Optimality

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190 V. H. de la Peña, R. Ibragimov and S. Sharakhmetov Remark 3.1. The functions g and ˜g in Theorems 2.1-3.3 are related in the following way: gi1,...,ic(xi1, . . . , xic) = ˜gi1,...,ic(Fi1(xi1), . . . , Fic(xic)). Theorems 2.1–3.3 provide a general device for constructing multivariate copulas and distributions. E.g., taking in (3.1) and (3.2) n = 2, ˜g1,2(t1, t2) = α(1− 2t1)(1−2t2), α∈[−1, 1], we get the family of bivariate Eyraud–Farlie–Gumbel– Morgenstern copulas Cα(u1, u2) = u1u2(1 + α(1−u1)(1−u2)) and corresponding distributions Fα(x1, x2) = F1(x1)F2(x2)(1+α(1−F1(x1))(1−F2(x2)). More generally, taking ˜gi1,...,ic(ti1, . . . , tic) = 0, 1≤i1
Copulas, information, dependence and decoupling 191 allows one to reduce problems for dependent r.v.’s to well-studied objects and to transfer results known for independent r.v.’s and U-statistics to the case of arbitrary dependence. In what follows, the joint distributions considered are assumed to be absolutely continuous with respect to the product of the marginal distributions � n k=1 Fk(xk). Theorem 4.1. The r.v.’s X1, . . . , Xn have one-dimensional cdf’s Fk(xk), xk∈ R, k = 1, . . . , n, if and only if there exists Un∈Gn such that for any Borel measurable function f : R n → R for which the expectations exist (4.2) Ef(X1, . . . , Xn) = Ef(ξ1, . . . , ξn)(1 + Un(ξ1, . . . , ξn)). Note that the above Theorem 4.1 holds for complex-valued functions f as well as for real-valued ones. That is, letting f(x1, . . . , xn) = exp(i �n k=1 tkxk), tk∈ R, k = 1, . . . , n, one gets the following representation for the joint characteristic function of the r.v.’s X1, . . . , Xn : � n� � � n� � � n� � E exp i = E exp i + E exp i Un(ξ1, . . . , ξn). k=1 tkXk k=1 tkξk k=1 tkξk 5. Characterizations of classes of dependent random variables The following Theorems 5.1–5.8 give characterizations of different classes of dependent r.v.’s in terms of functions g that appear in the representations for joint distributions obtained in Section 2. Completely similar results hold for the functions ˜g that enter corresponding representations for copulas in Section 3. Theorem 5.1. The r.v.’s X1, . . . , Xn with one-dimensional cdf’s Fk(xk), xk∈ R, k = 1, . . . , n, are independent if and only if the functions gi1,...,ic in representations (2.1) and (2.2) satisfy the conditions gi1,...,ic(ξi1, . . . , ξic) = 0 (a.s.), 1≤i1
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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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Semiparametric transformation model
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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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Optimality

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