Theory of Statistics - George Mason University

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888 Index gamma integral, 675 gamma-minimax Bayes action, 342 Gâteaux derivative, 601, 752 Gauss-Markov theorem, 423 Gaussian copula, 40 GEE (generalized estimating equation), 250, 482 generalized Bayes action, 341 generalized estimating equation (GEE), 250, 482 generalized inverse, 776, 791 generalized linear model, 488–494 generating function, 42–51 geometric Brownian motion, 765 geometric series, 676 Gibbs lemma, 41 Gibbs method, 663 Gini’s mean difference, 403 Glivenko-Cantelli theorem, 135, 244, 558 Godambe efficiency, 253 goodness of fit test, 532 gradient of a function, 799, 800, 863 Gram-Charlier series, 68, 746 Gram-Schmidt transformation, 679 Gramian matrix, 791 group, 624 transformation, 624, 747 group family, 178–183 Gumbel distribution, 99 Haar invariance, 722 Haar invariant measure, 702, 722 Hadamard derivative, 753 Hájek-Rènyi inequality, 133, 841 Hamburger moment problem, 142 Hammersley-Chapman-Robbins inequality, 846 Hausdorff moment problem, 142 Hausdorff space, 617, 619 hazard function, 572 Heaviside function, 730, 855 heavy-tailed family, 165 Heine-Borel theorem, 639 Heine-Cantor theorem, 716 Hellinger distance, 739 Helly-Bray theorem, 90 Helmert matrix, 429 Helmert transformation, 190 Theory of Statistics c○2000–2013 James E. Gentle Hermite polynomial, 68, 745 Hessian, 446, 652, 801, 863 hierarchical Bayes, 347 hierarchical Bayesian model, 353, 374 higher dimensions, 186, 198 highest posterior density credible set, 369 Hilbert space, 633, 642, 737 histospline, 585 Hodges’ superefficient estimator, 418 Hoeffding inequality, 840 Hölder norm, 637, 737 Hölder’s inequality, 636, 844 homogeneous process, 122 homomorphism, 625 Horowitz’s estimator (proportional hazards), 574 Horvitz-Thompson estimator, 437 HPD (highest posterior density) credible set, 369 hypergeometric series, 676 hyperparameter, 326, 331 hypothesis testing, 286–292, 358–368, 503–556 alternative hypothesis, 287 asymptotic significance, 523 Bayesian testing, 358–368 composite hypothesis, 287 consistency, 311, 523 invariant tests, 521–523 Lagrange multiplier test, 526 likelihood ratio test, 524–526 multiple tests, 532–534 Neyman-Pearson Lemma, 513 nonparametric tests, 531–532 nonrandomized test, 289, 505 null hypothesis, 287 observed significance level, 288 p-value, 288 randomized test, 289, 505, 509 Rao test, 526 score test, 526, 529 sequential tests, 534–535 significance level, 288 simple hypothesis, 287 size of test, 288, 291, 506 SPRT, 535 test statistic, 288 test with random component, 509
unbiased test, 292, 519 uniform consistency, 311, 523 Wald test, 526 i.o. (infinitely often), 71 convergence, 76 IAE (integrated absolute error), 567, 570 ideal bootstrap, 300 idempotent matrix, 787 identifiability, 13, 21 identity matrix, 775 iid (“independent and identically distributed”), 24, 849 IMAE (integrated mean absolute error), 569 image of a function, 619, 695 importance sampling, 678 improper integral, 730 improper prior, 326, 341 IMSE (integrated mean squared error), 569 inclusion-exclusion formula (“disjointification”), 613, 700 incomplete beta function, 858 incomplete gamma function, 858 independence, 5, 23, 74, 110, 120 independence of normal random variables, 187 index of stability, 62 indicator function, 713, 855 induced likelihood, 454 induced measure, 4, 706 induction (method of proof), 669 inductive probability, 139 inequalities, 837 infimum, 638 essential, 738 infinite divisibility, 61 infinitely divisible family, 185 infinitely often, 71 convergence, 76 influence function, 601–603 information, 40, 42, 225–231, 395, 842 information inequality, 230, 395, 417, 843, 846 information theory, 249 inner product, 630, 736, 773 inner product space, 630 Theory of Statistics c○2000–2013 James E. Gentle Index 889 integrable function, 721 integrable random variable, 26 integral, 719–731 double, 727 iterated, 727 integrated expectation integrated absolute bias, 568 integrated absolute error (IAE), 567, 570 integrated bias, 568 integrated mean absolute error (IMAE), 569 integrated mean squared error (IMSE), 569 integrated squared bias, 568 integrated squared error (ISE), 567 integrated variance, 569 integration, 719–731 integration by parts, 728 interior, 616, 639 interior point, 639 interquantile range, 53 interquartile range, 53 intersection of sets, 610 invariance, 216, 262, 275–285 Haar, 722 invariant family, 182 invariant function, 277, 747 invariant tests, 521–523 inverse CDF, 15 inverse CDF method, 657 inverse cumulative distribution function, 15 inverse function, 619 inverse image, 620, 696 inverse of a matrix, 776, 789 inverse of a partitioned matrix, 791 inverse probability, 207, 313, 376 inversion theorem, 48 IRLS (iteratively reweighted least squares), 492 irreducible Markov chain, 129 irreducible matrix, 810 ISE (integrated squared error), 567 isomorphism, 625 iterated integral, 727 iterated logarithm, law of, 104 iteratively reweighted least squares (IRLS), 492
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James E. Gentle Theory of Statistic
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Preface: Mathematical Statistics Af
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Preface vii The objective in the di
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x = ⎛ ⎜ ⎝ x1 . xd ⎞ ⎟ ⎠
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• State and prove Fatou’s lemma
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Contents Preface . . . . . . . . .
