Optimality

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Acknowledgement of referees’ services The efforts of the following referees are gratefully acknowledged Jose Luis Batun CIMAT, Mexico Roger Berger Arizona State University Prabir Burman University of California, Davis Ray Carroll Texas A&M University Cheng Cheng St. Jude’s Children’s Research Hospital David R. Cox Nuffield College, Oxford Dorota M. Dabrowska University of California, Los Angeles Victor H. de la Pena Columbia University Kjell Doksum University of Wisconsin, Madison Armando Dominguez CIMAT, Mexico Sandrine Dudoit University of California, Berkeley Richard Dykstra University of Iowa Bradley Efron Stanford University Harnmou El Barmi The City University of New York Luis Enrique Figueroa Purdue University Joseph L. Gastwirth George Washington University Marc G. Genton Texas A&M University Musie Ghebremichael Yale University Graciela Gonzalez Mexico Hannes Leeb Yale University Erich L. Lehmann University of California, Berkeley Ker-Chau Li University of California, Los Angeles Wei-Yin Loh University of Wisconsin, Madison Hari Mukerjee Wichita State University Loki Natarajan University of California, San Diego Ingram Olkin Stanford University Liang Peng Georgia Institute of Technology Joseph P. Romano Stanford University Louise Ryan Harvard University Sanat Sarkar Temple University xix William R. Schucany Southern Methodist University David W. Scott Rice University Juliet P. Shaffer University of California, Berkeley Nozer D. Singpurwalla George Washington University David Sprott CIMAT and University of Waterloo Jef L. Teugels Katholieke Universiteit Leuven Martin J. Wainwright University of California, Berkeley Jane-Ling Wang University of California, Davis Peter Westfall Texas Tech University Grace Yang University of Maryland, College Park Yannis Yatracos (2) National University of Singapore Guosheng Yin University of Texas MD Anderson Cancer Center Hongyu Zhao Yale University
IMS Lecture Notes–Monograph Series 2nd Lehmann Symposium – <strong>Optimality</strong> Vol. 49 (2006) 1–8 c○ Institute of Mathematical Statistics, 2006 DOI: 10.1214/074921706000000356 On likelihood ratio tests Erich L. Lehmann 1 University of California at Berkeley Abstract: Likelihood ratio tests are intuitively appealing. Nevertheless, a number of examples are known in which they perform very poorly. The present paper discusses a large class of situations in which this is the case, and analyzes just how intuition misleads us; it also presents an alternative approach which in these situations is optimal. 1. The popularity of likelihood ratio tests Faced with a new testing problem, the most common approach is the likelihood ratio (LR) test. Introduced by Neyman and Pearson in 1928, it compares the maximum likelihood under the alternatives with that under the hypothesis. It owes its popularity to a number of facts. (i) It is intuitively appealing. The likelihood of θ, Lx(θ) = pθ(x) i.e. the probability density (or probability) of x considered as a function of θ, is widely considered a (relative) measure of support that the observation x gives to the parameter θ. (See for example Royall [8]). Then the likelihood ratio (1.1) sup[pθ(x)]/ sup[pθ(x)] alt hyp compares the best explanation the data provide for the alternatives with the best explanations for the hypothesis. This seems quite persuasive. (ii) In many standard problems, the LR test agrees with tests obtained from other principles (for example it is UMP unbiased or UMP invariant). Generally it seems to lead to satisfactory tests. However, counter-examples are also known in which the test is quite unsatisfactory; see for example Perlman and Wu [7] and Menéndez, Rueda, and Salvador [6]. (iii) The LR test, under suitable conditons, has good asymptotic properties. None of these three reasons are convincing. (iii) tells us little about small samples. (i) has no strong logical grounding. (ii) is the most persuasive, but in these standard problems (in which there typically exist a complete set of sufficient statistics) all principles typically lead to tests that are the same or differ only by little. 1 Department of Statistics, 367 Evans Hall, University of California, Berkeley, CA 94720-3860, e-mail: shaffer@stat.berkeley.edu AMS 2000 subject classifications: 62F03. Keywords and phrases: likelihood ratio tests, average likelihood, invariance. 1
Page 1 and 2: Institute of Mathematical Statistic
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Page 5 and 6: iv Contents Copulas and Decoupling
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Page 9 and 10: SCIENTIFIC PROGRAM The Second Erich
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IMS Lecture Notes-Monograph Series
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Massive multiple hypotheses testing
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P value EDF qdf 0.0 0.2 0.4 0.6 0.8
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Proof. See Appendix. Massive multip
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X1 Massive multiple hypotheses test
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Then π0α ∗ cal π0α ∗ cal +
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Frequentist statistics: theory of i
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2.3. Failure and confirmation Frequ
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together with the probabilistic ass
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Statistical models: problem of spec
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Modeling inequality, spread in mult
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Semiparametric transformation model
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2.1. The model Semiparametric trans
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6.3. Part (ii) Put (6.1) (6.2) ˙Γ
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8. Proof of Proposition 2.3 Semipar
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Bayesian transformation hazard mode
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Cumulative Hazard 0.0 0.2 0.4 0.6 0
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Hazard function 0.0 0.5 1.0 1.5 2.0
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Copulas, information, dependence an
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Regression trees 211 right model in
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Abs(effects) 0.00 0.05 0.10 0.15 0.
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Regression trees 215 of the tree. T
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Regression trees 217 GUIDE methods
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Regression trees 219 and E appear a
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Regression trees 221 Table 3 Result
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Solder =thick 2.5 Regression trees
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Observed values 0 10 20 30 40 50 60
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Regression trees 227 effects and in
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On Competing risk and degradation p
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Restricted estimation in competing
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9. Conclusion 0.0 0.1 0.2 0.3 0.4 0
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2.1. Maximin efficiency robust test
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Robust tests for genetic associatio
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1 . . . 1 i . . . . . . R j . . . O
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Optimal sampling strategies 269 as
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Optimal sampling strategies 271 γ1
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Optimal sampling strategies 273 Def
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Optimal sampling strategies 275 Usi
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Optimal sampling strategies 277 Def
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optimal 1 2 3 4 leaf sets 5 6 (leaf
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6. Related work Optimal sampling st
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Optimal sampling strategies 283 Cla
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Optimal sampling strategies 285 Sin
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Optimal sampling strategies 287 µ
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Optimal sampling strategies 289 on
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Linear prediction after model selec
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y the P× 1 vector (3.1) ηn(p) = L
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Appendix B: Proofs for Section 5 Li
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Local asymptotic minimax risk/asymm
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Then (3.5) Local asymptotic minimax
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Moment-density estimation 323 may n
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3. The bias and MSE of ˆf ∗ α M
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Moment-density estimation 327 Corol
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Moment-density estimation 329 Suppo
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6. Simulations Moment-density estim
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Moment-density estimation 333 [7] M
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for positive α, and for α = 0 as
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Asymptotics of the MDPDE 337 Lemma
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Asymptotics of the MDPDE 339 asympt
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Optimality

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