Theory of Statistics - George Mason University

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800 0 Statistical Mathematics For a function f : S ⊆ IR n → IR m , we define ∂f T /∂x to be the n × m matrix, which is the natural extension of ∂/∂x applied to a scalar function, and ∂f/∂x T to be its transpose, the m×n matrix. Although the notation ∂f T /∂x is more precise because it indicates that the elements of f correspond to the columns of the result, we often drop the transpose in the notation. We have ∂f ∂x ∂fT = by convention ∂x ∂f1 ∂fm = . . . ∂x ∂x ⎡ ⎤ ⎢ = ⎢ ⎣ ∂f1 ∂x1 ∂f1 ∂x2 ∂f1 ∂xn ∂f2 ∂x1 ∂f2 ∂x2 · · · ∂f2 ∂xn · · · ∂fm ∂x1 · · · ∂fm ∂x2 · · · ∂fm ∂xn ⎥ ⎦ (0.3.48) if those derivatives exist. This derivative is called the matrix gradient and is denoted by Gf or ∇f for the vector-valued function f. (Note that the ∇ symbol can denote either a vector or a matrix, depending on whether the function being differentiated is scalar-valued or vector-valued.) The m × n matrix ∂f/∂x T = (∇f) T is called the Jacobian of f and is denoted by Jf: Jf = G T f = (∇f) T . (0.3.49) The absolute value of the determinant of the Jacobian appears in integrals involving a change of variables. (Occasionally, the term “Jacobian” is used to refer to the absolute value of the determinant rather than to the matrix itself.) To emphasize that the quantities are functions of x, we sometimes write ∂f(x)/∂x, Jf(x), Gf(x), or ∇f(x). 0.3.3.14 Derivatives of Matrices with Respect to Vectors The derivative of a matrix with respect to a vector is a three-dimensional object that results from applying equation (0.3.45) to each of the elements of the matrix. For this reason, it is simpler to consider only the partial derivatives of the matrix Y with respect to the individual elements of the vector x; that is, ∂Y/∂xi. The expressions involving the partial derivatives can be thought of as defining one two-dimensional layer of a three-dimensional object. Using the rules for differentiation of powers that result directly from the definitions, we can write the partial derivatives of the inverse of the matrix Y as ∂ ∂x Y −1 = −Y −1 ∂ ∂x Y Theory of Statistics c○2000–2013 James E. Gentle Y −1 . (0.3.50)
0.3 Some Basics of Linear Algebra 801 Beyond the basics of differentiation of constant multiples or powers of a variable, the two most important properties of derivatives of expressions are the linearity of the operation and the chaining of the operation. These yield rules that correspond to the familiar rules of the differential calculus. A simple result of the linearity of the operation is the rule for differentiation of the trace: ∂ tr(Y ) = tr ∂x ∂ ∂x Y 0.3.3.15 Higher-Order Derivatives with Respect to Vectors; The Hessian Higher-order derivatives are derivatives of lower-order derivatives. As we have seen, a derivative of a given function with respect to a vector is a more complicated object than the original function. The simplest higher-order derivative with respect to a vector is the second-order derivative of a scalar-valued function. Higher-order derivatives may become uselessly complicated. In accordance with the meaning of derivatives of vectors with respect to vectors, the second derivative of a scalar-valued function with respect to a vector is a matrix of the partial derivatives of the function with respect to the elements of the vector. This matrix is called the Hessian, and is denoted by Hf or sometimes by ∇∇f or ∇ 2 f: Hf = ∂2f = ∂x∂xT ⎡ ⎢ ⎣ ∂ 2 f ∂x 2 1 ∂ 2 f ∂x2∂x1 ∂ 2 f ∂xm∂x1 ∂ 2 f ∂x1∂x2 ∂ 2 f ∂x 2 2 · · · ∂ 2 f ∂xm∂x2 · · · · · · · · · . ∂ 2 f ∂x1∂xm ∂ 2 f ∂x2∂xm ∂ 2 f ∂x 2 m ⎤ ⎥ . (0.3.51) ⎥ ⎦ The Hessian is a function of x. We write Hf(x) or ∇∇f(x) or ∇ 2 f(x) for the value of the Hessian at x. 0.3.3.16 Summary of Derivatives with Respect to Vectors As we have seen, the derivatives of functions are complicated by the problem of measuring the change in the function, but often the derivatives of functions with respect to a vector can be determined by using familiar scalar differentiation. In general, we see that • the derivative of a scalar (a quadratic form) with respect to a vector is a vector and • the derivative of a vector with respect to a vector is a matrix. Table 0.3 lists formulas for the vector derivatives of some common expressions. The derivative ∂f/∂x T is the transpose of ∂f/∂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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1.6 Stochastic Processes 125 (Exerc
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1.6 Stochastic Processes 127 variab
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1.6 Stochastic Processes 129 in a d
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1.6 Stochastic Processes 131 We als
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1.6 Stochastic Processes 133 X0 has
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Convergence of Empirical Processes
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and Fn(x, ω) ≥ Fn(xm,k−1, ω)
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Notes and Further Reading 139 and f
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Notes and Further Reading 141 we ca
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Notes and Further Reading 143 After
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Exercises 145 of betting system. Do
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Exercises 147 1.22. Show that if th
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1.36. Consider the the distribution
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Exercises 151 1.59. A sufficient co
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Exercises 153 1.85. Show that Doob
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2 Distribution Theory and Statistic
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Example 2.1 complete and incomplete
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2.2 Shapes of the Probability Densi
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2.2 Shapes of the Probability Densi
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2.3.2 The Le Cam Regularity Conditi
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2.4 The Exponential Class of Famili
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or or 2.5 Parametric-Support Famili
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2.6 Transformation Group Families 1
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2.6 Transformation Group Families 1
