Transcription
Larry WassermanAll of StatisticsA Concise Course in Statistical InferenceWith 95 FiguresSpringer
ContentsIProbability1 Probability1.1 Introduction1.2 Sample Spaces and Events1.3 Probability1.4 Probability on Finite Sample Spaces1.5 Independent Events1.6 Conditional Probability1.7 Bayes' Theorem1.8 Bibliographic Remarks1.9 Appendix1.10 Exercises33357810121313132 Random Variables2.1 Introduction2.2 Distribution Functions and Probability Functions2.3 Some Important Discrete Random Variables2.4 Some Important Continuous Random Variables2.5 Bivariate Distributions2.6 Marginal Distributions2.7 Independent Random Variables2.8 Conditional Distributions191920252731333436
Contents2.92.102.112.122.132.14Multivariate Distributions and IID SamplesTwo Important Multivariate DistributionsTransformations of Random VariablesTransformations of Several Random 73.1 Expectation of a Random Variable473.2 Properties of Expectations503.3 Variance and Covariance503.4 Expectation and Variance of Important Random Variables . . . 523.5 Conditional Expectation543.6 Moment Generating Functions563.7 Appendix583.8 Exercises58II6Inequalities4.1 Probability Inequalities4.2 Inequalities For Expectations4.3 Bibliographic Remarks4.4 Appendix4.5 Exercises636366666768Convergence of Random Variables5.1 Introduction5.2 Types of Convergence5.3 The Law of Large Numbers5.4 The Central Limit Theorem5.5 The Delta Method5.6 Bibliographic Remarks5.7 Appendix5.7.1 Almost Sure and L\ Convergence5.7.2 Proof of the Central Limit Theorem5.8 Exercises7171727677798081818182Statistical InferenceModels, Statistical Inference and Learning6.1 Introduction6.2 Parametric and Nonparametric Models6.3 Fundamental Concepts in Inference6.3.1 Point Estimation6.3.2 Confidence Sets878787909092
Contents6.3.3 Hypothesis Testing6.4 Bibliographic Remarks6.5 Appendix6.6 Exercisesxv949595957 Estimating the CDF and Statistical Functionals7.1 The Empirical Distribution Function7.2 Statistical Functionals7.3 Bibliographic Remarks7.4 Exercises9797991041048 The8.18.28.38.48.5BootstrapSimulationBootstrap Variance EstimationBootstrap Confidence IntervalsBibliographic RemarksAppendix8.5.1 The Jackknife8.5.2 Justification For The Percentile Interval8.6 Exercises1071081081101151151151161169 Parametric Inference9.1 Parameter of Interest9.2 The Method of Moments9.3 Maximum Likelihood9.4 Properties of Maximum Likelihood Estimators9.5 Consistency of Maximum Likelihood Estimators9.6 Equivariance of the MLE9.7 Asymptotic Normality9.8 Optimality9.9 The Delta Method9.10 Multiparameter Models9.11 The Parametric Bootstrap9.12 Checking Assumptions9.13 Appendix9.13.1 Proofs9.13.2 Sufficiency9.13.3 Exponential Families9.13.4 Computing Maximum Likelihood Estimates9.14 513513714014214610 Hypothesis Testing and p-values10.1 The Wald Test10.2 p-values10.3 The x 2 Distribution149152156159
xviContents10.4 Pearson's x 2 Test For Multinomial Data10.5 The Permutation Test10.6 The Likelihood Ratio Test10.7 Multiple Testing10.8 Goodness-of-fit Tests10.9 Bibliographic RemarkslO.lOAppendix10.10.1 The Neyman-Pearson Lemma10.10.2The 1 Bayesian Inference17511.1 The Bayesian Philosophy17511.2 The Bayesian Method17611.3 Functions of Parameters18011.4 Simulation18011.5 Large Sample Properties of Bayes'Procedures18111.6 Flat Priors, Improper Priors, and "Noninformative" Priors . . . 18111.7 Multiparameter Problems18311.8 Bayesian Testing18411.9 Strengths and Weaknesses of Bayesian Inference185ll.lOBibliographic Remarks189ll.llAppendix19011.12Exercises19012 Statistical Decision Theory12.1 Preliminaries12.2 Comparing Risk Functions12.3 Bayes Estimators12.4 Minimax Rules12.5 Maximum Likelihood, Minimax, and Bayes12.6 Admissibility12.7 Stein's Paradox12.8 Bibliographic Remarks12.9 cal Models and Methods13 Linear and Logistic Regression13.1 Simple Linear Regression13.2 Least Squares and Maximum Likelihood13.3 Properties of the Least Squares Estimators13.4 Prediction13.5 Multiple Regression209209212214215216