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Contents xv 2.10 Multivariate Distr
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Contents xvii 5.2.1 Expectation Fun
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Contents xix 8.5.1 Nonparametric Pr
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1 Probability Theory Probability th
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1.1 Some Important Probability Fact
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Definition 1.8 (exchangeability) Le
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Theorem 1.5 (properties of a CDF) I
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.5 PDF p X(x;θ) 0 x θ=1 θ=5 1.1
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Joint and Marginal Distributions 1.
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Moment-Generating Functions 1.1 Som
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A Taylor series expansion of this g
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1.2 Series Expansions 65 X is the u
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κ1 = E(Z) κ2 = E(Z 2 ) − (E(Z))
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1.3 Sequences of Events and of Rand
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We write 1.3 Sequences of Events an
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1.3.6 Convergence of Functions 1.3
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we have for fixed k, 1.3 Sequences
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Multivariate Asymptotic Expectation
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1.4 Limit Theorems 103 The bn can b
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1.4 Limit Theorems 105 it applies t
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lim n→∞ max σ j≤kn 2 nj σ2
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1.5 Conditional Probability 109 The
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Conditional Expectation over a Sub-
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• monotone convergence: for 0 ≤
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Fn(t) = ⌊nt⌋ + 1 n + 1 ; 1.5 Co
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1.5 Conditional Probability 117 If
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1.5 Conditional Probability 119 1.5
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Conditional Entropy 1.6 Stochastic
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1.6 Stochastic Processes 123 Stoppi
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1.6 Stochastic Processes 125 (Exerc
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1.6 Stochastic Processes 127 variab
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1.6 Stochastic Processes 129 in a d
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1.6 Stochastic Processes 131 We als
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1.6 Stochastic Processes 133 X0 has
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Convergence of Empirical Processes
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and Fn(x, ω) ≥ Fn(xm,k−1, ω)
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Notes and Further Reading 139 and f
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Notes and Further Reading 141 we ca
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Notes and Further Reading 143 After
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Exercises 145 of betting system. Do
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Exercises 147 1.22. Show that if th
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1.36. Consider the the distribution
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Exercises 151 1.59. A sufficient co
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Exercises 153 1.85. Show that Doob
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2 Distribution Theory and Statistic
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Example 2.1 complete and incomplete
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2.2 Shapes of the Probability Densi
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2.2 Shapes of the Probability Densi
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2.3.2 The Le Cam Regularity Conditi
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2.4 The Exponential Class of Famili
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or or 2.5 Parametric-Support Famili
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2.6 Transformation Group Families 1
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2.6 Transformation Group Families 1
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2.7 Truncated and Censored Distribu
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2.9 Infinitely Divisible and Stable
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Higher Dimensions 2.11 The Family o
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2.11 The Family of Normal Distribut
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Exercises 195 are considered at som
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Exercises 197 This law has been use
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Exercises 199 a) Show that for d
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3 Basic Statistical Theory The fiel
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How Does PH Differ from P? 3 Basic
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3 Basic Statistical Theory 205 abov
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3.1 Inferential Information in Stat
