Theory of Statistics - George Mason University

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892 Index mean integrated absolute error (MIAE), 569, 570 mean integrated squared error (MISE), 569 mean square consistent, 570 mean squared error (MSE), 214, 565, 569 mean squared error, of series expansion, 743 mean squared prediction error, 232 mean sup absolute error (MSAE), 569 mean-value parameter, in exponential class, 172 mean-value theorem, 674 measurable function, 697 measurable set, 703 measurable space, 694 measure, 698 Borel, 711 complete, 701 counting, 702 Dirac, 702 dominating, 22, 705 Haar invariant, 702, 722 induced, 706 Lebesgue, 711 probability, 3, 701 pushforward, 706 Radon, 702 singular, 705, 860 measure space, 703 complete, 703 measure theory, 686–754 median-unbiasedness, 214, 255 method of moments, 243, 268, 412 metric, 617, 635, 773 in IR d , 635 in a function space, 738 metric space, 618 Metropolis algorithm, 660 Metropolis-Hastings algorithm, 662 MGF (moment-generating function), 43 MIAE (mean integrated absolute error), 569, 570 minimal complete, 261 minimal sufficiency, 220, 222 minimaxity, 264, 270–272, 349, 394 Bayes rule, 349 minimum risk equivariance (MRE), 263 Theory of Statistics c○2000–2013 James E. Gentle minimum risk equivariant estimation (MREE), 281–285 Minkowski distance, 637 Minkowski norm, 636, 737 Minkowski’s inequality, 636, 846 MISE (mean integrated squared error), 569 mixture distribution, 22, 184, 596–600, 747, 828 MLE (maximum likelihood estimator), 238–240, 444–498 MME, 243, 268, 412 mode, 18, 165 model algorithm, 210 equation, 209 moment, 25, 30, 52 uniqueness of, 32 moment problem, 142 moment-equivalent distribution, 33 moment-generating function (MGF), 43 moment-indeterminant distribution, 33 moments, method of, 243, 268, 412 monotone convergence theorem, 89, 643, 726 conditional, 112 monotone likelihood ratio, 165, 167, 241, 516 exponential class, 177 Monte Carlo, 373–376 Moore-Penrose inverse, 421, 425, 776, 793 morphism, 625 MRE (minimum risk equivariance), 263 MREE (minimum risk equivariant estimation), 281–285 MRIE (minimum risk invariant estimation), 281 MSAE (mean sup absolute error), 569 MSE (mean squared error), 214, 565, 569 MSPE (mean squared prediction error), 232 multiple tests, 532–534 multivariate central limit theorem, 107 n-divisibility, 60 natural exponential family, 173, 175, 195
natural parameter space, 173 negligible set, 704, 705 neighborhood, 617, 618 Newton’s method, 804, 817 Neyman structure, 516, 521 Neyman-Pearson Lemma, 513 Neyman-Scott problem, 486 no-data problem, 201, 326 nondegenerate random variable, 9 nonexistence of optimal statistical methods, 273 noninformative prior, 345 nonnegative definite matrix, 776, 782 nonparametric family, 13, 159 nonparametric inference, 211, 242, 495–498, 557–559 function estimation, 560–593 likelihood methods, 495 test, 531–532 nonparametric probability density estimation, 574–593 nonparametric test, 531–532 nonrandomized test, 289, 505 norm, 631, 635, 773, 838 Euclidean, 637, 774 Frobenius, 774, 787 in IR d , 635 Lp, 773 of a function, 737 of a matrix, 774, 787 of a vector, 635 normal distribution, characterizations of, 191 normal equations, 247, 252, 434 normal function, 738 normal integral, 675 normal vector, 679, 773 nuisance parameter, 219 null hypothesis, 287 O(·), 83, 646 o(·), 83, 646 OP(·), 83 oP(·), 83 objective prior, 345–346 observed significance level, 288 octile skewness, 53 one-parameter exponential family, 173, 267 Theory of Statistics c○2000–2013 James E. Gentle Index 893 one-sided confidence interval, 294 one-step MLE, 463 one-to-one function, 619 one-way AOV model, 430–432, 484–486 open cover, 616 open set, 616, 618, 707 optimization, 681, 814–824 optimization of vector/matrix functions, 803 orbit, 748 order of a field, 627 order of kernel or U statistic, 400 order statistic, 62–65, 96–99, 108–109, 218, 248, 405, 559–560 asymptotic distribution, 96–99 ordered field, 628 ordered set, 614, 638 ordering, 614 linear, 614 total, 614 well, 614 Ornstein-Ulenbeck process, 766 orthogonal matrix, 776 orthogonal polynomials, 563, 743–746 orthogonality, 631 orthogonalizing vectors, 679 orthogonally diagonalizable, 778 orthogonally similar, 778 orthonormal vectors, 679, 773 outer measure, 699 outlier-generating distribution, 166 over-dispersion, 494 P p distribution function space, 747 p-value, 288, 508 parameter space, 12, 159, 168 natural, 173 parametric family, 13, 159, 231 parametric inference, 211 parametric-support family, 177, 224, 495 Parseval’s theorem, 751 partial correlation, 118 partial likelihood, 497, 574 partition function in a PDF, 20, 172, 333 partition of a set, 612 PCER (per comparison error rate), 533 PDF (probability density function), 17
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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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0.1.12 Transforms 0.1 Measure, Inte
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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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B Useful Inequalities in Probabilit
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B.3 Pr(X ∈ A) and E(f(X)) 839 Pr(
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Page 867 and 868: C Notation and Definitions All nota
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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 and 906: Euclidean norm, 774 Euler’s formu
Page 907 and 908: unbiased test, 292, 519 uniform con
Page 909: sequence of sets, 620-623 limit poi
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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