribution Functions | |

| |

| |

| |

The Median of a Distribution | |

| |

| |

| |

Selected Reviews from Mathematics | |

| |

| |

| |

Some Standard Probability Distributions | |

| |

| |

| |

Discrete Distributions | |

| |

| |

| |

Continuous Distributions | |

| |

| |

| |

Exercises and Complements | |

| |

| |

| |

Expectations of Functions of Random Variables | |

| |

| |

| |

Introduction | |

| |

| |

| |

Expectation and Variance | |

| |

| |

| |

The Bernoulli Distribution | |

| |

| |

| |

The Binomial Distribution | |

| |

| |

| |

The Poisson Distribution | |

| |

| |

| |

The Uniform Distribution | |

| |

| |

| |

The Normal Distribution | |

| |

| |

| |

The Laplace Distribution | |

| |

| |

| |

The Gamma Distribution | |

| |

| |

| |

The Moments and Moment Generating Function | |

| |

| |

| |

The Binomial Distribution | |

| |

| |

| |

The Poisson Distribution | |

| |

| |

| |

The Normal Distribution | |

| |

| |

| |

The Gamma Distribution | |

| |

| |

| |

Determination of a Distribution via MGF | |

| |

| |

| |

The Probability Generating Function | |

| |

| |

| |

Exercises and Complements | |

| |

| |

| |

Multivariate Random Variables | |

| |

| |

| |

Introduction | |

| |

| |

| |

Discrete Distributions | |

| |

| |

| |

The Joint, Marginal and Conditional Distributions | |

| |

| |

| |

The Multinomial Distribution | |

| |

| |

| |

Continuous Distributions | |

| |

| |

| |

The Joint, Marginal and Conditional Distributions | |

| |

| |

| |

Three and Higher Dimensions | |

| |

| |

| |

Covariances and Correlation Coefficients | |

| |

| |

| |

The Multinomial Case | |

| |

| |

| |

Independence of Random Variables | |

| |

| |

| |

The Bivariate Normal Distribution | |

| |

| |

| |

Correlation Coefficient and Independence | |

| |

| |

| |

The Exponential Family of Distributions | |

| |

| |

| |

One-parameter Situation | |

| |

| |

| |

Multi-parameter Situation | |

| |

| |

| |

Some Standard Probability Inequalities | |

| |

| |

| |

Markov and Bernstein-Chernoff Inequalities | |

| |

| |

| |

Tchebysheff's Inequality | |

| |

| |

| |

Cauchy-Schwarz and Covariance Inequalities | |

| |

| |

| |

Jensen's and Lyapunov's Inequalities | |

| |

| |

| |

Holder's Inequality | |

| |

| |

| |

Bonferroni Inequality | |

| |

| |

| |

Central Absolute Moment Inequality | |

| |

| |

| |

Exercises and Complements | |

| |

| |

| |

Functions of Random Variables and Sampling Distribution | |

| |

| |

| |

Introduction | |

| |

| |

| |

Using Distribution Functions | |

| |

| |

| |

Discrete Cases | |

| |

| |

| |

Continuous Cases | |

| |

| |

| |

The Order Statistics | |

| |

| |

| |

The Convolution | |

| |

| |

| |

The Sampling Distribution | |

| |

| |

| |

Using the Moment Generating Function | |

| |

| |

| |

A General Approach with Transformations | |

| |

| |

| |

Several Variable Situations | |

| |

| |

| |

Special Sampling Distributions | |

| |

| |

| |

The Student's t Distribution | |

| |

| |

| |

The F Distribution | |

| |

| |

| |

The Beta Distribution | |

| |

| |

| |

Special Continuous Multivariate Distributions | |

| |

| |

| |

The Normal Distribution | |

| |

| |

| |

The t Distribution | |

| |

| |

| |

The F Distribution | |

| |

| |

| |

Importance of Independence in Sampling Distributions | |

| |

| |

| |

Reproductivity of Normal Distributions | |

| |

