Optimality

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198 V. H. de la Peña, R. Ibragimov and S. Sharakhmetov Theorem 7.5. The following inequalities hold: P (h(X1, . . . , Xn)>x)≤P(h(ξ1, . . . , ξn) > x) + φX1,...,Xn (P(h(ξ1, . . . , ξn) > x)) 1 2 , P (h(X1, . . . , Xn) > x)≤ � 1 + φ 2 X1,...,Xn � 1/2 (P(h(ξ1, . . . , ξn) > x)) 1/2 , P(h(X1, . . . , Xn) > x)≤(e−1)P(h(ξ1, . . . , ξn) > x) + δX1,...,Xn, P(h(X1, . . . , Xn) > x)≤ x∈R, where ψ(x) =|x| q/(q−1) − 1. 8. Appendix: Proofs � 1 + D ψ � 1 (1− q X1,...,Xn ) (P(h(ξ1, . . . , ξn) > x)) 1 q , q > 1, Proof of Theorem 2.1. Let us first prove the necessity part of the theorem. Denote T(x1, . . . , xn) = � x1 −∞ ··· � xn −∞ Let k∈{1, . . . , n}, xk∈ R. Let us show that (8.1) (1 + Un(t1, . . . , tn)) T(∞, . . . ,∞, xk,∞, . . . ,∞) = Fk(xk), n� dFi(ti). xk∈ R, k = 1, . . . , n. It suffices to consider the case k = 1. We have T(x1,∞, . . . ,∞) � x1 � ∞ � ∞ = . . . (1 + Un(t1, . . . , tn)) −∞ � −∞ �� −∞ � n = F1(x1) + n� = F1(x1) + Σ ′′ . � c=2 1≤i1
Copulas, information, dependence and decoupling 199 . . . , xn)−T(x1, . . . , xk−1, a, xk+1, . . . , xn), a < b. By integrability of the functions gi1,...,ic and condition A3 we obtain (I(·) denotes the indicator function) (8.3) δ 1 (a1,b1] δ2 (a2,b2] ···δn (an,bn] T(x1, . . . , xn) n� � = P(ai < ξi≤ bi) + E Un(ξi1, . . . , ξin) i=1 n� � I(ai < ξi≤ bi) ≥ 0 for all ai < bi, i = 1, . . . , n. 1 Right-continuity of T(x1, . . . , xn) and (8.1)–(8.3) imply that T(x1, . . . , xn) is a joint cdf of some r.v.’s X1, . . . , Xn with one-dimensional cdf’s Fk(xk), and the joint cdf T(x1, . . . , xn) satisfies (2.1). Let us now prove the sufficiency part. Consider the functions fi1,...,ic(xi1, . . . , xic) = c� s=2 (−1) c−s � j1
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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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Page 167 and 168: Semiparametric transformation model
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Page 199 and 200: Hazard function 0.0 0.5 1.0 1.5 2.0
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Page 205 and 206: Copulas, information, dependence an
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Page 231 and 232: Regression trees 211 right model in
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Page 241 and 242: Regression trees 221 Table 3 Result
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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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