Optimality

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14 G. J. Székely Table 1 � nx2 /(n − 1 + x2 )) = α Critical values for Gaussian scale mixture errors computed from t G n ( n − 1 0.125 0.100 0.050 0.025 2 1.625 1.886 2.920 4.303 3 1.495 1.664 2.353 3.182 4 1.440 1.579 2.132 2.776 5 1.410 1.534 2.015 2.571 6 1.391 1.506 1.943 2.447 7 1.378 1.487 1.895 2.365 8 1.368 1.473 1.860 2.306 9 1.361 1.462 1.833 2.262 10 1.355 1.454 1.812 2.228 11 1.351 1.448 1.796 2.201 12 1.347 1.442 1.782 2.179 13 1.344 1.437 1.771 2.160 14 1.341 1.434 1.761 2.145 15 1.338 1.430 1.753 2.131 16 1.336 1.427 1.746 2.120 17 1.335 1.425 1.740 2.110 18 1.333 1.422 1.735 2.101 19 1.332 1.420 1.730 2.093 20 1.330 1.419 1.725 2.086 21 1.329 1.417 1.722 2.080 22 1.328 1.416 1.718 2.074 23 1.327 1.414 1.715 2.069 24 1.326 1.413 1.712 2.064 25 1.325 1.412 1.709 2.060 100 1.311 1.392 1.664 1.984 500 1.307 1.387 1.652 1.965 1,000 1.307 1.386 1.651 1.962 Our approach can also be applied for two-sample tests. In a joint forthcoming paper with N. K. Bakirov the Behrens–Fisher problem will be discussed for Gaussian scale mixture errors with the help of our t G n (x) function. Acknowledgments The author also wants to thank many helpful suggestions of N. K. Bakirov, M. Rizzo, the referees of the paper, and the editor of the volume. References [1] Benjamini, Y. (1983). Is the t-test really conservative when the parent distribution is long-tailed? J. Amer. Statist. Assoc. 78, 645–654. [2] Eaton, M.L. (1974). A probability inequality for linear combinations of bounded random variables. Ann. Statist. 2,3, 609–614. [3] Edelman, D. (1990). An inequality of optimal order for the probabilities of the T statistic under symmetry. J. Amer. Statist. Assoc. 85, 120–123. [4] Efron, B. (1969). Student’s t-test under symmetry conditions. J. Amer. Statist. Assoc. 64, 1278–1302. [5] Efron, B. and Olshen, R. A. (1978). How broad is the class of normal scale mixtures? Ann. Statist.6, 5, 1159–1164.
Student’s t-test for scale mixture errors 15 [6] Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley, New York. [7] Gneiting, T. (1997). Normal scale mixtures and dual probability densities. J. Statist. Comput. Simul. 59, 375–384. [8] Kelker, D. (1971). Infinite divisibility and variance mixtures of the normal distribution. Ann. Math. Statist. 42, 2, 802–808. [9] Lehmann, E.L. (1999). ‘Student’ and small sample theory. Statistical Science 14, 4, 418–426. [10] Student (1929). Statistics in biological research. Nature 124, 93. [11] Székely, G.J. and N.K. Bakirov (under review). Generalized t-tests for unimodal and normal scale mixture errors.
Page 1 and 2: Institute of Mathematical Statistic
Page 3 and 4: Institute of Mathematical Statistic
Page 5 and 6: iv Contents Copulas and Decoupling
Page 7 and 8: vi Finally, thanks go the Statistic
Page 9 and 10: SCIENTIFIC PROGRAM The Second Erich
Page 11 and 12: x Recent Advances in Longitudinal D
Page 13 and 14: The Second Lehmann Symposium—Opti
Page 15 and 16: xiv Richard Davis Colorado State Un
Page 17 and 18: xvi Matthias Matheas Rice Universit
Page 19 and 20: xviii Hui Zhao University of Texas
Page 21 and 22: IMS Lecture Notes-Monograph Series
Page 23 and 24: Proof. The maximum LR test rejects
Page 25 and 26: 4. Location-scale families On likel
Page 27 and 28: 6. Conclusions On likelihood ratio
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Page 47 and 48: Optimality in multiple testing 27 M
Page 49 and 50: 6. Summary Optimality in multiple t
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Proof. See Appendix. Massive multip
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X1 Massive multiple hypotheses test
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Massive multiple hypotheses testing
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0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4
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Then π0α ∗ cal π0α ∗ cal +
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Massive multiple hypotheses testing
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IMS Lecture Notes-Monograph Series
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Frequentist statistics: theory of i
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2.3. Failure and confirmation Frequ
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3.1. Types of null hypothesis 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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