| |

| |

Reproductivity of Chi-square Distributions | |

| |

| |

| |

The Student's t Distribution | |

| |

| |

| |

The F Distribution | |

| |

| |

| |

Selected Review in Matrices and Vectors | |

| |

| |

| |

Exercises and Complements | |

| |

| |

| |

Concepts of Stochastic Convergence | |

| |

| |

| |

Introduction | |

| |

| |

| |

Convergence in Probability | |

| |

| |

| |

Convergence in Distribution | |

| |

| |

| |

Combination of the Modes of Convergence | |

| |

| |

| |

The Central Limit Theorems | |

| |

| |

| |

Convergence of Chi-square, t, and F Distributions | |

| |

| |

| |

The Chi-square Distribution | |

| |

| |

| |

The Student's t Distribution | |

| |

| |

| |

The F Distribution | |

| |

| |

| |

Convergence of the PDF and Percentage Points | |

| |

| |

| |

Exercises and Complements | |

| |

| |

| |

Sufficiency, Completeness, and Ancillarity | |

| |

| |

| |

Introduction | |

| |

| |

| |

Sufficiency | |

| |

| |

| |

The Conditional Distribution Approach | |

| |

| |

| |

The Neyman Factorization Theorem | |

| |

| |

| |

Minimal Sufficiency | |

| |

| |

| |

The Lehmann-Scheffe Approach | |

| |

| |

| |

Information | |

| |

| |

| |

One-parameter Situation | |

| |

| |

| |

Multi-parameter Situation | |

| |

| |

| |

Ancillarity | |

| |

| |

| |

The Location, Scale, and Location-Scale Families | |

| |

| |

| |

Its Role in the Recovery of Information | |

| |

| |

| |

Completeness | |

| |

| |

| |

Complete Sufficient Statistics | |

| |

| |

| |

Basu's Theorem | |

| |

| |

| |

Exercises and Complements | |

| |

| |

| |

Point Estimation | |

| |

| |

| |

Introduction | |

| |

| |

| |

Finding Estimators | |

| |

| |

| |

The Method of Moments | |

| |

| |

| |

The Method of Maximum Likelihood | |

| |

| |

| |

Criteria to Compare Estimators | |

| |

| |

| |

Unbiasedness, Variance and Mean Squared Error | |

| |

| |

| |

Best Unbiased and Linear Unbiased Estimators | |

| |

| |

| |

Improved Unbiased Estimator via Sufficiency | |

| |

| |

| |

The Rao-Blackwell Theorem | |

| |

| |

| |

Uniformly Minimum Variance Unbiased Estimator | |

| |

| |

| |

The Cramer-Rao Inequality and UMVUE | |

| |

| |

| |

The Lehmann-Scheffe Theorems and UMVUE | |

| |

| |

| |

A Generalization of the Cramer-Rao Inequality | |

| |

| |

| |

Evaluation of Conditional Expectations | |

| |

| |

| |

Unbiased Estimation Under Incompleteness | |

| |

| |

| |

Does the Rao-Blackwell Theorem Lead to UMVUE? | |

| |

| |

| |

Consistent Estimators | |

| |

| |

| |

Exercises and Complements | |

| |

| |

| |

Tests of Hypotheses | |

| |

| |

| |

Introduction | |

| |

| |

| |

Error Probabilities and the Power Function | |

| |

| |

| |

The Concept of a Best Test | |

| |

| |

| |

Simple Null Versus Simple Alternative Hypotheses | |

| |

| |

| |

Most Powerful Test via the Neyman-Pearson Lemma | |

| |

| |

| |

Applications: No Parameters Are Involved | |

| |

| |

| |

Applications: Observations Are Non-IID | |

| |

| |

| |

One-Sided Composite Alternative Hypothesis | |

| |

| |

| |

UMP Test via the Neyman-Pearson Lemma | |

| |

| |

| |

Monotone Likelihood Ratio Property | |

| |

| |

| |

UMP Test via MLR Property | |

| |

| |

| |

Simple Null Versus Two-Sided Alternative Hypotheses | |

| |

| |

| |

An Example Where UMP Test Does Not Exist | |

| |

| |

| |

An Example Where UMP Test Exists | |

