Theory of Statistics - George Mason University

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168 2 Distribution Theory and Statistical Models Totally Positive Family A totally positive family of distributions is defined in terms of the total positivity of the PDF, treating it as a function of two variables, θ and x. In this sense, a family is totally positive of order r iff for all x1 < · · · < xn and θ1 < · · · < θn, fθ1(x1) · · · fθ1(xn) . . . . . ≥ 0 ∀ n = 1, . . ., r. (2.5) fθn(x1) · · · fθn(xn) A totally positive family with r = 2 is a monotone likelihood ratio family. 2.3 “Regular” Families Conditions that characterize a set of objects for which a theorem applies are called “regularity conditions”. I do not know the origin of this term, but it occurs in many areas of mathematics. In statistics there are a few sets of regularity conditions that define classes of interesting probability distributions. We will often use the term “regularity conditions” to refer to continuity and differentiability of the PDF wrt the parameter. 2.3.1 The Fisher Information Regularity Conditions The most important set of regularity conditions in statistics are some that allow us to put a lower bound on the variance of an unbiased estimator (see inequality (B.25) and Sections 3.1.3 and 5.1). Consider the family of distributions P = {Pθ; θ ∈ Θ} that have densities fθ. There are generally three conditions that together are called the Fisher information regularity conditions: • The parameter space Θ ⊆ IR k is convex and contains an open set. • For any x in the support and θ ∈ Θ ◦ , ∂fθ(x)/∂θ and ∂ 2 fθ(x)/∂θ 2 exist and are finite, and ∂ 2 fθ(x)/∂θ 2 is continuous in θ. • The support is independent of θ; that is, all Pθ have a common support. The latter two conditions ensure that the operations of integration and differentiation can be interchanged twice. Because the Fisher information regularity conditions are so important, the phrase “regularity conditions” is often taken to mean “Fisher information regularity conditions”. The phrase “Fisher regularity conditions” is also used synonymously, as is “FI regularity conditions”. Theory of Statistics c○2000–2013 James E. Gentle
2.3.2 The Le Cam Regularity Conditions 2.4 The Exponential Class of Families 169 The Le Cam regularity conditions are the first two of the usual FI regularity conditions plus the following. • The Fisher information matrix (see equation (1.80)) is positive definite for any fixed θ ∈ Θ. • There exists a positive number cθ and a positive function hθ such that E(hθ(X)) < ∞ and sup ∂ 2 log fγ(x) ∂γ(∂γ) T ≤ hθ(x) a.e. (2.6) γ:γ−θ
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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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Page 141 and 142: 1.6 Stochastic Processes 123 Stoppi
Page 143 and 144: 1.6 Stochastic Processes 125 (Exerc
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Page 149 and 150: 1.6 Stochastic Processes 131 We als
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Page 153 and 154: Convergence of Empirical Processes
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Page 157 and 158: Notes and Further Reading 139 and f
Page 159 and 160: Notes and Further Reading 141 we ca
Page 161 and 162: Notes and Further Reading 143 After
Page 163 and 164: Exercises 145 of betting system. Do
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Page 169 and 170: Exercises 151 1.59. A sufficient co
Page 171 and 172: Exercises 153 1.85. Show that Doob
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Page 181 and 182: Example 2.1 complete and incomplete
Page 183 and 184: 2.2 Shapes of the Probability Densi
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Page 205 and 206: Higher Dimensions 2.11 The Family o
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Page 213 and 214: Exercises 195 are considered at som
Page 215 and 216: Exercises 197 This law has been use
Page 217 and 218: Exercises 199 a) Show that for d
Page 219 and 220: 3 Basic Statistical Theory The fiel
Page 221 and 222: How Does PH Differ from P? 3 Basic
Page 223 and 224: 3 Basic Statistical Theory 205 abov
Page 225 and 226: 3.1 Inferential Information in Stat
Page 231 and 232: The Basic Paradigm of Point Estimat
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3.1 Inferential Information in Stat
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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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B.4 E(f(X)) and f(E(X)) 841 Theorem
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B.4 E(f(X)) and f(E(X)) 843 A relat
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B.5 E(f(X,Y )) and E(g(X)) and E(h(
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Now assume p > 1. Now, E(|X + Y | p
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C Notation and Definitions All nota
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C.1 General Notation 851 constant;
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C.2 General Mathematical Functions
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Functions of Convenience C.2 Genera
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C.2 General Mathematical Functions
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A1 ∩ A2 A1 − A2 A1∆A2 A1 × A
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C.4 Linear Spaces and Matrices 861
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a(ij) C.4 Linear Spaces and Matrice
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References The number(s) following
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References 867 Lyle D. Broemeling.
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References 869 William Feller. An I
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References 871 H. O. Hartley. In Dr
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References 873 Feldman, and Murad S
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References 875 Roger B. Nelsen. An
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References 877 Glenn Shafer. A Math
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References 879 Abraham Wald. Sequen
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Index a.e. (almost everywhere), 704
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cartesian product measurable space,
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derivative of a functional, 752-753
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Euclidean norm, 774 Euler’s formu
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unbiased test, 292, 519 uniform con
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sequence of sets, 620-623 limit poi
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natural parameter space, 173 neglig
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proportional hazards, 573 pseudoinv
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square root matrix, 783 squared-err
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weak convergence in mean square, 56
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