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The Basic Paradigm of Point Estimat
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Completeness 3.1 Inferential Inform
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and so by completeness, 3.1 Inferen
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and so I(µ, σ) = Eθ = E(µ,σ) =
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Robustness 3.1 Inferential Informat
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3.2 Statistical Inference: Approach
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3.3 The Decision Theory Approach to
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Risk 0.005 0.010 0.015 0.020 0.025
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3.4 Invariant and Equivariant Stati
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Equivariant Point Estimation 3.4 In
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3.5 Probability Statements in Stati
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Test Rules 3.5 Probability Statemen
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if 3.6 Variance Estimation 297 Give
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3.6 Variance Estimation 299 J(T) =
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3.7 Applications 3.7.1 Inference in
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3.8 Asymptotic Inference 303 The ca
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3.8 Asymptotic Inference 305 The mo
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3.8 Asymptotic Inference 307 wherea
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3.8 Asymptotic Inference 309 tendin
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Definition 3.20 (asymptotic signifi
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Proof. *************** Notes and Fu
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Notes and Further Reading 315 Least
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Approximations and Asymptotic Infer
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Exercises 319 b) the expectation of
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4 Bayesian Inference We have used a
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4.1 The Bayesian Paradigm 323 For m
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4.1 The Bayesian Paradigm 325 A qua
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4.2 Bayesian Analysis 327 even if t
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4.2 Bayesian Analysis 329 Hence P(B
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4.2 Bayesian Analysis 331 B1. ∀θ
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4.2 Bayesian Analysis 333 This mean
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Prior Prior 0.0 0.5 1.0 1.5 2.0 0 2
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4.2 Bayesian Analysis 337 We go thr
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4.2 Bayesian Analysis 339 We constr
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4.2 Bayesian Analysis 341 as the mo
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Assessing the Problem Formulation 4
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Posterior Posterior 0 5 10 15 0 1 2
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4.2 Bayesian Analysis 347 distribut
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4.3 Bayes Rules 349 • the nature
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Then we have E(g(θ)T(X)) = E(T(X)E
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4.3 Bayes Rules 353 equivariance Fo
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4.3 Bayes Rules 355 Example 4.9 (Co
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p0 = Pr(Θ ∈ Θ0) and p1 = Pr(Θ
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The Weighted 0-1 or α0-α1 Loss Fu
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The term ˆp0 ˆp1 = p0 p1 BF(x) =
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4.5 Bayesian Testing 365 • Bayes
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where fΘ|x(θ|x) = p1 if θ = θ0
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4.6 Bayesian Confidence Sets 369 as
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Posterior 0 1 2 3 95% Credible Regi
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Markov Chain Monte Carlo 4.7 Comput
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4.7 Computational Methods 375 β/(t
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Notes and Further Reading 377 An al
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4.4. Given the conditional PDF a) U
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Exercises 381 4.14. Consider again
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Exercises 383 4.26. As in Exercise
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5 Unbiased Point Estimation In a de
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Estimability 5 Unbiased Point Estim
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s1π + s0(1 − π) = π, 5.1 UMVUE
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5.1 UMVUE 391 Let T be UMVUE for g(
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5.1 UMVUE 393 Example 5.7 UMVUE of
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5.1 UMVUE 395 The risk of T ∗ is
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5.1 UMVUE 397 estimator of θ that
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(Compare this with Tfdx = g(θ).)