| |

| |

| |

Unbiased and UMP Unbiased Tests | |

| |

| |

| |

Exercises and Complements | |

| |

| |

| |

Confidence Interval Estimation | |

| |

| |

| |

Introduction | |

| |

| |

| |

One-Sample Problems | |

| |

| |

| |

Inversion of a Test Procedure | |

| |

| |

| |

The Pivotal Approach | |

| |

| |

| |

The Interpretation of a Confidence Coefficient | |

| |

| |

| |

Ideas of Accuracy Measures | |

| |

| |

| |

Using Confidence Intervals in the Tests of Hypothesis | |

| |

| |

| |

Two-Sample Problems | |

| |

| |

| |

Comparing the Location Parameters | |

| |

| |

| |

Comparing the Scale Parameters | |

| |

| |

| |

Multiple Comparisons | |

| |

| |

| |

Estimating a Multivariate Normal Mean Vector | |

| |

| |

| |

Comparing the Means | |

| |

| |

| |

Comparing the Variances | |

| |

| |

| |

Exercises and Complements | |

| |

| |

| |

Bayesian Methods | |

| |

| |

| |

Introduction | |

| |

| |

| |

Prior and Posterior Distributions | |

| |

| |

| |

The Conjugate Priors | |

| |

| |

| |

Point Estimation | |

| |

| |

| |

Credible Intervals | |

| |

| |

| |

Highest Posterior Density | |

| |

| |

| |

Contrasting with the Confidence Intervals | |

| |

| |

| |

Tests of Hypotheses | |

| |

| |

| |

Examples with Non-Conjugate Priors | |

| |

| |

| |

Exercises and Complements | |

| |

| |

| |

Likelihood Ratio and Other Tests | |

| |

| |

| |

Introduction | |

| |

| |

| |

One-Sample Problems | |

| |

| |

| |

LR Test for the Mean | |

| |

| |

| |

LR Test for the Variance | |

| |

| |

| |

Two-Sample Problems | |

| |

| |

| |

Comparing the Means | |

| |

| |

| |

Comparing the Variances | |

| |

| |

| |

Bivariate Normal Observations | |

| |

| |

| |

Comparing the Means: The Paired Difference t Method | |

| |

| |

| |

LR Test for the Correlation Coefficient | |

| |

| |

| |

Tests for the Variances | |

| |

| |

| |

Exercises and Complements | |

| |

| |

| |

Large-Sample Inference | |

| |

| |

| |

Introduction | |

| |

| |

| |

The Maximum Likelihood Estimation | |

| |

| |

| |

Confidence Intervals and Tests of Hypothesis | |

| |

| |

| |

The Distribution-Free Population Mean | |

| |

| |

| |

The Binomial Proportion | |

| |

| |

| |

The Poisson Mean | |

| |

| |

| |

The Variance Stabilizing Transformations | |

| |

| |

| |

The Binomial Proportion | |

| |

| |

| |

The Poisson Mean | |

| |

| |

| |

The Correlation Coefficient | |

| |

| |

| |

Exercises and Complements | |

| |

| |

| |

Sample Size Determination: Two-Stage Procedures | |

| |

| |

| |

Introduction | |

| |

| |

| |

The Fixed-Width Confidence Interval | |

| |

| |

| |

Stein's Sampling Methodology | |

| |

| |

| |

Some Interesting Properties | |

| |

| |

| |

The Bounded Risk Point Estimation | |

| |

| |

| |

The Sampling Methodology | |

| |

| |

| |

Some Interesting Properties | |

| |

| |

| |

Exercises and Complements | |

| |

| |

| |

Appendix | |

| |

| |

| |

Abbreviations and Notation | |

| |

| |

| |

A Celebration of Statistics: Selected Biographical Notes | |

| |

| |

| |

Selected Statistical Tables | |

| |

| |

| |

The Standard Normal Distribution Function | |

| |

| |

| |

Percentage Points of the Chi-Square Distribution | |

| |

| |

| |

Percentage Points of the Student's t Distribution | |

| |

| |

| |

Percentage Points of the F Distribution | |

| |

| |

References | |

| |

| |

Index | |