Theory of Statistics - George Mason University

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156 2 Distribution Theory and Statistical Models assume that the data-generating process giving rise to a particular set of data is in the Poisson family of distributions, and based on our methods of inference decide it is the Poisson distribution with θ = 5. (See Appendix A for how θ parameterizes the Poisson family.) A very basic distinction is the nature of the values the random variable assumes. If the set of values is countable, we call the distribution “discrete”; otherwise, we call it “continuous”. With a family of probability distributions is associated a random variable space whose properties depend on those of the family. For example, the random variable space associated with a location-scale family (defined below) is a linear space. Discrete Distributions The probability measures of discrete distributions are dominated by the counting measure. One of the simplest types of discrete distribution is the discrete uniform. In this distribution, the random variable assumes one of m distinct values with probability 1/m. Another basic discrete distribution is the Bernoulli, in which random variable takes the value 1 with probability π and the value 0 with probability 1 − π. There are two common distributions that arise from the Bernoulli: the binomial, which is the sum of n iid Bernoullis, and the negative binomial, which is the number of Bernoulli trials before r 1’s are obtained. A special version of the negative binomial with r = 1 is called the geometric distribution. A generalization of the binomial to sums of multiple independent Bernoullis with different values of π is called the multinomial distribution. The random variable in the Poisson distribution takes the number of events within a finite time interval that occur independently and with constant probability in any infinitesimal period of time. A hypergeometric distribution models the number of selections of a certain type out of a given number of selections. A logarithmic distribution (also called a logarithmic series distribution) models phenomena with probabilities that fall off logarithmically, such as first digits in decimal values representing physical measures. Continuous Distributions The probability measures of continuous distributions are dominated by the Lebesgue measure. Continuous distributions may be categorized first of all by the nature of their support. The most common and a very general distribution with a finite interval as support is the beta distribution. Although we usually think of the support as [0, 1], it can easily be scaled into any finite interval [a, b]. Two parameters determine the shape of the PDF. It can have a U shape, a J shape, Theory of Statistics c○2000–2013 James E. Gentle
2 Distribution Theory and Statistical Models 157 a backwards J shape, or a unimodal shape with the mode anywhere in the interval and with or without an inflection point on either side of the mode. A special case of the beta has a constant PDF over the support. Another distribution with a finite interval as support is the von Mises distribution, which provides useful models of a random variable whose value is to be interpreted as an angle. One of the most commonly-used distributions of all is the normal or Gaussian distribution. Its support is ]−∞, ∞[. There are a number of distributions, called stable distributions, that are similar to the normal. The normal has a multivariate extension that is one of the simplest multivariate distributions, in the sense that the second-order moments have intuitive interpretations. It is the prototypic member of an important class of multivariate distributions, the elliptically symmetric family. From the normal distribution, several useful sampling distributions can be derived. These include the chi-squared, the t, the F, and the Wishart, which come from sample statistics from a normal distribution with zero mean. There are noncentral analogues of these that come from statistics that have not been centered on zero. Two other common distributions that are related to the normal are the lognormal and the inverse Gaussian distribution. These are related by applications. The lognormal is an exponentiated normal. (Recall two interesting properties of the lognormal: the moments do not determine the distribution (Exercise 1.28) and although the moments of all orders exist, the moment-generating function does not exist (Example 1.10).) The inverse Gaussian models the length of time required by Brownian motion to achieve a certain distance, while the normal distribution models the distance achieved in a certain time. Two other distributions that relate to the normal are the inverted chisquared and the inverted Wishart. They are useful because they are conjugate priors and they are also related to the reciprocal or inverse of statistics formed from samples from a normal. They are also called “inverse” distributions, but their origins are not at all related to that of the standard inverse Gaussian distribution. Continuous Distributions with Point Masses We can form a mixture of a continuous distribution and a discrete distribution. Such a distribution is said to be continuous with point masses. The probability measure is not dominated by either a counting measure or Lebesgue measure. A common example of such a distribution is the ɛ-mixture distribution, whose CDF is given in equation (2.27) on page 185. Entropy It is often of interest to know the entropy of a given probability distribution. In some applications we seek distributions with maximum or “large” entropy to 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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Page 127 and 128: 1.5 Conditional Probability 109 The
Page 129 and 130: Conditional Expectation over a Sub-
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Page 135 and 136: 1.5 Conditional Probability 117 If
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Page 139 and 140: Conditional Entropy 1.6 Stochastic
Page 141 and 142: 1.6 Stochastic Processes 123 Stoppi
Page 143 and 144: 1.6 Stochastic Processes 125 (Exerc
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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
Page 167 and 168: 1.36. Consider the the distribution
Page 169 and 170: Exercises 151 1.59. A sufficient co
Page 171 and 172: Exercises 153 1.85. Show that Doob
Page 173: 2 Distribution Theory and Statistic
Page 177 and 178: 2 Distribution Theory and Statistic
Page 179 and 180: 2 Distribution Theory and Statistic
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 187 and 188: 2.3.2 The Le Cam Regularity Conditi
Page 189 and 190: 2.4 The Exponential Class of Famili
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Page 201 and 202: 2.7 Truncated and Censored Distribu
Page 203 and 204: 2.9 Infinitely Divisible and Stable
Page 205 and 206: Higher Dimensions 2.11 The Family o
Page 207 and 208: 2.11 The Family of Normal Distribut
Page 213 and 214: Exercises 195 are considered at som
Page 215 and 216: Exercises 197 This law has been use
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
Page 223 and 224: 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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0.1.12 Transforms 0.1 Measure, Inte
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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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0.2.2.2 Ito’s Lemma 0.2 Stochasti
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0.3 Some Basics of Linear Algebra 0
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0.3.2.2 The Trace and Some of Its P
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The Fourier Expansion 0.3 Some Basi
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0.3.2.8 Spectral Decomposition 0.3
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0.3.2.11 Inverses of Matrices 0.3 S
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we have 0.3 Some Basics of Linear A
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f(x) ≈ f(x∗) + (x − x∗) T
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and so E T π (I + A) n−1 Eπ =
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0.4.1.2 Methods 0.4 Optimization 81
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0.4.1.7 The Steps in Iterative Algo
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0.4.1.11 Modifications of Newton’
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We generally require g x (k) ; x (
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0.4 Optimization 823 3. Generate st
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Theory of Statistics c○2000-2013
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A Important Probability Distributio
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Appendix A. Important Probability D
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B Useful Inequalities in Probabilit
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B.3 Pr(X ∈ A) and E(f(X)) 839 Pr(
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B.4 E(f(X)) and f(E(X)) 841 Theorem
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B.4 E(f(X)) and f(E(X)) 843 A relat
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B.5 E(f(X,Y )) and E(g(X)) and E(h(
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Now assume p > 1. Now, E(|X + Y | p
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C Notation and Definitions All nota
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C.1 General Notation 851 constant;
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C.2 General Mathematical Functions
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Functions of Convenience C.2 Genera
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C.2 General Mathematical Functions
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A1 ∩ A2 A1 − A2 A1∆A2 A1 × A
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C.4 Linear Spaces and Matrices 861
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a(ij) C.4 Linear Spaces and Matrice
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References The number(s) following
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References 867 Lyle D. Broemeling.
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References 869 William Feller. An I
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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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