Stat 5101 Lecture Notes

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1 Stat 5101 Lecture Notes Charles J. Geyer Copyright 1998, 1999, 2000, 2001 by Charles J. Geyer May 7, 2001

2 ii Stat 5101 (Geyer) Course Notes

3 Contents 1 Random Variables and Change of Variables Random Variables Variables Functions Random Variables: Informal Intuition Random Variables: Formal Definition Functions of Random Variables Change of Variables General Definition Discrete Random Variables Continuous Random Variables Random Vectors Discrete Random Vectors Continuous Random Vectors The Support of a Random Variable Joint and Marginal Distributions Multivariable Change of Variables The General and Discrete Cases Continuous Random Vectors Expectation Introduction The Law of Large Numbers Basic Properties Axioms for Expectation (Part I) Derived Basic Properties Important Non-Properties Moments First Moments and Means Second Moments and Variances Standard Deviations and Standardization Mixed Moments and Covariances Exchangeable Random Variables Correlation iii

4 iv Stat 5101 (Geyer) Course Notes 2.5 Probability Theory as Linear Algebra The Vector Space L Two Notions of Linear Functions Expectation on Finite Sample Spaces Axioms for Expectation (Part II) General Discrete Probability Models Continuous Probability Models The Trick of Recognizing a Probability Density Probability Zero How to Tell When Expectations Exist L p Spaces Probability is a Special Case of Expectation Independence Two Definitions The Factorization Criterion Independence and Correlation Conditional Probability and Expectation Parametric Families of Distributions Conditional Probability Distributions Axioms for Conditional Expectation Functions of Conditioning Variables The Regression Function Iterated Expectations Joint, Conditional, and Marginal Joint Equals Conditional Times Marginal Normalization Renormalization Renormalization, Part II Bayes Rule Conditional Expectation and Prediction Parametric Families of Distributions Location-Scale Families The Gamma Distribution The Beta Distribution The Poisson Process Spatial Point Processes The Poisson Process One-Dimensional Poisson Processes Multivariate Theory Random Vectors Vectors, Scalars, and Matrices Random Vectors Random Matrices...128

5 CONTENTS v Variance Matrices What is the Variance of a Random Matrix? Covariance Matrices Linear Transformations Characterization of Variance Matrices Degenerate Random Vectors Correlation Matrices The Multivariate Normal Distribution The Density Marginals Partitioned Matrices Conditionals and Independence Bernoulli Random Vectors Categorical Random Variables Moments The Multinomial Distribution Categorical Random Variables Moments Degeneracy Density Marginals and Sort Of Marginals Conditionals Convergence Concepts Univariate Theory Convergence in Distribution The Central Limit Theorem Convergence in Probability The Law of Large Numbers The Continuous Mapping Theorem Slutsky s Theorem Comparison of the LLN and the CLT Applying the CLT to Addition Rules The Cauchy Distribution Sampling Theory Empirical Distributions The Mean of the Empirical Distribution The Variance of the Empirical Distribution Characterization of the Mean Review of Quantiles Quantiles of the Empirical Distribution The Empirical Median Characterization of the Median Samples and Populations Finite Population Sampling

6 vi Stat 5101 (Geyer) Course Notes Repeated Experiments Sampling Distributions of Sample Moments Sample Moments Sampling Distributions Moments Asymptotic Distributions The t Distribution The F Distribution Sampling Distributions Related to the Normal Sampling Distributions of Sample Quantiles Convergence Concepts Continued Multivariate Convergence Concepts Convergence in Probability to a Constant The Law of Large Numbers Convergence in Distribution The Central Limit Theorem Slutsky and Related Theorems The Delta Method The Univariate Delta Method The Multivariate Delta Method Asymptotics for Sample Moments Asymptotics of Independent Sequences Asymptotics of Sample Quantiles Frequentist Statistical Inference Introduction Inference The Sample and the Population Frequentist versus Bayesian Inference Models, Parameters, and Statistics Parametric Models Nonparametric Models Semiparametric Models Interest and Nuisance Parameters Statistics Point Estimation Bias Mean Squared Error Consistency Asymptotic Normality Method of Moments Estimators Relative Efficiency Asymptotic Relative Efficiency (ARE) Interval Estimation Exact Confidence Intervals for Means...252