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¯h(X1, . . ., Xm) = 1 m! 5.2 U-Sta
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where Pn is the ECDF. U(X1, . . .,
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and T 2 r,S = C = Tr,S = C C TnT
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Generalized U-Statistics 5.2 U-Stat
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5.2 U-Statistics 409 Theorem 5.4 (H
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5.3 Asymptotically Unbiased Estimat
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5.3.2 Ratio Estimators 5.3 Asymptot
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5.4 Asymptotic Efficiency 415 of a
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⎧ ⎨ Tn = ⎩ n2 with probabilit
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5.5 Applications 5.5 Applications 4
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5.5 Applications 421 one aspect of
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Proof. Because l ∈ span(X T ) = s
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Optimal Properties of the Moore-Pen
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5.5 Applications 427 implies implie
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The “Sum of Squares” Quadratic
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5.5 Applications 431 The question o
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We now write the original model as
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5.5 Applications 435 it from other
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(it’s Bernoulli), and for i = j,
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Fisher Efficient Estimators and Exp
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6 Statistical Inference Based on Li
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6.1 The Likelihood Function 443 is
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6.2 Maximum Likelihood Parametric E
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6.2.2 Finite Sample Properties of M
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where Now, consider This has two pa
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6.3 Asymptotic Properties of MLEs,
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6.4 Application: MLEs in Generalize
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Residuals 6.4 Application: MLEs in
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6.5 Variations on the Likelihood 49
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6.5 Variations on the Likelihood 49
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Maximum Likelihood in Linear Models
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Exercises 501 b) Assume that σ 2 1
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7 Statistical Hypotheses and Confid
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Tests of Hypotheses 7.1 Statistical
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7.1 Statistical Hypotheses 507 Know
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Power of a Statistical Test 7.1 Sta
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An Optimal Test in a Simple Situati
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7.2 Optimal Tests 513 All of these
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Use of Sufficient Statistics 7.2 Op
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7.2 Optimal Tests 517 Syppose we as
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Nonexistence of UMP Tests 7.2 Optim
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H0 : θ = θ0 versus H1 : θ = θ0.
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7.2.6 Equivariance, Unbiasedness, a
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7.3 Likelihood Ratio Tests, Wald Te
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7.3 Likelihood Ratio Tests, Wald Te
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Minimize 7.3 Likelihood Ratio Tests
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7.4 Nonparametric Tests 531 I(0, ˆ
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Yij = µ + αi + ɛij, i = 1, . . .
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7.7 The Likelihood Principle and Te
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7.8 Confidence Sets 537 Monte Carlo
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7.8 Confidence Sets 539 The concept
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7.8 Confidence Sets 541 For any giv
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7.9 Optimal Confidence Sets 543 Thi
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7.9 Optimal Confidence Sets 545 Con
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7.10 Asymptotic Confidence sets 547
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7.11 Bootstrap Confidence Sets 549
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Bias in Intervals Due to Bias in th
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7.12 Simultaneous Confidence Sets 5
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Sequential Tests Exercises 555 Wald
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8 Nonparametric and Robust Inferenc
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8.2 Inference Based on Order Statis
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f ^ (x) Nonparametric Regression 0.
