Theory of Statistics - George Mason University

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884 Index of tests, 311, 523 strong, 303 weak, 303 consistent estimator, 303, 566 continuity theorem, 87 continuous function, 620, 714–717, 795 absolutely continuous, 716 Lipschitz-continuous, 717 Lipschitz-continuous PDF, 579 continuous random variable, 19 contradiction (method of proof), 669 contrast, 431, 553 convergence, 75–102 almost sure, 76, 718 in Lr, 76 in absolute mean, 77 in distribution, 78 in law, 78 in mean, 77 in mean square, 77, 566 in probability, 77 in quadratic mean, 566 in second moment, 77 of function estimators, 566–567, 570–571 of probability density functions, 82 pointwise, 718 uniform, 718 weak, 78, 80 with probability 1, 76 wp1, 76 convergence of a sequence of sets, 621 convergence of powers of a matrix, 812 convergence-determining class, 79 convex function, 651, 841 convex loss, 257, 260, 263, 265 convex set, 651 convexity, 651 convolution, 57, 734 convolution theorem, 751 copula, 39–40, 120 Cor(·, ·), 37 correctness of confidence sets, 542 correlation, 36, 40 correlation of functions, 734 correlation theorem, 751 countable, 610 counting measure, 702 Cov(·, ·), 36 Theory of Statistics c○2000–2013 James E. Gentle covariance, 36 covariance inequality, 395, 845 covariance of functions, 734 cover (by a collection of sets), 614 coverage probability, 293 Cox proportional hazards model, 574 Cox-Ingersoll-Ross (CIR) process, 766 Cramér von Mises test, 532 Cramér-Rao lower bound, 231, 395, 417 Cramér-Wold device, 91 credible set, 368–372 Cressie-Read divergence measure, 249 critical region, 288 CRLB (information inequality), 231, 395, 417 cumulant, 34 cumulant-generating function, 50 cumulative distribution function (CDF), 14 inverse, 15 notation, 137 relation to uniform distribution, 15 tail, 14, 166 curved exponential families, 175 Darmois theorem, 192 data-generating process, 201, 233 de Finetti’s representation theorem, 74, 114, 329 de Moivre Laplace central limit theorem, 104 de Moivre’s formula, 673 de Moivre’s martingale, 152 De Morgan’s law, 611 decision rule, 256 randomized, 256 decision theory, 255–272, 274–285 decomposable matrix, 810 decomposition of a function, 563 Dedekind completeness, 638 degenerate random variable, 9 degree of statistical function, 388, 401 delete d jackknife, 298 δ-field, 688 delta method, 94, 312, 477, 553 second order, 94, 477 dense set, 616, 635 density function, 19 derivative, 731
derivative of a functional, 752–753 derivative with respect to a vector or matrix, 793 det(·), 775 determinant of a square matrix, 775 determining class, 3, 79 deviance, 241, 490 DF see CDF, 14 DFT (discrete Fourier transform), 680 diag(·), 791 difference equation, 681 differential, 797 differential equation, 681 differentiation of vectors and matrices, 793 diffusion process, 757 digamma function, 462, 857 dimensionality, 186 problems in higher dimensions, 186, 198 Dirac delta function, 730, 856 Dirac measure, 702 direct product, 611, 785 Dirichlet function, 715 discrete Fourier transform (DFT), 680 discrete random variable, 19 disjoint sets, 612 disjointification, 613 distribution family Benford’s, 831 Bernoulli, 94, 233, 265, 277, 329, 336, 386, 390, 393, 394, 443, 448, 455, 477, 478, 514, 516, 535, 537, 830 beta, 63, 170, 228, 333, 351, 353, 356, 370, 379, 381, 835 binomial, 58, 167, 170, 179, 233, 272, 333, 336, 351, 356, 361, 370, 379, 381, 386, 443, 455, 527, 830 Cauchy, 26, 43, 65, 150, 171, 174, 181, 457, 835 chi-squared, 56, 59, 60, 190, 833 complex multivariate normal, 189, 832 conditional, 111, 119 Dirichlet, 170, 835 discrete uniform, 830 double exponential, 167, 170, 171, 181, 309, 836 doubly noncentral F, 191 Theory of Statistics c○2000–2013 James E. Gentle Index 885 elliptical, 187 ɛ-mixture distribution, 157, 184, 457, 596–600, 747 exponential, 20, 55, 63, 97, 99, 129, 167, 170, 171, 181, 224, 378, 393, 446, 447, 452, 467, 479, 506, 508, 517, 525, 836 spacings, 64, 129 exponential class, 169–177 attainment of CRLB, 396 conjugate priors, 333, 378 extreme value, 99, 836 F, 59, 191, 833 families, 155–194, 827–836 gamma, 58, 98, 170, 181, 227, 380, 383, 451, 462, 463, 836 geometric, 830 hypergeometric, 159, 380, 464, 830 infinitely divisible, 185 inverse Gaussian, 58, 170, 181, 834 inverted chi-squared, 170, 338, 834 inverted gamma, 378, 834 inverted Wishart, 834 location-scale, 179 logarithmic, 831 logistic, 170, 181, 835 lognormal, 43, 148, 170, 834 multinomial, 170, 466, 830 multivariate matrix normal, 188, 832 multivariate normal, 188, 268, 832 negative binomial, 58, 170, 233, 336, 353, 380, 386, 439, 443, 455, 830 noncentral chi-squared, 190, 833 noncentral F, 191, 833 noncentral t, 833 normal, 56, 58, 60, 96, 167, 170, 181, 187–194, 223, 226, 238, 285, 294, 308, 337, 338, 355, 362, 367, 392, 396, 469, 664, 832 Pareto, 164, 170, 171, 835 Poisson, 57, 167, 170, 374, 380, 383, 397, 439, 831 positive Poisson, 183, 197, 198 power function, 164, 835 power law, 164 power series, 170, 174, 831 regular, 168–169 skew normal, 834 spherical, 187
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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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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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Appendix A. Important Probability D
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Page 855 and 856: B Useful Inequalities in Probabilit
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Page 863 and 864: B.5 E(f(X,Y )) and E(g(X)) and E(h(
Page 865 and 866: Now assume p > 1. Now, E(|X + Y | p
Page 867 and 868: C Notation and Definitions All nota
Page 869 and 870: C.1 General Notation 851 constant;
Page 871 and 872: C.2 General Mathematical Functions
Page 873 and 874: Functions of Convenience C.2 Genera
Page 875 and 876: C.2 General Mathematical Functions
Page 877 and 878: A1 ∩ A2 A1 − A2 A1∆A2 A1 × A
Page 879 and 880: C.4 Linear Spaces and Matrices 861
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: cartesian product measurable space,
Page 905 and 906: Euclidean norm, 774 Euler’s formu
Page 907 and 908: unbiased test, 292, 519 uniform con
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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