Optimality

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166 D. M. Dabrowska Part (i). For u > s, set c((s, u]) = c(u)−c(s). The n-the term of the series Ψ0n(s, t) is given by the multiple integral � c((s, s1])b(ds1)c((s1, s2])b(ds2)···c((sn−1, sn])b(dsn), s
Semiparametric transformation models 167 The first pair of equations for Ψ0 and Ψ2 in part (i) follows by setting g1(s, t) = 1 = g3(s, t). With s fixed, the equations � ¯h1(s, t)− ¯h1(s, u+)b(du)c1((u, t)) = ¯g1(s, t), have solutions � ¯h3(s, t)− [s,t) (s,t] � ¯h1(s, t) = ¯g1(s, t) + � ¯h3(s, t) = ¯g3(s, t) + ¯h3(s, u−)c(du)b3([u, t]) = ¯g3(s, t), [s,t) (s,t] ¯g1(s, u+)b(du)Ψ1(u, t−), ¯g3(s, u−)c(du)Ψ3(u, t+). The second pair of equations for Ψ0 and Ψ2 in part (i) follows by setting ¯g1(s, t)≡ 1 ≡ ¯g3(s, t). Next, the “odd” functions can be represented in terms of “even” functions using Fubini. 9. Gronwall’s inequalities Following Gill and Johansen [18], recall that if b is a cadlag function of bounded variation,�b�v≤ r1 then the associated product integralP(s, t) =π(s,t] (1+b(du)) satisfies the bound|P(s, t)|≤π(s,t] (1 +�b�v(dw))≤exp�b�v(s, t] uniformly in 0 < s < t≤τ. Moreover, the functions s→P(s, t), s≤t≤τ and t→P(s, t), t∈ (s, τ] are of bounded variation with variation norm bounded by r1er1 . The proofs use the following consequence of Gronwall’s inequalities in Beesack [3] and Gill and Johansen [18]. If b is a nonnegative measure and y∈ D([0, τ]) is a nonnegative function then for any x∈D([0, τ]) satisfying � 0≤x(t)≤y(t) + x(u−)b(du), t∈[0, τ], we have � 0≤x(t)≤y(t) + Pointwise in t,|x(t)| is bounded by max{�y�∞,�y − � �∞}[1 + (0,t] (0,t] (0,t] y(u−)b(du)P(u, t), t∈[0, τ]. b(du)P(u, t)]≤{�y�∞,�y − � t �∞} exp[ b(du)]. 0 We also have�e −b |x|�∞≤ max{�y�∞,�y − �∞}. Further, if 0�≡ y∈ D([0, τ]) and b is a function of bounded variation then the solution to the linear Volterra equation x(t) = y(t) + is unique and given by � x(t) = y(t) + (0,t] � t 0 x(u−)b(du) y(u−)b(du)P(u, t).
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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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Page 135 and 136: Statistical models: problem of spec
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Page 153 and 154: Semiparametric transformation model
Page 155 and 156: 2.1. The model Semiparametric trans
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Page 191 and 192: Bayesian transformation hazard mode
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Page 205 and 206: Copulas, information, dependence an
Page 231 and 232: Regression trees 211 right model in
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Page 235 and 236: 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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