Contentsxvii13.6 Model Selection13.7 Logistic Regression13.8 Bibliographic Remarks13.9 Appendix13.10Exercises21822322522522614 Multivariate Models14.1 Random Vectors14.2 Estimating the Correlation14.3 Multivariate Normal14.4 Multinomial14.5 Bibliographic Remarks14.6 Appendix14.7 Exercises23123223323423523723723815 Inference About Independence15.1 Two Binary Variables15.2 Two Discrete Variables15.3 Two Continuous Variables15.4 One Continuous Variable and One Discrete15.5 Appendix15.6 Exercises23923924324424424524816 Causal Inference16.1 The Counterfactual Model16.2 Beyond Binary Treatments16.3 Observational Studies and Confounding16.4 Simpson's Paradox16.5 Bibliographic Remarks16.6 Exercises25125125525725926126117 Directed Graphs and Conditional Independence17.1 Introduction17.2 Conditional Independence17.3 DAGs17.4 Probability and DAGs17.5 More Independence Relations17.6 Estimation for DAGs17.7 Bibliographic Remarks17.8 Appendix17.9 Exercises26326326426426626727227227227618 Undirected Graphs18.1 Undirected Graphs18.2 Probability and Graphs281281282
xviii18.318.418.518.6ContentsCliques and PotentialsFitting Graphs to DataBibliographic RemarksExercises28528628628619 Log-Linear Models19.1 The Log-Linear Model19.2 Graphical Log-Linear Models19.3 Hierarchical Log-Linear Models19.4 Model Generators19.5 Fitting Log-Linear Models to Data19.6 Bibliographic Remarks19.7 Exercises29129129429629729830030120 Nonparametric Curve Estimation20.1 The Bias-Variance Tradeoff20.2 Histograms20.3 Kernel Density Estimation20.4 Nonparametric Regression20.5 Appendix20.6 Bibliographic Remarks20.7 Exercises30330430531231932432532521 Smoothing Using Orthogonal Functions21.1 Orthogonal Functions and L2 Spaces21.2 Density Estimation21.3 Regression21.4 Wavelets21.5 Appendix21.6 Bibliographic Remarks21.7 Exercises32732733133534034534634622 Classification22.1 Introduction22.2 Error Rates and the Bayes Classifier22.3 Gaussian and Linear Classifiers22.4 Linear Regression and Logistic Regression22.5 Relationship Between Logistic Regression and LDA22.6 Density Estimation and Naive Bayes22.7 Trees22.8 Assessing Error Rates and Choosing a Good Classifier22.9 Support Vector Machines22.10 Kernelization22.11 Other Classifiers22.12 Bibliographic Remarks349349350353356358359360362368371375377
Contents22.13 Exercises37723 Probability Redux: Stochastic Processes23.1 Introduction23.2 Markov Chains23.3 Poisson Processes23.4 Bibliographic Remarks23.5 Exercises38138138339439739824 Simulation Methods24.1 Bayesian Inference Revisited24.2 Basic Monte Carlo Integration24.3 Importance Sampling24.4 MCMC Part I: The Metropolis-Hastings Algorithm24.5 MCMC Part II: Different Flavors24.6 Bibliographic Remarks24.7 Exercises403403404408411415420420IndexLxix434
All of Statistics A Concise Course in Statistical Inference With 95 Figures Springer. Contents I Probability 1 Probability 3 1.1 Introduction 3 1.2 Sample Spaces and Events 3 . 6 Models, Statistical Inference and Learning 87 6.1 Introduction 87 6.2 Parametric and Nonparametric Models 87 6.3 Fundamental Concepts in Inference 90
