Theory of Statistics - George Mason University

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516 7 Statistical Hypotheses and Confidence Sets hence, the test is similar wrt P0. This condition on the critical region is called Neyman structure. The concepts of similarity and Neyman structure are relevant for unbiased tests, which we will consider in Section 7.2.3. Now suppose that U is boundedly complete sufficient for θu. If E(EH0(δ(X)|U)) = α, then the test has Neyman structure. While the power may still depend on θu, this fact may allow us to determine optimal tests of given size without regard to the nuisance parameters. 7.2.2 Uniformly Most Powerful Tests The probability of a type I error is limited to α or less. We seek a procedure that yields the minimum probability of a type II error, given that bound on the probability of a type I error. This would be a “most powerful” test. Ideally, the test would be most powerful for all values of θ ∈ Θ1. We call such a procedure a uniformly most powerful or UMP α-level test. For a given problem, finding such tests, or establishing that they do not exist, will be one of our primary objectives. The Neyman-Pearson Lemma gives us a way of determining whether a UMP test exists, and if so how to find one. The main issue is the likelihood ratio as a function of the parameter in the region specified by a composite H1. If the likelihood ratio is monotone, then we have a UMP based on the ratio. Generalizing the Optimal Test to Hypotheses of Intervals: UMP Tests Although it applies to a simple alternative (and hence “uniform” properties do not make much sense), the Neyman-Pearson Lemma gives us a way of determining whether a uniformly most powerful (UMP) test exists, and if so how to find one. We are often interested in testing hypotheses in which either or both of Θ0 and Θ1 are convex regions of IR (or IR k ). We must look at the likelihood ratio as a function both of a scalar parameter θ and of a scalar function of x. The question is whether, for given θ0 and any θ1 > θ0 (or equivalently any θ1 < θ0), the likelihood is monotone in some scalar function of x; that is, whether the family of distributions of interest is parameterized by a scalar in such a way that it has a monotone likelihood ratio (see page 167 and Exercise 2.5). In that case, it is clear that we can extend the test in (7.15) to be uniformly most powerful for testing H0 : θ = θ0 against an alternative H1 : θ > θ0 (or θ1 < θ0). Example 7.5 Testing hypotheses about the parameter in a Bernoulli distribution (continuation of Example 7.4) Theory of Statistics c○2000–2013 James E. Gentle
7.2 Optimal Tests 517 Syppose we assume a Bernoulli(π) distribution, and we wish to test H0 : π = 1/4 versus H1 : π > 1/4 at the level α = 0.05. The alternative hypothesis is certainly more reasonable than the one in Example 7.4. ************* modify Example 7.4 ****** also mention randomization *** don’t do it In this case, we do not have the problem of conflicting evidence mentioned on page 537. The exponential class of distributions is important because UMP tests are easy to find for families of distributions in that class. Discrete distributions are especially simple, but there is nothing special about them. Example 7.1 developed a test for H0 : θ ≤ θ0 versus the alternative H1 : θ > θ0 in a one-parameter exponential distribution, that is clearly UMP, as we can see by using the formulation (7.15) in a pointwise fashion. (The one-parameter exponential distribution, with density over the positive reals θ −1 e −x/θ is a member of the exponential class. Recall that the two-parameter exponential distribution used is not a member of the exponential family.) Let us first identify some classes of hypotheses. • simple versus simple • one-sided • two-sided; null on extremes H0 : θ = θ0 versus H1 : θ = θ1. (7.19) H0 : θ ≤ θ0 versus H1 : θ > θ0. (7.20) H0 : θ ≤ θ1 or θ ≥ θ2 versus H1 : θ1 < θ < θ2. (7.21) • two-sided; null in center H0 : θ1 ≤ θ ≤ θ2 versus H1 : θ < θ1 or θ > θ2. (7.22) We can examine tests of these hypotheses in the context of the exponential family of Example 7.1. Example 7.6 Testing in the exponential family Suppose we have observations X1, . . ., Xn iid ∼exponential(θ). In Example 7.1, we developed a “reasonable” test of the hypothesis H0 : θ ≤ θ0 versus H1 : θ > θ0. We will now use the test of equation (7.15) for these hypotheses, as well as the other hypotheses listed above. We will consider a test at the α level for each type of hypothesis. We will develop a test based on the statistic T(X) = X, which is sufficient for θ. For the given observations, the likelihood is L(θ; X) = θ −n e − P Xi/θ I]0,∞[(θ). • First, we wish to test H0 : θ = θ0 versus H1 : θ = θ1. 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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Page 517 and 518: Maximum Likelihood in Linear Models
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8.3 Nonparametric Estimation of Fun
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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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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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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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Variation of Functionals 0.2 Stocha
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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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0.3.2.8 Spectral Decomposition 0.3
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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,
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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