Optimality

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150 D. M. Dabrowska define regular parametric models, but we expect that in such models parameters are estimated subject to added constraints in both parametric and semiparametric setting. Evidently, our results do not apply to such complicated problems. The choice of g(x) = [1 + ηx] ξ , ξ
Semiparametric transformation models 151 n −1 � n i=1 1(Xi ≥ t) and Further, let �·� be the supremum norm in the set ℓ ∞ ([0, τ]×Θ), and let�·�∞ be the supremum norm ℓ ∞ ([0, τ]). We assume that the point τ satisfies Condition 2.0(iii). Define Rn(t, θ) = Rpn(t, θ) = Rpn(t, θ) = R9n(t, θ) = R10n(t, θ) = Rpn(t, θ) = Bpn(t, θ) = � t 0 � t 0 � t 0 N.(du) S(Γθ(u−), θ, u) − � t EN(du) 0 s(Γθ(u−), θ, u) , h(Γθ(u−), θ, u)[N.− EN](du), p = 5,6, Hn(Γθ(u−), θ, u)N(du) � t − h(Γθ(u−), θ, u)EN(du), p = 7, 8, � 0 � EN(du)| Pθ(u, w)R5n(dw, θ)|, [0,t) � t 0 � t 0 � t 0 (u,t] √ nRn(u−, θ)R5n(du, θ), Hn(Γnθ(u−), θ, u)N.(du) − � t 0 h(Γθ(u−), θ, u)EN(du), p = 11,12, Fpn(u, θ)Rn(du, θ), p = 1, 2, wherePθ(u, w) is given by (2.3). In addition, Hn = K ′ n for p = 7 or p = 11, Hn = ˙ Kn for p = 8 or p = 12, h = k ′ for p = 5, 7 or p = 11, and h = ˙ k for p = 6, 8 or 12. Here k ′ =−[s ′ /s 2 ], ˙ k =−[˙s/s 2 ], K ′ n =−[S ′ /S 2 ], ˙ Kn =−[ ˙ S/S 2 ]. Further, set F1n(u, θ) = [ ˙ S− ēS](Γθ(u), θ, u) and F2n(u, θ) = [S ′ − eS ′ ](Γθ(u), θ, u). Lemma 5.1. Suppose that Conditions 2.0 and 2.1 are satisfied. (i) √ nRn(t, θ) converges weakly in ℓ ∞ ([0, τ]×Θ) to a mean zero Gaussian process R whose covariance function is given below. (ii)�Rpn�→0 a.s., for p = 5, . . . ,12. (iii) √ n�Bpn�→0 a.s. for p = 1, 2. (iv) The processes Vn(Γθ(t−), θ, t) and Vn(Γθ(t), θ, t), where Vn = S/s−1 satisfy �Vn� = O(bn) a.s. In addition,�Vn�→0a.s. for Vn = [S ′ − s ′ ]/s,[S ′′ − s ′′ ]/s,[ ˙ S− ˙s]/s,[ ¨ S−¨s]/s and [ ˙ S ′ − ˙s ′ ]/s. Proof. The Volterra identity (2.2), which defines Γθ as a parameter dependent on P, is used in the foregoing to compute the asymptotic covariance function of the process R1n. In Section 6 we show that the solution to the identity (2.2) is unique and, for some positive constants d0, d1, d2, we have (5.1) Γθ(t)≤d0AP(t), |Γθ(t)−Γθ ′(t)|≤|θ− θ′ |d1 exp[d2AP(t)], |Γθ(t)−Γθ(t ′ )| ≤ d0|AP(t)−AP(t ′ )| ≤ d0 EPY (τ) P(X∈ (t∧t′ , t∨t ′ ], δ = 1), with similar inequalities holding for the left continuous version of Γθ = Γθ,P. Here AP(t) is the cumulative hazard function corresponding to observations (X, δ).
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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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Page 119 and 120: together with the probabilistic ass
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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 179 and 180: 6.3. Part (ii) Put (6.1) (6.2) ˙Γ
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Page 197 and 198: Cumulative Hazard 0.0 0.2 0.4 0.6 0
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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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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