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874 References E. L. Lehmann. A general concept of unbiasedness. Annals of Mathematical Statistics, 22:587–592, 1951. 316 E. L. Lehmann. Elements of Large-Sample Theory. Springer, New York, 1999. 317 E. L. Lehmann and George Casella. Theory of Point Estimation. Springer, New York, second edition, 1998. vii E. L. Lehmann and Joseph P. Romano. Testing Statistical Hypotheses. Springer, New York, third edition, 2005. vii D. V. Lindley. A statistical paradox. Biometrika, 44:187–192, 1957. 367 D. V. Lindley and L. D. Phillips. Inference for a Bernoulli process (A Bayesian view). The American Statistician, 30:112–119, 1976. 1, 377, 554 J. E. Littlewood. Lectures on the Theory of Functions. Oxford University Press, Oxford, UK, 1944. 719 Thomas A. Louis. Finding the observed information matrix when using the EM algorithm. Journal of the the Royal Statistical Society, Series B, 44: 226–233, 1982. 475 Eugene Lukacs. Characteristic Functions. Hafner Publishing Company, New York, second edition, 1970. 142 Charles F. Manski. Analog Estimation Methods in Econometrics. Chapman & Hall, New York, 1988. 315 P. Massart. The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Annals of Probability, 18:1269–1283, 1990. 135, 145 Sharon Bertsch McGrayne. The Theory that Would Not Die. Yale University Press, New Haven, 2011. 377 X.-L. Meng and D. B. Rubin. Maximum likelihood estimation via the ECM algorithm: A general framework. Biometrika, 80:267–278, 1993. 475 N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller. Equations of state calculation by fast computing machines. Journal of Chemical Physics, 21:1087–1092, 1953. (Reprinted in Samuel Kotz and Norman L. Johnson (Editors) (1997), Breakthroughs in Statistics, Volume III, Springer-Verlag, New York, 127–139.). 822 Sean Meyn and Richard L. Tweedie. Markov Chains and Stochastic Stability. Cambridge University Press, Cambridge, United Kingdom, second edition, 2009. 144 Thomas Mikosch. Copulas: Tales and facts (with discussion). Extremes, 9: 3–66, 2006. 142 B. J. T. Morgan, K. J. Palmer, and M. S. Ridout. Negative score test statistic. The American Statistician, 61:285–288, 2007. 529, 531 Carl N. Morris. Natural exponential families with quadratic variance functions. Annals of Statistics, 10:65–80, 1982. 195 Carl N. Morris and Kari F. Lock. Unifying the named natural exponential families and their relatives. The American Statistician, 63:247–253, 2009. 173, 195, 829 MS2. Shao (2003). 228, 267, 271, 283, 312, 393, 394, 409, 414, 415, 416, 436, 438, 482, 521, 525, 526, 527, 542, 543, 547, 554, 604 Theory of Statistics c○2000–2013 James E. Gentle
References 875 Roger B. Nelsen. An Introduction to Copulas. Springer, New York, second edition, 2006. 40 J. Neyman and E. L. Scott. Consistent estimates based on partially consistent observations. Econometrica, 16:1–32, 1948. 499 Shu Kay Ng, Thriyambakam Krishnan, and Geoffrey J. McLachlan. The EM algorithm. In James E. Gentle, Wolfgang Härdle, and Yuichi Mori, editors, Handbook of Computational Statistics; Concepts and Methods, pages 139– 172, Berlin, 2012. Springer. 498 Bernt Øksendal. Stochastic Differential Equations. An Introduction with Applications. Springer, Heidelberg, fifth edition, 1998. 772 Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark, editors. NIST Handbook of Mathematical Functions. National Institute of Standards and Technology and Cambridge University Press, Gaithersburg, Maryland and Cambridge, United Kingdom, 2010. Available at http://dlmf.nist.gov/. 462, 856 Art B. Owen. Empirical Likelihood. Chapman & Hall/CRC, Boca Raton, 2001. 499 Leandro Pardo. Statistical Inference Based on Divergence Measures. Chapman & Hall/CRC, Boca Raton, 2005. 315, 739 Valentin V. Petrov. Limit Theorems of Probability Theory. Oxford University Press, Oxford, United Kingdom, 1995. 102, 143, 848 E. J. G. Pitman. Subexponential distribution functions. Journal of the Australian Mathematical Society (Series A), 29:337–347, 1980. 196 E. J. G. Pitman. The “closest” estimates of statistical parameters. Communications in Statistics — Theory and Methods, 20:3423–3437, 1991. 315 David Pollard. A User’s Guide to Mearure Theoretic Probability. Cambridge University Press, Cambridge, United Kingdom, 2003. second printing with corrections. 145 Raquel Prado and Mike West. Time Series: Modelling, Computation and Inference. Chapman & Hall/CRC Press, Boca Raton, 2010. 378 S. James Press. Estimation in univariate and multivariate stable distributions. Journal of the American Statistical Association, 67:842–846, 1972. 248 S. James Press and Judith M. Tanur. The Subjectivity of Scientists and the Bayesian Approach. Wiley-Interscience, New York, 2001. 313, 314 A. R. Rajwade and A. K. Bhandari. Surprises and Counterexamples in Real Function Theory. Hindustan Book Agency, New Delhi, 2007. 754 C. R. Rao. Some comments on the minimum mean square as a criterion of estimation. In M. Csörgö, D. A. Dawson, J. N. K. Rao, and A. K. Md. E. Saleh, editors, Statistics and Related Topics, pages 123–143, Amsterdam, 1981. North-Holland. 214, 315 J. N. K. Rao. Impact of frequentist and Bayesian methods on survey sampling practice: A selective appraisal (with discussion). Statistical Science, 26:240– 270, 2011. 378 J. N. K. Rao and J. T. Webster. On two methods of bias reduction in the estimation of ratios. Biometrika, 53:571–577, 1966. 299 Theory of Statistics c○2000–2013 James E. Gentle
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James E. Gentle Theory of Statistic
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Preface: Mathematical Statistics Af
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Preface vii The objective in the di
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x = ⎛ ⎜ ⎝ x1 . xd ⎞ ⎟ ⎠
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• State and prove Fatou’s lemma
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Contents Preface . . . . . . . . .
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Contents xv 2.10 Multivariate Distr
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Contents xvii 5.2.1 Expectation Fun
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Contents xix 8.5.1 Nonparametric Pr
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1 Probability Theory Probability th
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1.1 Some Important Probability Fact
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Definition 1.8 (exchangeability) Le
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Theorem 1.5 (properties of a CDF) I
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.5 PDF p X(x;θ) 0 x θ=1 θ=5 1.1
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Joint and Marginal Distributions 1.
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Moment-Generating Functions 1.1 Som
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A Taylor series expansion of this g
