Theory of Statistics - George Mason University

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158 2 Distribution Theory and Statistical Models use as probability models. It is interesting to note that of all distributions with given first and second moments and having a PDF dominated by Lebesgue measure, the one with maximum entropy is the normal (Exercise 1.88). Characterizing a Family of Distributions A probability family or family of distributions, P = {Pθ, θ ∈ Θ}, is a set of probability distributions of a random variable that is defined over a given sample space Ω. The index of the distributions may be just that, an arbitrary index in some given set Θ which may be uncountable, or it may be some specific point in a given set Θ in which the value of θ carries some descriptive information about the distribution; for example, θ may be a 2-vector in which one element is the mean of the distribution and the other element is the variance of the distribution. The distribution functions corresponding to the members of most interesting families of distributions that we will discuss below do not constitute a distribution function space as defined on page 746. This is because mixtures of distributions in most interesting families of distributions are not members of the same family. That is, distributions defined by convex linear combinations of CDFs generally are not members of the same family of distributions. On the other hand, often linear combinations of random variables do have distributions in the same family of distributions as that of the individual random variables. (The sum of two normals is normal; but a mixture of two normals is not normal.) Table 1.1 on page 58 lists a number of families of distributions that are closed under addition of independent random variables. Likelihood Functions The problem of fundamental interest in statistics is to identify a particular distribution within some family of distributions, given observed values of the random variable. Hence, in statistics, we may think of θ or Pθ as a variable. A likelihood function is a function of that variable. Definition 2.1 (likelihood function) Given a PDF fθ, which is a function whose argument is a value of a random variable x, we define a likelihood function as a function of θ for the fixed x: L(θ | x) = fθ(x). The PDF fθ(x) is a function whose argument is a value of a random variable x for a fixed θ; the likelihood function L(θ | x) is a function of θ for a fixed x; see Figure 1.2 on page 21. In statistical applications we may be faced with the problem of choosing between two distributions Pθ1 and Pθ2. For a given value of x, we may base Theory of Statistics c○2000–2013 James E. Gentle
2 Distribution Theory and Statistical Models 159 our choice on the two likelihoods, L(θ1 | x) and L(θ2 | x), perhaps using the likelihood ratio λ(θ1, θ2|x) = L(θ2 | x) L(θ1 | x) . We have seen in equation (1.63) that the expectation of the likelihood ratio, taken wrt to distribution in the denominator, is 1. Parametric Families, Parameters, and Parameter Spaces In Definition 1.13, we say that a family of distributions on a measurable space (Ω, F) with probability measures Pθ for θ ∈ Θ is called a parametric family if Θ ⊆ IR d for some fixed positive integer d and θ fully determines the measure. In that case, we call θ the parameter and Θ the parameter space. A family that cannot be indexed in this way is called a nonparametric family. In nonparametric methods, our analysis usually results in some general description of the distribution, such as that the CDF is continuous or that the distribution has finite moments or is continuous, rather than in a specification of the distribution. The type of a family of distributions depends on the parameters that characterize the distribution. A “parameter” is a real number that can take on more than one value within a parameter space. If the parameter space contains only one point, the corresponding quantity characterizing the distribution is not a parameter. In most cases of interest the parameter space Θ is an open convex subset of IR d . In the N(µ, σ 2 ) family, for example, Θ = IR × IR+. In the binomial(n, π) family Θ =]0, 1[ and n is usually not considered a “parameter” because in most applications it is assumed to be fixed and known. In many cases, for a family with PDF pθ(x), the function ∂pθ(x)/∂θ exists and is an important characteristic of the family (see page 168). An example of a family of distributions whose parameter space is neither open nor convex is the hypergeometric(N, M, n) family (see page 830). In this family, as in the binomial, n is usually not considered a parameter because in most applications it is assumed known. Also, in most applications, either N or M is assumed known, but if they are both taken to be parameters, then Θ = {(i, j) : i = 2, 3, . . ., j = 1, . . ., i}. Obviously, in the case of the hypergeometric family, the function ∂pθ(x)/∂θ does not exist. Many common families are multi-parameter, and specialized subfamilies are defined by special values of one or more parameters. As we have mentioned and illustrated, a certain parameter may be referred to as a “location parameter” because it identifies a point in the support that generally locates the support within the set of reals. A location parameter may be a boundary point of the support of it may be the mean or a median of the distribution. Another parameter may be referred to as a “scale parameter” because it is associated with scale transformations of the random variable. The standard deviation of a normal random variable, for example, is the scale parameter of 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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Page 135 and 136: 1.5 Conditional Probability 117 If
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Page 149 and 150: 1.6 Stochastic Processes 131 We als
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Page 153 and 154: Convergence of Empirical Processes
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Page 157 and 158: Notes and Further Reading 139 and f
Page 159 and 160: Notes and Further Reading 141 we ca
Page 161 and 162: Notes and Further Reading 143 After
Page 163 and 164: Exercises 145 of betting system. Do
Page 165 and 166: Exercises 147 1.22. Show that if th
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Page 169 and 170: Exercises 151 1.59. A sufficient co
Page 171 and 172: Exercises 153 1.85. Show that Doob
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Page 181 and 182: Example 2.1 complete and incomplete
Page 183 and 184: 2.2 Shapes of the Probability Densi
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Page 213 and 214: Exercises 195 are considered at som
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Page 217 and 218: Exercises 199 a) Show that for d
Page 219 and 220: 3 Basic Statistical Theory The fiel
Page 221 and 222: How Does PH Differ from P? 3 Basic
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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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The sequence of sets {An} is said t
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The proof of this is a classic: fir
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• ∀x ∈ IR ∪ {∞}, x × ±
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0.0.5.2 Sets of Reals; Open, Closed
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We have 0.0 Some Basic Mathematical
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Definition 0.0.14 (superharmonic fu
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Ordering the Complex Numbers 0.0 So
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∞ 0.0 Some Basic Mathematical Con
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a) Evaluate the integral (0.0.84):
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0.1 Measure, Integration, and Funct
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3. if A1, A2, . . . ∈ F are disjo
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(why?), and so We also have λ 0.1
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Proof. Exercise. 0.1 Measure, Integ
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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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0.4.1.11 Modifications of Newton’
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We generally require g x (k) ; x (
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0.4 Optimization 823 3. Generate st
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Theory of Statistics c○2000-2013
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A Important Probability Distributio
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Appendix A. Important Probability D
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B Useful Inequalities in Probabilit
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B.3 Pr(X ∈ A) and E(f(X)) 839 Pr(
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B.4 E(f(X)) and f(E(X)) 841 Theorem
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B.4 E(f(X)) and f(E(X)) 843 A relat
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B.5 E(f(X,Y )) and E(g(X)) and E(h(
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Now assume p > 1. Now, E(|X + Y | p
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C Notation and Definitions All nota
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C.1 General Notation 851 constant;
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C.2 General Mathematical Functions
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Functions of Convenience C.2 Genera
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C.2 General Mathematical Functions
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A1 ∩ A2 A1 − A2 A1∆A2 A1 × A
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C.4 Linear Spaces and Matrices 861
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a(ij) C.4 Linear Spaces and Matrice
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References The number(s) following
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References 867 Lyle D. Broemeling.
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References 869 William Feller. An I
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References 871 H. O. Hartley. In Dr
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References 873 Feldman, and Murad S
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References 875 Roger B. Nelsen. An
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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