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2.7 Truncated and Censored Distribu
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2.9 Infinitely Divisible and Stable
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Higher Dimensions 2.11 The Family o
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2.11 The Family of Normal Distribut
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Exercises 195 are considered at som
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Exercises 197 This law has been use
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Exercises 199 a) Show that for d
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3 Basic Statistical Theory The fiel
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How Does PH Differ from P? 3 Basic
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3 Basic Statistical Theory 205 abov
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3.1 Inferential Information in Stat
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The Basic Paradigm of Point Estimat
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Completeness 3.1 Inferential Inform
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and so by completeness, 3.1 Inferen
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and so I(µ, σ) = Eθ = E(µ,σ) =
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Robustness 3.1 Inferential Informat
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3.2 Statistical Inference: Approach
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3.3 The Decision Theory Approach to
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Risk 0.005 0.010 0.015 0.020 0.025
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3.4 Invariant and Equivariant Stati
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Equivariant Point Estimation 3.4 In
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3.5 Probability Statements in Stati
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Test Rules 3.5 Probability Statemen
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if 3.6 Variance Estimation 297 Give
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3.6 Variance Estimation 299 J(T) =
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3.7 Applications 3.7.1 Inference in
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3.8 Asymptotic Inference 303 The ca
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3.8 Asymptotic Inference 305 The mo
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3.8 Asymptotic Inference 307 wherea
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3.8 Asymptotic Inference 309 tendin
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Definition 3.20 (asymptotic signifi
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Proof. *************** Notes and Fu
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Notes and Further Reading 315 Least
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Approximations and Asymptotic Infer
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Exercises 319 b) the expectation of
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4 Bayesian Inference We have used a
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4.1 The Bayesian Paradigm 323 For m
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4.1 The Bayesian Paradigm 325 A qua
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4.2 Bayesian Analysis 327 even if t
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4.2 Bayesian Analysis 329 Hence P(B
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4.2 Bayesian Analysis 331 B1. ∀θ
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4.2 Bayesian Analysis 333 This mean
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Prior Prior 0.0 0.5 1.0 1.5 2.0 0 2
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4.2 Bayesian Analysis 337 We go thr
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4.2 Bayesian Analysis 339 We constr
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4.2 Bayesian Analysis 341 as the mo
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Assessing the Problem Formulation 4
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Posterior Posterior 0 5 10 15 0 1 2
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4.2 Bayesian Analysis 347 distribut
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4.3 Bayes Rules 349 • the nature
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Then we have E(g(θ)T(X)) = E(T(X)E
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4.3 Bayes Rules 353 equivariance Fo
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4.3 Bayes Rules 355 Example 4.9 (Co
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p0 = Pr(Θ ∈ Θ0) and p1 = Pr(Θ
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The Weighted 0-1 or α0-α1 Loss Fu
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The term ˆp0 ˆp1 = p0 p1 BF(x) =
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4.5 Bayesian Testing 365 • Bayes
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where fΘ|x(θ|x) = p1 if θ = θ0
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4.6 Bayesian Confidence Sets 369 as
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Posterior 0 1 2 3 95% Credible Regi
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Markov Chain Monte Carlo 4.7 Comput
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4.7 Computational Methods 375 β/(t
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Notes and Further Reading 377 An al
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4.4. Given the conditional PDF a) U
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Exercises 381 4.14. Consider again
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Exercises 383 4.26. As in Exercise
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5 Unbiased Point Estimation In a de
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Estimability 5 Unbiased Point Estim
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s1π + s0(1 − π) = π, 5.1 UMVUE
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5.1 UMVUE 391 Let T be UMVUE for g(
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5.1 UMVUE 393 Example 5.7 UMVUE of
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5.1 UMVUE 395 The risk of T ∗ is
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5.1 UMVUE 397 estimator of θ that
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(Compare this with Tfdx = g(θ).)
