Optimality

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336 S. Juárez and W. R. Schucany for all θ∈Θ. Then the MDPDE is an M-estimator with criterion function given by (3.1) and it is obtained by maximizing mn(θ) = 1 n n� m(Xi, θ) over the parameter space Θ. Let ΘG⊆ Θ be the set where � (3.2) |m(x, θ)|g(x)dx
Asymptotics of the MDPDE 337 Lemma 3. M(θ) as given by (3.3) is twice continuous differentiable in a neighborhood B of θ0 with second derivative (Hessian matrix) HθM(θ) =−(1+α)Jα(θ), if: 1. The integral � X f1+α (x;θ)dx is twice continuously differentiable with respect to θ in B, and the derivative can be taken under the integral sign. 2. The order of integration with respect to x and differentiation with respect to θ can be interchanged in M(θ), for θ∈B. Proof. Consider the (transpose) score function S t (x; θ) = Dθ log f(x; θ) and the information matrix i(x;θ) = −Hθ log f(x; θ) = −DθS(x;θ). Also note that [Dθf(x;θ)]f α−1 (x;θ) = S t (x;θ)f α (x;θ). Use the previous expressions and condition 1 to obtain the first derivative of θ↦→ m(x; θ) (3.4) Dθm(x, θ) = (1 + α)S t (x; θ)f α � (x;θ)−(1 + α) S X t (x; θ)f 1+α (x; θ)dx. Proceeding in a similar way, the second derivative of θ↦→ m(x; θ) is Hθm(x, θ) = (1 + α){−i(x;θ) + αS(x;θ)S t (x; θ)}f α (x; θ)−(1 + α) (3.5) �� × −i(x; θ)f X 1+α (x;θ) + (1 + α)S(x; θ)S t (x; θ)f 1+α (x; θ)dx Then using condition 2 we can compute the second derivative of M(θ) under the integral sign and, after some algebra, obtain � HθM(θ) = {Hθm(x, θ)}g(x)dx =−(1 + α)Jα(θ). X The second result is an elementary fact about differentiable mappings. Proposition 4. Suppose the function θ↦→ m(x, θ) is differentiable at θ0 for x a.e. with derivative Dθm(x, θ). Suppose there exists an open ball B∈ Θ and a constant M
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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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6.3. Part (ii) Put (6.1) (6.2) ˙Γ
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8. Proof of Proposition 2.3 Semipar
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Bayesian transformation hazard mode
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Cumulative Hazard 0.0 0.2 0.4 0.6 0
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Hazard function 0.0 0.5 1.0 1.5 2.0
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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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