Theory of Statistics - George Mason University

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176 2 Distribution Theory and Statistical Models An exponential family that is not of full rank may also be degenerate, meaning that there exists a vector a and a constant r such that pθ(x)dx = 1. a T x=r (The term “degenerate” in this sense is also applied to any distribution, whether in an exponential family or not.) The support of a degenerate distribution within IR d is effectively within IR k for k < d. An example of a nonfull rank exponential family that is also a degenerate family is the family of multinomial distributions (page 830). A continuous degenerate distribution is also called a singular distribution. An example of a family of distributions that is a nonfull rank exponential family is the normal family N(µ, µ 2 ). A nonfull rank exponential family is also called a curved exponential family. 2.4.7 Properties of Exponential Families Exponential families have a number of useful properties. First of all, we note that an exponential family satisfies the Fisher information regularity conditions. This means that we can interchange the operations of differentiation and integration, a fact that we will exploit below. Other implications of the Fisher information regularity conditions allow us to derive optimal statistical inference procedures, a fact that we will exploit in later chapters. In the following, we will use the usual form of the PDF, fθ(x) = exp(η(θ) T T(x) − ξ(θ))h(x), and we will assume that it is of full rank. We first of all differentiate both sides of the identity, wrt θ, fθ(x)dx = 1 (2.17) Carrying the differentiation on the left side under the integral, we have Jη(θ)T(x) − ∇ξ(θ) exp η(θ)T(x) − ξ(θ) h(x)dx = 0. Hence, because by assumption Jη(θ) is of full rank, by rearranging terms under the integral and integrating out terms not involving x, we get the useful fact E(T(X)) = (Jη(θ)) −1 ∇ξ(θ). (2.18) We now consider E(T(X)). As it turns out, this is a much more difficult situation. Differentiation yields more complicated objects. (See Gentle (2007), page 152, for derivatives of a matrix wrt a vector.) Let us first consider the scalar case; that is, η(θ) and T(x) are scalars, so η(θ) T T(x) is just η(θ)T(x). Theory of Statistics c○2000–2013 James E. Gentle
or or 2.5 Parametric-Support Families 177 In this case, differentiating a second time with respect to θ, we get T(x)(η ′ (θ)T(x)−ξ ′ (θ))exp(η(θ)T(x)−ξ(θ))h(x)dx = ξ ′′ (θ)η ′ (θ)/(η ′ (θ)) 2 −ξ ′ (θ)η ′′ (θ)/(η ′ (θ)) 2 , η ′ (θ)E((T(X)) 2 ) − ξ ′ (θ)E(T(X)) = ξ ′′ (θ)/η ′ (θ) − ξ ′ (θ)η ′′ (θ)/(η ′ (θ)) 2 E((T(X)) 2 ) = (ξ ′ (θ)) 2 /(η ′ (θ)) 2 − ξ ′′ (θ)/(η ′ (θ)) 2 − ξ ′ (θ)η ′′ (θ)/(η ′ (θ)) 3 . Finally, collecting terms, we have V(T(X)) = E((T(X)) 2 ) − (E((T(X))) 2 = ξ′′ (θ) (η ′ (θ)) 2 − ξ′ (θ)η ′′ (θ) (η ′ (θ)) 3 . (2.19) ********* proposition 3.2 in shao, page 171. look at two parameterizations; natural and mean. V(T(X)) = Hζ(η) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 8, where Hζ(η) is the matrix of second derivatives of ζ with respect to η. It is often a simple matter to determine if a member of the exponential class of distributions is a monotone likelihood ratio family. If η(θ) and T(x) in equation (2.7) for the PDF of a distribution in the exponential class are scalars, and if η(θ) is monotone in θ, then the family has a monotone likelihood ratio in T(x). 2.5 Parametric-Support Families Parametric-support families have simple range dependencies, that is, these are distributions whose supports depend on parameters. A distribution in any of these families has a PDF in the general form pθ(x) = f(x, θ)I[f1(θ),f2(θ)](x). (2.20) These families are also called “truncation families”, but most people use the term “truncated family” to refer to a family that is artificially truncated (for example, due to censoring; see Section 2.7). For example, to refer to the threeparameter gamma as a truncated distribution would be to confuse it with the more standard terminology in which a truncated gamma is the distribution formed from a two-parameter distribution with PDF c Γ(α)β α xα−1 e −x/β I[τ1,τ2](x), Theory of Statistics c○2000–2013 James E. Gentle
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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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Page 153 and 154: Convergence of Empirical Processes
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Page 159 and 160: Notes and Further Reading 141 we ca
Page 161 and 162: Notes and Further Reading 143 After
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Page 181 and 182: Example 2.1 complete and incomplete
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and so I(µ, σ) = Eθ = E(µ,σ) =
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3.1 Inferential Information in Stat
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3.1 Inferential Information in Stat
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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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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 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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0.2 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
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 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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