Mathematical Statistics

Richard J. Rossi, PhD, is Director of the Statistics Program and Co-Director of the Data Science Program at Montana Tech of The University of Montana, in Butte, MT. He acted as President of the Montana Chapter of the American Statistical Association in 2001 and as Associate Editor for Biometrics from 1997-2000. Dr. Rossi is a member of the American Mathematical Society, the Institute of Mathematical Statistics, and the American Statistical Association.

• Preface xiiiAcknowledgments xvii1 Probability 11.1 Sample Spaces, Events, and ;;-Algebras 1Problems 71.2 Probability Axioms and Rules 9Problems 141.3 Probability with Equally Likely Outcomes 16Problems 181.4 Conditional Probability 19Problems 251.5 Independence 28Problems 311.6 Counting Methods 33Problems 381.7 Case Study –The Birthday Problem 41Problems 442 Random Variables and Random Vectors 452.1 Random Variables 452.1.1 Properties of Random Variables 46Problems 502.2 Random Vectors 532.2.1 Properties of Random Vectors 53Problems 602.3 Independent Random Variables 63Problems 662.4 Transformations of Random Variables 682.4.1 Transformations of Discrete Random Variables 682.4.2 Transformations of Continuous Random Variables 692.4.3 Transformations of Continuous Bivariate Random Vectors 73Problems 752.5 Expected Values for Random Variables 772.5.1 Expected Values and Moments of Random Variables 772.5.2 The Variance of a Random Variable 812.5.3 Moment Generating Functions 86Problems 892.6 Expected Values for Random Vectors 942.6.1 Properties of Expectation with Random Vectors 962.6.2 Covariance and Correlation 992.6.3 Conditional Expectation and Variance 106Problems 1102.7 Sums of Random Variables 114Problems 1202.8 Case Study – HowMany Times Was the Coin Tossed? 1232.8.1 The Probability Model 124Problems 1263 Probability Models 1293.1 Discrete Probability Models 1293.1.1 The Binomial Model 1293.1.1.1 Binomial Setting 1303.1.2 The HypergeometricModel 1323.1.2.1 Hypergeometric Setting 1323.1.3 The Poisson Model 1343.1.4 The Negative BinomialModel 1353.1.4.1 Negative Binomial Setting 1353.1.5 The MultinomialModel 1383.1.5.1 Multinomial Setting 139Problems 1403.2 Continuous Probability Models 1473.2.1 The Uniform Model 1473.2.2 The Gamma Model 1493.2.3 The Normal Model 1523.2.4 The Log-normal Model 1553.2.5 The Beta Model 156Problems 1583.3 Important Distributional Relationships 1633.3.1 Sums of Random Variables 1633.3.2 The T and F Distributions 166Problems 1703.4 Case Study –The Central LimitTheorem 1723.4.1 Convergence in Distribution 1723.4.2 The Central LimitTheorem 173Problems 1764 Parametric Point Estimation 1774.1 Statistics 1774.1.1 Sampling Distributions 1784.1.2 Unbiased Statistics and Estimators 1794.1.3 Standard Error and Mean Squared Error 1814.1.4 The Delta Method 186Problems 1864.2 Sufficient Statistics 1904.2.1 Exponential Family Distributions 195Problems 2004.3 Minimum Variance Unbiased Estimators 2034.3.1 Cramér–Rao Lower Bound 205Problems 2124.4 Case Study –The Order Statistics 214Problems 2195 Likelihood-based Estimation 2235.1 Maximum Likelihood Estimation 2265.1.1 Properties of MLEs 2265.1.2 One-parameter Probability Models 2285.1.3 Multiparameter Probability Models 235Problems 2405.2 Bayesian Estimation 2475.2.1 The Bayesian Setting 2475.2.2 Bayesian Estimators 250Problems 2555.3 Interval Estimation 2585.3.1 Exact Confidence Intervals 2595.3.2 Large Sample Confidence Intervals 2645.3.3 Bayesian Credible Intervals 267Problems 2695.4 Case Study – Modeling Obsidian Rind Thicknesses 2735.4.1 Finite Mixture Model 274Problems 2786 Hypothesis Testing 2816.1 Components of a Hypothesis Test 282Problems 2866.2 Most Powerful Tests 288Problems 2936.3 Uniformly Most Powerful Tests 2966.3.1 Uniformly Most Powerful Unbiased Tests 299Problems 3016.4 Generalized Likelihood Ratio Tests 305Problems 3116.5 Large Sample Tests 3146.5.1 Large Sample Tests Based on the MLE 3146.5.2 Score Tests 316Problems 3206.6 Case Study – Modeling Survival of the Titanic Passengers 3236.6.1 Exploring the Data 3246.6.2 Modeling the Probability of Survival 3256.6.3 Analysis of the Fitted Survival Model 327Problems 3287 Generalized Linear Models 3317.1 Generalized LinearModels 332Problems 3347.2 Fitting a Generalized LinearModel 3367.2.1 Estimating ⃗ ;; 3367.2.2 Model Deviance 338Problems 3407.3 Hypothesis Testing in a Generalized Linear Model 3417.3.1 Asymptotic Properties 3417.3.2 Wald Tests and Confidence Intervals 3427.3.3 Likelihood Ratio Tests 343Problems 3467.4 Generalized LinearModels for a Normal Response Variable 3487.4.1 Estimation 3497.4.2 Properties of the MLEs 3537.4.3 Deviance 3577.4.4 Hypothesis Testing 359Problems 3627.5 Generalized LinearModels for a Binomial Response Variable 3657.5.1 Estimation 3667.5.2 Properties of the MLEs 3687.5.3 Deviance 3707.5.4 Hypothesis Testing 371Problems 3737.6 Case Study – IDNAP Experimentwith Poisson Count Data 3757.6.1 The Model 3767.6.2 StatisticalMethods 3767.6.3 Results of the First Experiment 379Problems 381References 383A Probability Models 385B DataSets 387Problem Solutions 389Index 413