Optimality

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338 S. Juárez and W. R. Schucany So far we have not given explicit conditions for the existence of the matrices Jα and Kα as defined by (2.3) and (2.2), respectively. In order to complete the asymptotic analysis of the MDPDE we now do that. Condition 2 in Lemma 3 implicitly assumes the existence of Jα. This can be justified by observing that the condition that allows interchanging the order integration and differentiation in M(θ) is equivalent to the existence of Jα. For Jα to exist we need ijk(x; θ), the jk-element of the information matrix i(x;θ), to be such that � ijk(x; θ)f 1+α � (x;θ)dx
Asymptotics of the MDPDE 339 asymptotic concavity of mn(θ), would also give consistency of the MDPDE without requiring compactness of Θ, see Giurcanu and Trindade [3]. To decide which set of conditions are easier to verify seems to be more conveniently handled on a case by case basis. Acknowledgements The authors thank professor Javier Rojo for the invitation to present this work at the Second Symposium in Honor of Erich Lehmann held at Rice University. They are also indebted to the editor for his comments and suggestions which led to a substantial improvement of the article. Finally, the first author is deeply grateful to Professor Rojo for his proverbial patience during the preparation of this article. References [1] Basu, A., Harris, I. R., Hjort, N. L. and Jones, M. C. (1997). Robust and efficient estimation by minimising a density power divergence. Statistical Report No. 7, Department of Mathematics, University of Oslo. [2] Basu, A., Harris, I. R., Hjort, N. L., and Jones, M. C. (1998). Robust and efficient estimation by minimising a density power divergence. Biometrika, 85 (3), 549–559. [3] Giurcanu, M., and Trindade, A. A. (2005). Establishing consistency of Mestimators under concavity with an application to some financial risk measures. Paper available athttp:www.stat.ufl.edu/ trindade/papers/concave.pdf. [4] Juárez, S. F. (2003). Robust and efficient estimation for the generalized Pareto distribution. Ph.D. dissertation. Statistical Science Department, Southern Methodist University. Available at http://www.smu.edu/statistics/ faculty/SergioDiss1.pdf. [5] Lehmann, E. L. and Casella, G. (1998). Theory of Point Estimation. Springer, New York. [6] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press, New York.
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Institute of Mathematical Statistic
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Institute of Mathematical Statistic
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iv Contents Copulas and Decoupling
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vi Finally, thanks go the Statistic
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SCIENTIFIC PROGRAM The Second Erich
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x Recent Advances in Longitudinal D
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The Second Lehmann Symposium—Opti
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xiv Richard Davis Colorado State Un
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xvi Matthias Matheas Rice Universit
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xviii Hui Zhao University of Texas
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IMS Lecture Notes-Monograph Series
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Proof. The maximum LR test rejects
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4. Location-scale families On likel
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6. Conclusions On likelihood ratio
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Student’s t-test for scale mixtur
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Optimality in multiple testing 17 w
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Optimality in multiple testing 19 I
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power 0.02 0.03 0.04 0.05 0.06 0.07
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Optimality in multiple testing 23 i
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Optimality in multiple testing 25 (
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Optimality in multiple testing 27 M
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6. Summary Optimality in multiple t
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Optimality in multiple testing 31 [
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On the false discovery proportion 3
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≤ N� � N� P m=1 On the fals
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Since j≤ s, it follows that On th
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0.025 0.02 0.015 0.01 0.005 On the
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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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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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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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