Theory of Statistics - George Mason University

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618 0 Statistical Mathematics The structure (Ω, ρ) is called a metric space. A common example of a metric space is the set IR together with ρ(x, y) = |x − y|, where | · | denotes the ordinary absolute value. The concept of a metric allows us to redefine the topological properties introduced above in terms of the metric. The definitions in terms of a metric are generally more useful, and also a metric allows us to define additional important properties, such as continuity. Rather than define special sets for general metric spaces here, we will discuss these sets and their properties in the context of IR in Section 0.0.5. 0.0.2.7 Neighborhoods Defined by Metrics We have defined neighborhoods in general topological spaces, but the concept of a metric allows us to give a more useful definition of a neighborhood of a point in a set. For a point x ∈ Ω, a metric ρ on Ω, and any positive number ɛ, an ɛ-neighborhood of x, denoted by Nρ(x, ɛ), is the set of y ∈ Ω whose distance from x is less than ɛ; that is, Nρ(x, ɛ) def = {y : ρ(x, y) < ɛ}. (0.0.9) Notice that the meaning of a neighborhood depends on the metric, but in any case it is an open set, in the sense made more precise below. Usually, we assume that a metric is given and just denote the neighborhood as N(x, ɛ) or, with the size in place of the metric, as Nɛ(x). We also often refer to some (unspecified) ɛ-neighborhood of x just as a “neighborhood” and denote it as N(x). The concept of a neighborhood allows us to give a more meaningful definition of open sets and to define such things as continuity. These definitions are consistent with the definitions of the same concepts in a general topological space, as discussed above. Theorem 0.0.2 Every metric space is a Hausdorff space. Proof. Exercise. 0.0.2.8 Open and Closed Sets in a Metric Space The specification of a topology defines the open sets of the structure and consequently neighborhoods of points. It is often a more useful approach, however, first to define a metric, then to define neighborhoods as above, and finally to define open sets in terms of neighborhoods. In this approach, a subset G of Ω is said to be open if each member of G has a neighborhood that is contained in G. Note that with each metric space (Ω, ρ), we can associate a topological space (Ω, T ), where T is the collection of open sets in (Ω, ρ) that are defined Theory of Statistics c○2000–2013 James E. Gentle
0.0 Some Basic Mathematical Concepts 619 in terms of the metric. The topology provides the definition of a closed set, as above; that is, a set A ⊆ Ω is said to be closed iff Ω ∩ A c ∈ T , where T is the collection of open sets defined in terms of the metric. As with the definitions above for general topological spaces, some sets are both open and closed, and such sets are said to be clopen. We note that (IR, ρ) is a Hausdorff space because, given x, y ∈ IR and x = y we have ρ(x, y) > 0 and so N(x, ρ(x, y)/2) and N(y, ρ(x, y)/2) are disjoint open sets. We also note that IR is connected, as is any interval in IR. (Connectedness is a topological property that is defined on page 617.) We will defer further discussion of openness and related concepts to page 639 in Section 0.0.5 where we discuss the real number system. 0.0.2.9 Relations and Functions A relation is a set of doubletons, or pairs of elements; that is, a relation is a subset of a cartesian product of two sets. We use “relation” and “mapping” synonymously. A function is a relation in which no two different pairs have the same first element. To say that f is a function from Ω to Λ, written f : Ω ↦→ Λ, means that for every ω ∈ Ω there is a pair in f whose first member is ω. We use the notation f(ω) to represent the second member of the pair in f whose first member is ω, and we call ω the argument of the function. We call Ω the domain of the function and we call {λ|λ = f(ω) for some ω ∈ Ω} the range of the function. Variations include functions that are onto, meaning that for every λ ∈ Λ there is a pair in f whose second member is λ; and functions that are one-toone, often written as 1 : 1, meaning that no two pairs have the same second member. A function that is one-to-one and onto is called a bijection. A function f that is one-to-one has an inverse, written f −1 , that is a function from Λ to Ω, such that if f(ω0) = λ0, then f −1 (λ0) = ω0. If (a, b) ∈ f, we may write a = f −1 (b), although sometimes this notation is restricted to the cases in which f is one-to-one. If f is not one-to-one and if the members of the pairs in f are reversed, the resulting relation is not a function. We say f −1 does not exist; yet for convenience we may write a = f −1 (b), with the meaning above. If A ⊆ Ω, the image of A, denoted by f[A], or just by f(A), is the set of all λ ∈ Λ for which λ = f(ω) for some ω ∈ Ω. (The notation f[A] is preferable, but we will often just use f(A).) Similarly, if C is a collection of sets (see below), the notation f[C] denotes the collection of sets {f[C] : C ∈ C}. For the function f that maps from Ω to Λ, Ω is called the domain of the function and and f[Ω] is called the range of the function. 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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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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8.3 Nonparametric Estimation of Fun
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0.0 Some Basic Mathematical Concept
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0.0.9.5 Working with Real-Valued Fu
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∞ 0.0 Some Basic Mathematical Con
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a) Evaluate the integral (0.0.84):
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0.1 Measure, Integration, and Funct
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3. if A1, A2, . . . ∈ F are disjo
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(why?), and so We also have λ 0.1
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Proof. Exercise. 0.1 Measure, Integ
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Hence, from inequality (0.1.47),
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0.1.9.3 Norms of Functions 0.1 Meas
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0.2 Stochastic Processes and the St
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Variation of Functionals 0.2 Stocha
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0.2.1.3 Ito Processes 0.2 Stochasti
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0.3 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
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 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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