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¯h(X1, . . ., Xm) = 1 m! 5.2 U-Sta
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where Pn is the ECDF. U(X1, . . .,
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and T 2 r,S = C = Tr,S = C C TnT
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Generalized U-Statistics 5.2 U-Stat
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5.2 U-Statistics 409 Theorem 5.4 (H
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5.3 Asymptotically Unbiased Estimat
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5.3.2 Ratio Estimators 5.3 Asymptot
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5.4 Asymptotic Efficiency 415 of a
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⎧ ⎨ Tn = ⎩ n2 with probabilit
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5.5 Applications 5.5 Applications 4
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5.5 Applications 421 one aspect of
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Proof. Because l ∈ span(X T ) = s
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Optimal Properties of the Moore-Pen
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5.5 Applications 427 implies implie
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The “Sum of Squares” Quadratic
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5.5 Applications 431 The question o
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We now write the original model as
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5.5 Applications 435 it from other
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(it’s Bernoulli), and for i = j,
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Fisher Efficient Estimators and Exp
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6 Statistical Inference Based on Li
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6.1 The Likelihood Function 443 is
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6.2 Maximum Likelihood Parametric E
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6.2.2 Finite Sample Properties of M
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where Now, consider This has two pa
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6.3 Asymptotic Properties of MLEs,
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6.4 Application: MLEs in Generalize
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Residuals 6.4 Application: MLEs in
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6.5 Variations on the Likelihood 49
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6.5 Variations on the Likelihood 49
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Maximum Likelihood in Linear Models
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Exercises 501 b) Assume that σ 2 1
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7 Statistical Hypotheses and Confid
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Tests of Hypotheses 7.1 Statistical
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7.1 Statistical Hypotheses 507 Know
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Power of a Statistical Test 7.1 Sta
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An Optimal Test in a Simple Situati
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7.2 Optimal Tests 513 All of these
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Use of Sufficient Statistics 7.2 Op
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7.2 Optimal Tests 517 Syppose we as
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Nonexistence of UMP Tests 7.2 Optim
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H0 : θ = θ0 versus H1 : θ = θ0.
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7.2.6 Equivariance, Unbiasedness, a
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7.3 Likelihood Ratio Tests, Wald Te
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7.3 Likelihood Ratio Tests, Wald Te
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Minimize 7.3 Likelihood Ratio Tests
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7.4 Nonparametric Tests 531 I(0, ˆ
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Yij = µ + αi + ɛij, i = 1, . . .
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7.7 The Likelihood Principle and Te
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7.8 Confidence Sets 537 Monte Carlo
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7.8 Confidence Sets 539 The concept
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7.8 Confidence Sets 541 For any giv
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7.9 Optimal Confidence Sets 543 Thi
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7.9 Optimal Confidence Sets 545 Con
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7.10 Asymptotic Confidence sets 547
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7.11 Bootstrap Confidence Sets 549
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Bias in Intervals Due to Bias in th
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7.12 Simultaneous Confidence Sets 5
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Sequential Tests Exercises 555 Wald
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8 Nonparametric and Robust Inferenc
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8.2 Inference Based on Order Statis
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f ^ (x) Nonparametric Regression 0.
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8.3 Nonparametric Estimation of Fun
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ISE θ = 8.3 Nonparametric Estim
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8.4 Semiparametric Methods and Part
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8.5 Nonparametric Estimation of PDF
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Some Properties of the Histogram Es
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AISB 1 pH = 12 8.5 Nonparametric
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and the integrated variance is 8.5
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and 8.5 Nonparametric Estimation o
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h ∗ = 8.5 Nonparametric Estimatio
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Computation of Kernel Density Estim
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8.6 Perturbations of Probability Di
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CDF 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
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8.6 Perturbations of Probability Di
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8.7 Robust Inference 8.7 Robust Inf
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p(y) (1-ε)p(y) The Influence Funct
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8.7 Robust Inference 603 Notice tha
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Nonparametric Statistics Exercises
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0 Statistical Mathematics Statistic
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0 Statistical Mathematics 609 x may
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0.0 Some Basic Mathematical Concept
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Because each is a subset of the oth
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0.0.2 Sets and Spaces 0.0 Some Basi
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0.0.2.4 Point Sequences in a Topolo
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The sequence of sets {An} is said t
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The proof of this is a classic: fir
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• ∀x ∈ IR ∪ {∞}, x × ±
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0.0.5.2 Sets of Reals; Open, Closed
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We have 0.0 Some Basic Mathematical
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Definition 0.0.14 (superharmonic fu
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Ordering the Complex Numbers 0.0 So
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0.0.9.5 Working with Real-Valued Fu
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∞ 0.0 Some Basic Mathematical Con
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a) Evaluate the integral (0.0.84):
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0.1 Measure, Integration, and Funct
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3. if A1, A2, . . . ∈ F are disjo
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(why?), and so We also have λ 0.1
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Proof. Exercise. 0.1 Measure, Integ
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Hence, from inequality (0.1.47),
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0.1.9.3 Norms of Functions 0.1 Meas
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Page 845 and 846: A Important Probability Distributio
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Page 855 and 856: B Useful Inequalities in Probabilit
Page 857 and 858: B.3 Pr(X ∈ A) and E(f(X)) 839 Pr(
Page 859 and 860: B.4 E(f(X)) and f(E(X)) 841 Theorem
Page 861 and 862: B.4 E(f(X)) and f(E(X)) 843 A relat
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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
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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