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1.2 Series Expansions 65 X is the u
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κ1 = E(Z) κ2 = E(Z 2 ) − (E(Z))
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1.3 Sequences of Events and of Rand
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We write 1.3 Sequences of Events an
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1.3.6 Convergence of Functions 1.3
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we have for fixed k, 1.3 Sequences
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Multivariate Asymptotic Expectation
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1.4 Limit Theorems 103 The bn can b
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1.4 Limit Theorems 105 it applies t
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lim n→∞ max σ j≤kn 2 nj σ2
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1.5 Conditional Probability 109 The
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Conditional Expectation over a Sub-
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• monotone convergence: for 0 ≤
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Fn(t) = ⌊nt⌋ + 1 n + 1 ; 1.5 Co
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1.5 Conditional Probability 117 If
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1.5 Conditional Probability 119 1.5
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Conditional Entropy 1.6 Stochastic
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1.6 Stochastic Processes 123 Stoppi
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1.6 Stochastic Processes 125 (Exerc
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1.6 Stochastic Processes 127 variab
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1.6 Stochastic Processes 129 in a d
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1.6 Stochastic Processes 131 We als
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1.6 Stochastic Processes 133 X0 has
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Convergence of Empirical Processes
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and Fn(x, ω) ≥ Fn(xm,k−1, ω)
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Notes and Further Reading 139 and f
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Notes and Further Reading 141 we ca
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Notes and Further Reading 143 After
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Exercises 145 of betting system. Do
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Exercises 147 1.22. Show that if th
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1.36. Consider the the distribution
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Exercises 151 1.59. A sufficient co
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Exercises 153 1.85. Show that Doob
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2 Distribution Theory and Statistic
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Example 2.1 complete and incomplete
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2.2 Shapes of the Probability Densi
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2.2 Shapes of the Probability Densi
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2.3.2 The Le Cam Regularity Conditi
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2.4 The Exponential Class of Famili
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or or 2.5 Parametric-Support Famili
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2.6 Transformation Group Families 1
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2.6 Transformation Group Families 1
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2.7 Truncated and Censored Distribu
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2.9 Infinitely Divisible and Stable
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Higher Dimensions 2.11 The Family o
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2.11 The Family of Normal Distribut
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Exercises 195 are considered at som
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Exercises 197 This law has been use
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Exercises 199 a) Show that for d
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3 Basic Statistical Theory The fiel
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How Does PH Differ from P? 3 Basic
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3 Basic Statistical Theory 205 abov
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3.1 Inferential Information in Stat
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The Basic Paradigm of Point Estimat
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Completeness 3.1 Inferential Inform
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and so by completeness, 3.1 Inferen
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and so I(µ, σ) = Eθ = E(µ,σ) =
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Robustness 3.1 Inferential Informat
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3.2 Statistical Inference: Approach
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3.3 The Decision Theory Approach to
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Risk 0.005 0.010 0.015 0.020 0.025
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3.4 Invariant and Equivariant Stati
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Equivariant Point Estimation 3.4 In
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3.5 Probability Statements in Stati
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Test Rules 3.5 Probability Statemen
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if 3.6 Variance Estimation 297 Give
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3.6 Variance Estimation 299 J(T) =
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3.7 Applications 3.7.1 Inference in
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3.8 Asymptotic Inference 303 The ca
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3.8 Asymptotic Inference 305 The mo
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3.8 Asymptotic Inference 307 wherea
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3.8 Asymptotic Inference 309 tendin
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Definition 3.20 (asymptotic signifi
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Proof. *************** Notes and Fu
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Notes and Further Reading 315 Least
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Approximations and Asymptotic Infer
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Exercises 319 b) the expectation of
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4 Bayesian Inference We have used a
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4.1 The Bayesian Paradigm 323 For m
