Optimality

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158 D. M. Dabrowska Next suppose that τ0 is a continuity point of this survival function, and let T = [0, τ0). We have sup t∈T|exp[−Apn(t)−exp[−Ap(t)| = oP(1). In addition, for any τ < τ0, we have exp[−A1n(τ)] ≤ exp[−Γnθ(τ)] ≤ exp[−A2n(τ)]. Standard monotonicity arguments imply sup t∈T|exp[−Γnθ(t)−exp[−Γθ(t)| = oP(1), because Γθ(τ)↑∞ as τ↑∞. 6.2. Part (iii) The process ˆ W(t, θ) = √ n[Γnθ− Γθ](t) satisfies where Define ˆW(t, θ) = √ � nRn(t, θ)− b ∗ nθ(u) = �� 1 0 [0,t] ˆW(u−, θ)N.(du)b ∗ nθ(u), � ′ 2 S /S � � (θ, Γθ(u−) + λ[Γnθ− Γθ](u−), u)dλ . ˜W(t, θ) = √ nRn(t, θ)− � t where bθ(u) = [s ′ /s 2 ](Γθ(u), θ, u). We have and where ˜W(t, θ) = √ nRn(t, θ)− ˆW(t, θ)− ˜ W(t, θ) =− � rem(t, θ) =− [0,t] � t The remainder term is bounded by � τ 0 0 � t 0 0 ˜W(u−, θ)bθ(u)EN(du), √ nRn(u−, θ)bθ(u)EN(du)Pθ(u, t) [ ˆ W− ˜ W](u−, θ)b ∗ nθ(u)N.(du) + rem(t, θ), ˜W(u−, θ)[b ∗ nθ(u)N.(du)−bθ(u)EN(du)]. | ˜ W(u−, θ)||[b ∗ nθ− bθ](u)|N.(du) + R10n(t, θ) + � t− 0 | √ nRn(u−, θ)||bθ(u)|R9n(du, θ). By noting that R9n(·, θ) is a nonnegative increasing process, we have�rem� = oP(1) +�R10n� + OP(1)�R9n� = oP(1). Finally, | ˆ W(t, θ)− ˜ W(t, θ)|≤|rem(t, θ)| + � t 0 | ˆ W− ˜ W|(u−, θ)ρn(du). By Gronwall’s inequality (Section 9), we have ˆ W(t, θ) = ˜ W(t, θ) + oP(1) uniformly in t ≤ τ, θ ∈ Θ. This verifies that the process √ n[Γnθ− Γθ] is asymptotically Gaussian, under the assumption that observations are iid, but Condition 2.2 does not necessarily hold.
6.3. Part (ii) Put (6.1) (6.2) ˙Γnθ(t) = ˙Γθ(t) = Semiparametric transformation models 159 � t 0 � t + � t ˙Kn(Γnθ(u−), θ, u)N.(du) 0 0 � t + K ′ n(Γnθ(u−), θ, u) ˙ Γnθ(u−)N.(du), ˙k(Γθ(u−), θ, u)EN(du) 0 k ′ (Γθ(u−), θ, u) ˙ Γθ(u−)EN(du). Here ˙ K = ˙ S/S2 , K ′ =−S ′ /S2 , k ˙ 2 ′ ′ 2 = ˙s/s and k =−s /s . Assumption 2.0(iii) implies that Γθ(τ) ≤ m −1 1 (τ) < ∞. For G = k′ , ˙ k, Conditions 2.1 imply that � t supθ,t 0 |G(Γθ(u−), θ, u)|EN(du)
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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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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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