Theory of Statistics - George Mason University

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876 References Herbert Robbins. Statistical methods related to the law of the iterated logarithm. Annals of Mathematical Statistics, 41:1397–1409, 1970. 132 Christian P. Robert. The Bayesian Choice. Springer, New York, second edition, 2001. 377 L. C. G. Rogers and David Williams. Diffusions, Markov Processes, and Martingales: Volume 1, Foundations. Cambridge University Press, Cambridge, United Kingdom, second edition, 2000a. 772 L. C. G. Rogers and David Williams. Diffusions, Markov Processes, and Martingales: Volume 2, Itô Calculus. Cambridge University Press, Cambridge, United Kingdom, second edition, 2000b. 772 Joseph P. Romano and Andrew F. Siegel. Counterexamples in probability and statistics. Wadsworth & Brooks/Cole, Monterey, California, 1986. 79, 80, 82, 387, 397, 416, 417, 457, 480, 682, 754 Jutta Roosen and David A. Hennessy. Testing for the monotone likelihood ratio assumption. Journal of Business & Economic Statistics, 22:358–366, 2004. 554 Richard M. Royall. Statistical Evidence: A Likelihood Paradigm. Chapman & Hall/CRC, Boca Raton, 1997. 314, 537, 554 H. L. Royden. Real Analysis. MacMillan, New York, third edition, 1988. 754 Francisco J. Samaniego. A Comparison of the Bayesian and Frequentist Approaches to Estimation. Springer-Verlag, New York, 2010. 314 Gennady Samorodnitsky and Murad S. Taqqu. Stable Non-Gaussian Random Processes. Chapman & Hall/CRC, Boca Raton, 1994. 195 Carl-Erik Särndal, Bengt Swensson, and Jan Wretman. Model Assisted Survey Sampling. Springer-Verlag, New York, 1997. 435 Leonard J. Savage. The Foundations of Statistics. Dover Publications, New York, second revised edition, 1972. (First edition by John Wiley & Sons, New York, 1954.). 376 Henry Scheffé. A useful convergence theorem for probability distributions. Annals of Mathematical Statistics, 18:434–438, 1947. 83 Mark J. Schervish. Theory of Statistics. Springer, New York, 1995. 75, 114 Robert Schlaifer. Probability and Statistics for Business Decisions. McGraw- Hill Book Company, New York, 1959. 376 David W. Scott. Multivariate density estimation and visualization. In James E. Gentle, Wolfgang Härdle, and Yuichi Mori, editors, Handbook of Computational Statistics; Concepts and Methods, pages 549–570, Berlin, 2012. Springer. 574 David W. Scott. Multivariate Density Estimation. John Wiley & Sons, New York, 1992. 574 Shayle R. Searle, George Casella, and Charles E. McCulloch. Variance Components. Wiley-Interscience, New York, 1992. 499 Robert J. Serfling. Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York, 1980. 144, 317, 408, 439 J. Sethuraman. A constructive definition of dirichlet priors. Statistica Sinica, 4:639–650, 1994. 378 Theory of Statistics c○2000–2013 James E. Gentle
References 877 Glenn Shafer. A Mathematical Theory of Evidence. Princeton University Press, Princeton, New Jersey, 1976. 377 Glenn Shafer. Lindley’s paradox (with discussion by D. V. Lindley, Morris H. DeGroot, I. J. Good, Bruce M. Hill, and Robert E. Kass). Journal of the American Statistical Association, 77:325–351, 1982. 368 Jun Shao. Mathematical Statistics. Springer, New York, second edition, 2003. vii Jun Shao. Mathematical Statistics: Exercises and Solutions. Springer, New York, 2005. Jun Shao and Dongsheng Tu. The Jackknife and Bootstrap. Springer-Verlag, New York, 1995. 317 J. A. Shohat and J. D. Tamarkin. The Problem of Moments. American Mathematical Society, New York, 1943. 142, 143 Galen R. Shorack and Jon A. Wellner. Empirical Processes With Applications to Statistics (Classics in Applied Mathematics). Society for Industrial and Applied Mathematics, Philadelphia, 2009. (Originally published in 1986 by John Wiley & Sons. This “Classic Edition” includes errata as well as some updated material.). 145 Christopher G. Small. Expansions and Asymptotics for Statistics. Chapman & Hall/CRC, Boca Raton, 2010. 605 Christopher G. Small and Jinfang Wang. Numerical Methods for Nonlinear Estimating Functions. Oxford University Press, Oxford, 2003. 316 Daniel Solow. How to Read and Do Proofs. John Wiley & Sons, New York, third edition, 2003. 682 Ehsan S. Soofi. Capturing the intangible concept of information. Journal of the American Statistical Association, 89:1243–1254, 1994. 314 James C. Spall. Stochastic optimization. In James E. Gentle, Wolfgang Härdle, and Yuichi Mori, editors, Handbook of Computational Statistics; Concepts and Methods, pages 173–202, Berlin, 2012. Springer. 821 Robert G. Staudte and Simon J. Sheather. Robust Estimation and Testing. John Wiley & Sons, New York, 1990. 605 J. Michael Steele. Stochastic Calculus and Financial Applications. Springer- Verlag, New York, 2001. 772 J. Michael Steele. The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Cambridge University Press, Cambridge, United Kingdom, 2004. 848 Lynn Arthur Steen and J. Arthur Seebach Jr. Counterexamples in Topology. Dover Publications, Inc., Mineola, New York, 1995. (reprint of the second edition published by Springer-Verlag, New York, 1978). 754 Frederick F. Stephan. The expected value and variance of the reciprocal and other negative powers of a positive bernoullian variate. Annals of Mathematical Statistics, 16:50–61, 1945. 456 Fred W. Steutel. Preservation of Infinite Divisibility under Mixing and Related Topics. Mathematisch Centrum Amsterdam, Amsterdam, The Netherlands, 1970. 195 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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0.2.1.3 Ito Processes 0.2 Stochasti
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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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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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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 and 892: References 873 Feldman, and Murad S
Page 893: References 875 Roger B. Nelsen. An
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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