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4.1 The Bayesian Paradigm 325 A qua
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4.2 Bayesian Analysis 327 even if t
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4.2 Bayesian Analysis 329 Hence P(B
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4.2 Bayesian Analysis 331 B1. ∀θ
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4.2 Bayesian Analysis 333 This mean
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Prior Prior 0.0 0.5 1.0 1.5 2.0 0 2
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4.2 Bayesian Analysis 337 We go thr
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4.2 Bayesian Analysis 339 We constr
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4.2 Bayesian Analysis 341 as the mo
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Assessing the Problem Formulation 4
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Posterior Posterior 0 5 10 15 0 1 2
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4.2 Bayesian Analysis 347 distribut
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4.3 Bayes Rules 349 • the nature
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Then we have E(g(θ)T(X)) = E(T(X)E
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4.3 Bayes Rules 353 equivariance Fo
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4.3 Bayes Rules 355 Example 4.9 (Co
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p0 = Pr(Θ ∈ Θ0) and p1 = Pr(Θ
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The Weighted 0-1 or α0-α1 Loss Fu
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The term ˆp0 ˆp1 = p0 p1 BF(x) =
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4.5 Bayesian Testing 365 • Bayes
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where fΘ|x(θ|x) = p1 if θ = θ0
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4.6 Bayesian Confidence Sets 369 as
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Posterior 0 1 2 3 95% Credible Regi
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Markov Chain Monte Carlo 4.7 Comput
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4.7 Computational Methods 375 β/(t
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Notes and Further Reading 377 An al
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4.4. Given the conditional PDF a) U
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Exercises 381 4.14. Consider again
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Exercises 383 4.26. As in Exercise
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5 Unbiased Point Estimation In a de
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Estimability 5 Unbiased Point Estim
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s1π + s0(1 − π) = π, 5.1 UMVUE
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5.1 UMVUE 391 Let T be UMVUE for g(
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5.1 UMVUE 393 Example 5.7 UMVUE of
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5.1 UMVUE 395 The risk of T ∗ is
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5.1 UMVUE 397 estimator of θ that
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(Compare this with Tfdx = g(θ).)
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¯h(X1, . . ., Xm) = 1 m! 5.2 U-Sta
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where Pn is the ECDF. U(X1, . . .,
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and T 2 r,S = C = Tr,S = C C TnT
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Generalized U-Statistics 5.2 U-Stat
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5.2 U-Statistics 409 Theorem 5.4 (H
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5.3 Asymptotically Unbiased Estimat
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5.3.2 Ratio Estimators 5.3 Asymptot
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5.4 Asymptotic Efficiency 415 of a
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⎧ ⎨ Tn = ⎩ n2 with probabilit
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5.5 Applications 5.5 Applications 4
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5.5 Applications 421 one aspect of
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Proof. Because l ∈ span(X T ) = s
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Optimal Properties of the Moore-Pen
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5.5 Applications 427 implies implie
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The “Sum of Squares” Quadratic
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5.5 Applications 431 The question o
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We now write the original model as
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5.5 Applications 435 it from other
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(it’s Bernoulli), and for i = j,
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Fisher Efficient Estimators and Exp
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6 Statistical Inference Based on Li
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6.1 The Likelihood Function 443 is
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6.2 Maximum Likelihood Parametric E
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6.2.2 Finite Sample Properties of M
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where Now, consider This has two pa
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6.3 Asymptotic Properties of MLEs,
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6.4 Application: MLEs in Generalize
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Residuals 6.4 Application: MLEs in
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6.5 Variations on the Likelihood 49
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6.5 Variations on the Likelihood 49
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Maximum Likelihood in Linear Models
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Exercises 501 b) Assume that σ 2 1
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7 Statistical Hypotheses and Confid
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Tests of Hypotheses 7.1 Statistical
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7.1 Statistical Hypotheses 507 Know
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Power of a Statistical Test 7.1 Sta
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An Optimal Test in a Simple Situati
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7.2 Optimal Tests 513 All of these
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Use of Sufficient Statistics 7.2 Op
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7.2 Optimal Tests 517 Syppose we as
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Nonexistence of UMP Tests 7.2 Optim
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H0 : θ = θ0 versus H1 : θ = θ0.
