Theory of Statistics - George Mason University

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542 7 Statistical Hypotheses and Confidence Sets 7.9 Optimal Confidence Sets Just as we refer to a test as being “valid” if the significance level of the test is not exceeded (that is, if the probability of rejecting a true null hypothesis is bounded by the level of significance), we refer to a a confidence interval that has a probability of at least the confidence level as being “correct”. We often evaluate a confidence set using a family of distributions that does not include the true parameter. For example, “accuracy” refers to the (true) probability of the set including an incorrect value. A confidence that is more accurate has a smaller probability of including a distribution that did not give rise to the sample. This is a general way relates to the size of the confidence set. Size of Confidence Sets The “volume” (or “length”) of a confidence set is the Lebesgue measure of the set: vol(ΘS) = d˜ θ. This may not be finite. If the volume is finite, we have (Theorem 7.6 in MS2) Eθ(vol(ΘS)) = ΘS θ= ˜ Prθ(ΘS ∋ θ ˜ θ)d˜ θ. We see this by a simple application of Fubini’s theorem to handle the integral over the product space, and then an interchange of integration: Want to minimize volume (if appropriate; i.e., finite.) 7.9.1 Most Accurate Confidence Set Accuracy of Confidence Sets **********Want to maximize accuracy.??????????? Confidence sets can be thought of a a family of tests of hypotheses of the form θ ∈ H0( ˜ θ) versus θ ∈ H1( ˜ θ). A confidence set of size 1 − α is equivalent to a critical region S(X) such that Pr S(X) ∋ ˜ θ ≥ 1 − α ∀ θ ∈ H0 The power of the related tests is just Pr S(X) ∋ ˜ θ ˜θ . for any θ. In testing hypotheses, we are concerned about maximizing this for θ ∈ H1( ˜ θ). Theory of Statistics c○2000–2013 James E. Gentle
7.9 Optimal Confidence Sets 543 This is called the accuracy of the confidence set, and so in this terminology, we seek the most accurate confidence set, and, of course, the uniformly most accurate confidence region. Similarly to the case of UMP tests, the uniformly most accurate confidence region may or may not exist. The question of existence of uniformly most accurate confidence intervals also depends on whether or not there are nuisance parameters. Just as with UMP tests, in the presence of nuisance parameters, usually uniformly most accurate confidence intervals do not exist. (We must add other restrictions on the intervals, as we see below.) The nonexistence of uniformly most accurate confidence sets can also be addressed by imposing unbiasedness. Uniformly most accurate 1 − α level set: Prθ(ΘS ∋ ˜ θ) is minimum among all 1 − α level sets and ∀ ˜ θ = θ. This definition of UMA may not be so relevant in the case of a one-sided confidence interval. If Θ is a subset of Θ that does not include θ, and Prθ(ΘS ∋ ˜ θ) ≤ Prθ(Θ eS ∋ ˜ θ) for any 1 − α level set Θ eS and ∀˜ θ ∈ Θ, then ΘS is said to be Θ-uniformly most accurate. A confidence set formed by inverting a nonrandomized UMP test is UMA. We see this easily from the definitions of UMP and UMA. (This is Theorem 7.4 in MS2.) 7.9.2 Unbiased Confidence Sets With tests, sometimes no UMP exists, and hence we added a criterion, such as unbiasedness or invariance. Likewise, sometimes we cannot form a UMA confidence interval, so we add some criterion. We define unbiasedness in terms of a subset Θ that does not include the true θ. A 1 − α level confidence set C(X) is said to be Θ-unbiased if Prθ(ΘS ∋ ˜ θ) ≤ 1 − α ∀ ˜ θ ∈ Θ. If Θ = {θ} c , we call the set unbiased. A Θ-unbiased set that is uniformly more accurate (“more” is defined similarly to “most”) than any other Θ-unbiased set is said to be a uniformly most accurate unbiased (UMAU) set. ********************* The concept of unbiasedness in tests carries over immediately to confidence sets. A family of confidence sets of size 1 − α is said to be unbiased if Pr S(X) ∋ ˜ θ ≤ 1 − α ∀ θ ∈ H1 ˜θ . In the case of nuisance parameters θu, unbiasedness means that this holds for all values of the nuisance parameters. In this case, similar regions and Neyman structure also are relevant, just as in the case of testing. 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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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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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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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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The proof of this is a classic: fir
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Definition 0.0.14 (superharmonic 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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(why?), and so We also have λ 0.1
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0.3 Some Basics of Linear Algebra 0
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The Fourier Expansion 0.3 Some Basi
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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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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
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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