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8.3 Nonparametric Estimation of Fun
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ISE θ = 8.3 Nonparametric Estim
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8.4 Semiparametric Methods and Part
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8.5 Nonparametric Estimation of PDF
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Some Properties of the Histogram Es
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AISB 1 pH = 12 8.5 Nonparametric
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and the integrated variance is 8.5
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and 8.5 Nonparametric Estimation o
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h ∗ = 8.5 Nonparametric Estimatio
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Computation of Kernel Density Estim
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8.6 Perturbations of Probability Di
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CDF 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
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8.6 Perturbations of Probability Di
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8.7 Robust Inference 8.7 Robust Inf
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p(y) (1-ε)p(y) The Influence Funct
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8.7 Robust Inference 603 Notice tha
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Nonparametric Statistics Exercises
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0 Statistical Mathematics Statistic
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0 Statistical Mathematics 609 x may
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0.0 Some Basic Mathematical Concept
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Because each is a subset of the oth
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0.0.2 Sets and Spaces 0.0 Some Basi
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0.0.2.4 Point Sequences in a Topolo
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The sequence of sets {An} is said t
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The proof of this is a classic: fir
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• ∀x ∈ IR ∪ {∞}, x × ±
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0.0.5.2 Sets of Reals; Open, Closed
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We have 0.0 Some Basic Mathematical
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Definition 0.0.14 (superharmonic fu
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Ordering the Complex Numbers 0.0 So
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0.0.9.5 Working with Real-Valued Fu
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∞ 0.0 Some Basic Mathematical Con
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a) Evaluate the integral (0.0.84):
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0.1 Measure, Integration, and Funct
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3. if A1, A2, . . . ∈ F are disjo
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(why?), and so We also have λ 0.1
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Proof. Exercise. 0.1 Measure, Integ
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Hence, from inequality (0.1.47),
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0.1.9.3 Norms of Functions 0.1 Meas
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0.2 Stochastic Processes and the St
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Variation of Functionals 0.2 Stocha
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0.2.1.3 Ito Processes 0.2 Stochasti
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0.2.2.2 Ito’s Lemma 0.2 Stochasti
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0.3 Some Basics of Linear Algebra 0
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0.3.2.2 The Trace and Some of Its P
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The Fourier Expansion 0.3 Some Basi
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0.3.2.8 Spectral Decomposition 0.3
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0.3.2.11 Inverses of Matrices 0.3 S
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we have 0.3 Some Basics of Linear A
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f(x) ≈ f(x∗) + (x − x∗) T
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and so E T π (I + A) n−1 Eπ =
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0.4.1.2 Methods 0.4 Optimization 81
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0.4.1.7 The Steps in Iterative Algo
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0.4.1.11 Modifications of Newton’
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We generally require g x (k) ; x (
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0.4 Optimization 823 3. Generate st
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Theory of Statistics c○2000-2013
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A Important Probability Distributio
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Appendix A. Important Probability D
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Page 855 and 856: B Useful Inequalities in Probabilit
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Page 859 and 860: B.4 E(f(X)) and f(E(X)) 841 Theorem
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Page 867 and 868: C Notation and Definitions All nota
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Page 871 and 872: C.2 General Mathematical Functions
Page 873 and 874: Functions of Convenience C.2 Genera
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Page 881 and 882: a(ij) C.4 Linear Spaces and Matrice
Page 883 and 884: References The number(s) following
Page 885 and 886: References 867 Lyle D. Broemeling.
Page 887 and 888: References 869 William Feller. An I
Page 889 and 890: References 871 H. O. Hartley. In Dr
Page 891 and 892: References 873 Feldman, and Murad S
Page 893 and 894: References 875 Roger B. Nelsen. An
Page 895 and 896: References 877 Glenn Shafer. A Math
Page 897 and 898: References 879 Abraham Wald. Sequen
Page 899 and 900: Index a.e. (almost everywhere), 704
Page 901 and 902: cartesian product measurable space,
Page 903 and 904: derivative of a functional, 752-753
Page 905: Euclidean norm, 774 Euler’s formu
Page 909 and 910: sequence of sets, 620-623 limit poi
Page 911 and 912: natural parameter space, 173 neglig
Page 913 and 914: proportional hazards, 573 pseudoinv
Page 915 and 916: square root matrix, 783 squared-err
Page 917: weak convergence in mean square, 56
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Theory of Statistics - George Mason University

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