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7.2.6 Equivariance, Unbiasedness, a
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7.3 Likelihood Ratio Tests, Wald Te
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7.3 Likelihood Ratio Tests, Wald Te
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Minimize 7.3 Likelihood Ratio Tests
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7.4 Nonparametric Tests 531 I(0, ˆ
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Yij = µ + αi + ɛij, i = 1, . . .
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7.7 The Likelihood Principle and Te
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7.8 Confidence Sets 537 Monte Carlo
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7.8 Confidence Sets 539 The concept
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7.8 Confidence Sets 541 For any giv
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7.9 Optimal Confidence Sets 543 Thi
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7.9 Optimal Confidence Sets 545 Con
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7.10 Asymptotic Confidence sets 547
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7.11 Bootstrap Confidence Sets 549
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Bias in Intervals Due to Bias in th
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7.12 Simultaneous Confidence Sets 5
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Sequential Tests Exercises 555 Wald
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8 Nonparametric and Robust Inferenc
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8.2 Inference Based on Order Statis
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f ^ (x) Nonparametric Regression 0.
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8.3 Nonparametric Estimation of Fun
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ISE θ = 8.3 Nonparametric Estim
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8.4 Semiparametric Methods and Part
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8.5 Nonparametric Estimation of PDF
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Some Properties of the Histogram Es
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AISB 1 pH = 12 8.5 Nonparametric
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and the integrated variance is 8.5
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and 8.5 Nonparametric Estimation o
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h ∗ = 8.5 Nonparametric Estimatio
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Computation of Kernel Density Estim
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8.6 Perturbations of Probability Di
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CDF 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
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8.6 Perturbations of Probability Di
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8.7 Robust Inference 8.7 Robust Inf
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p(y) (1-ε)p(y) The Influence Funct
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8.7 Robust Inference 603 Notice tha
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Nonparametric Statistics Exercises
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0 Statistical Mathematics Statistic
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0 Statistical Mathematics 609 x may
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0.0 Some Basic Mathematical Concept
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Because each is a subset of the oth
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0.0.2 Sets and Spaces 0.0 Some Basi
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0.0.2.4 Point Sequences in a Topolo
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The sequence of sets {An} is said t
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The proof of this is a classic: fir
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• ∀x ∈ IR ∪ {∞}, x × ±
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0.0.5.2 Sets of Reals; Open, Closed
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We have 0.0 Some Basic Mathematical
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Definition 0.0.14 (superharmonic fu
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Ordering the Complex Numbers 0.0 So
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0.0.9.5 Working with Real-Valued Fu
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∞ 0.0 Some Basic Mathematical Con
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a) Evaluate the integral (0.0.84):
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0.1 Measure, Integration, and Funct
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3. if A1, A2, . . . ∈ F are disjo
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(why?), and so We also have λ 0.1
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Proof. Exercise. 0.1 Measure, Integ
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0.2 Stochastic Processes and the St
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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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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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Page 861 and 862: B.4 E(f(X)) and f(E(X)) 843 A relat
Page 863 and 864: B.5 E(f(X,Y )) and E(g(X)) and E(h(
Page 865 and 866: Now assume p > 1. Now, E(|X + Y | p
Page 867 and 868: C Notation and Definitions All nota
Page 869 and 870: C.1 General Notation 851 constant;
Page 871 and 872: C.2 General Mathematical Functions
Page 873 and 874: Functions of Convenience C.2 Genera
Page 875 and 876: C.2 General Mathematical Functions
Page 877 and 878: A1 ∩ A2 A1 − A2 A1∆A2 A1 × A
Page 879 and 880: C.4 Linear Spaces and Matrices 861
Page 881 and 882: a(ij) C.4 Linear Spaces and Matrice
Page 883 and 884: References The number(s) following
Page 885 and 886: References 867 Lyle D. Broemeling.
Page 887 and 888: References 869 William Feller. An I
Page 889 and 890: References 871 H. O. Hartley. In Dr
Page 891: References 873 Feldman, and Murad S
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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