7 CONTENTS vii Pivotal Quantities Approximate Confidence Intervals for Means Paired Comparisons Independent Samples Confidence Intervals for Variances The Role of Asymptotics Robustness Tests of Significance Interest and Nuisance Parameters Revisited Statistical Hypotheses Tests of Equality-Constrained Null Hypotheses P -values One-Tailed Tests The Duality of Tests and Confidence Intervals Sample Size Calculations Multiple Tests and Confidence Intervals Likelihood Inference Likelihood Maximum Likelihood Sampling Theory Derivatives of the Log Likelihood The Sampling Distribution of the MLE Asymptotic Relative Efficiency Estimating the Variance Tests and Confidence Intervals Multiparameter Models Maximum Likelihood Sampling Theory Likelihood Ratio Tests Change of Parameters Invariance of Likelihood Invariance of the MLE Invariance of Likelihood Ratio Tests Covariance of Fisher Information Bayesian Inference Parametric Models and Conditional Probability Prior and Posterior Distributions Prior Distributions Posterior Distributions The Subjective Bayes Philosophy More on Prior and Posterior Distributions Improper Priors Conjugate Priors The Two-Parameter Normal Distribution...362

8 viii Stat 5101 (Geyer) Course Notes 11.5 Bayesian Point Estimates Highest Posterior Density Regions Bayes Tests Bayesian Asymptotics Regression The Population Regression Function Regression and Conditional Expectation Best Prediction Best Linear Prediction The Sample Regression Function Sampling Theory The Regression Model The Gauss-Markov Theorem The Sampling Distribution of the Estimates Tests and Confidence Intervals for Regression Coefficients The Hat Matrix Polynomial Regression The F -Test for Model Comparison Intervals for the Regression Function The Abstract View of Regression Categorical Predictors (ANOVA) Categorical Predictors and Dummy Variables ANOVA Residual Analysis Leave One Out Quantile-Quantile Plots Model Selection Overfitting Mean Square Error The Bias-Variance Trade-Off Model Selection Criteria All Subsets Regression Bernoulli Regression A Dumb Idea (Identity Link) Logistic Regression (Logit Link) Probit Regression (Probit Link) Generalized Linear Models Parameter Estimation Fisher Information, Tests and Confidence Intervals Poisson Regression Overdispersion A Greek Letters 469

9 CONTENTS ix B Summary of Brand-Name Distributions 471 B.1 Discrete Distributions B.1.1 The Discrete Uniform Distribution B.1.2 The Binomial Distribution B.1.3 The Geometric Distribution, Type II B.1.4 The Poisson Distribution B.1.5 The Bernoulli Distribution B.1.6 The Negative Binomial Distribution, Type I B.1.7 The Negative Binomial Distribution, Type II B.1.8 The Geometric Distribution, Type I B.2 Continuous Distributions B.2.1 The Uniform Distribution B.2.2 The Exponential Distribution B.2.3 The Gamma Distribution B.2.4 The Beta Distribution B.2.5 The Normal Distribution B.2.6 The Chi-Square Distribution B.2.7 The Cauchy Distribution B.2.8 Student s t Distribution B.2.9 Snedecor s F Distribution B.3 Special Functions B.3.1 The Gamma Function B.3.2 The Beta Function B.4 Discrete Multivariate Distributions B.4.1 The Multinomial Distribution B.5 Continuous Multivariate Distributions B.5.1 The Uniform Distribution B.5.2 The Standard Normal Distribution B.5.3 The Multivariate Normal Distribution B.5.4 The Bivariate Normal Distribution C Addition Rules for Distributions 485 D Relations Among Brand Name Distributions 487 D.1 Special Cases D.2 Relations Involving Bernoulli Sequences D.3 Relations Involving Poisson Processes D.4 Normal, Chi-Square, t, and F D.4.1 Definition of Chi-Square D.4.2 Definition of t D.4.3 Definition of F D.4.4 t as a Special Case of F...489

10 x Stat 5101 (Geyer) Course Notes E Eigenvalues and Eigenvectors 491 E.1 Orthogonal and Orthonormal Vectors E.2 Eigenvalues and Eigenvectors E.3 Positive Definite Matrices F Normal Approximations for Distributions 499 F.1 Binomial Distribution F.2 Negative Binomial Distribution F.3 Poisson Distribution F.4 Gamma Distribution F.5 Chi-Square Distribution G Maximization of Functions 501 G.1 Functions of One Variable G.2 Concave Functions of One Variable G.3 Functions of Several Variables G.4 Concave Functions of Several Variables H Projections and Chi-Squares 509 H.1 Orthogonal Projections H.2 Chi-Squares

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