Optimality

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162 D. M. Dabrowska �[ˆr1− r](·, θ0)�∞ = oP(1), so that the same integration by parts argument implies that √ nUn2(θ0) = √ n Ũn2(θ0) + oP(1). Finally, √ nUn1(θ0) = √ n Ũn1(θ0) + oP(1), by Lemma 5.1 and Fubini theorem. Suppose now that θ varies over a ball B(θ0, εn) centered at θ0 and having radius εn, εn ↓ 0, √ nεn → ∞. It is easy to verify that for θ, θ ′ ∈ B(θ0, εn) we have Un(θ ′ )−Un(θ) = −(θ ′ − θ) T Σ1n(θ0) + (θ ′ − θ) T Rn(θ, θ ′ ), where Rn(θ, θ ′ ) is a remainder term satisfying sup{|Rn(θ, θ ′ )| : θ, θ ′ ∈ B(θ0, εn)} = oP(1). The matrix Σ1n(θ) is equal to the sum Σ1n(θ) = Σ11n(θ) + Σ12n(θ), Σ11n(θ) = 1 n Σ12n(θ) =− 1 n n� i=1 � τ n� i=1 0 � τ [g1ig T 2i](Γnθ(u), θ, u) T Ni(du), 0 [fi− Sf/S](Γnθ(u), θ, u)Ni(du), where Sf(Γnθ(u), θ, u) = n −1 � n i=1 Yi(u)[αifi](Γnθ(u), θ, u) and g1i(θ, Γnθ(u), u) = b1i(Γnθ(u), θ)−b2i(Γnθ(u), θ)ϕθ0(u), g2i(θ, Γnθ(u), u) = b1i(Γnθ(u), θ) + b2i(Γnθ(u), θ) ˙ Γnθ(u) fi(θ, Γnθ(u), u) = ¨α α (Γnθ(u), θ, Zi)− ˙α′ α (Γnθ(u), θ, Zi)ϕθ0(u) T + ˙ Γnθ(u)[ ˙α′ α (Γnθ(u), θ, Zi)] T + α′′ α (Γnθ(u), θ, Zi) ˙ Γnθ(u)ϕθ0(u) T . These matrices satisfy Σ11n(θ0) →P Σ1,ϕ(θ0, τ) and Σ12n(θ0) →P 0, and Σ1,ϕ(θ0, τ) = Σ1(θ0) is defined in the statement of Proposition 2.2. By assumption this matrix is non-singular. Finally, set hn(θ) = θ + Σ1(θ0) −1 Un(θ). It is easy to verify that this mapping forms a contraction on the set{θ :|θ−θ0|≤An/(1−an)}, where An =|Σ1(θ0) −1 Un(θ0)| = OP(n −1/2 ) and an = sup{|I− Σ1(θ0) −1 Σ1n(θ0) + Σ1(θ0) −1 Rn(θ, θ ′ )| : θ, θ ′ ∈ B(θ0, εn)} = oP(1). The argument is similar to Bickel et al. ([6], p.518), though note that we cannot apply their mean value theorem arguments. Next consider Condition 2.3(v.2). In this case we have Ûn(θ ′ )− Ûn(θ) =−(θ ′ − θ) T Σ1n(θ0) + (θ ′ − θ) T ˆ Rn(θ, θ ′ ), where sup{| ˆ Rn(θ, θ ′ ) : θ, θ ′ ∈ B(θ0, εn)} = oP(1). In addition, for θ∈Bn(θ0, ε), we have the expansion Ūn(θ) = [ Ūn(θ)− Ūn(θ0)] + Ūn(θ0) = oP(|θ− θ0| + n −1/2 ). The same argument as above shows that the equation Ûn(θ) has, with probability tending to 1, a unique root in the ball B(θ0, εn). But then, we also have Un( ˆ θn) = Ûn( ˆ θn) + Ūn( ˆ θn) = oP(| ˆ θn− θ0| + n −1/2 ) = oP(OP(n −1/2 ) + n −1/2 ) = op(n −1/2 ). Part (iv) can be verified analogously, i.e. it amounts to showing that if √ n[ ˆ θ−θ0] is bounded in probability, then the remainder term ˆ Rn( ˆ θ, θ0) is of order oP(| ˆ θ−θ0|), and Ūn( ˆ θ) = oP(| ˆ θ− θ0| + n −1/2 ).
8. Proof of Proposition 2.3 Semiparametric transformation models 163 Part (i) is verified at the end of the proof. To show part (ii), define D(λ) = � D(t, u, λ) = � (−1) m m≥0 (−1) m m≥0 m! λm dm, m! λm Dm(t, u). The numbers dm and the functions Dm(t, u) are given by dm = 1, Dm(t, u) = k(t, u) for m = 0. For m≥1 set � � dm = . . . det ¯ dm(s)b(ds1)·. . .·b(dsm), Dm(t, u) = � � . . . (s1,...,sm)∈(0,τ] (s1,...,sm)∈(0,τ] det ¯ Dm(t, u;s)b(ds1)·. . .·b(dsm), where for any s = (s1, . . . , sm), ¯ dm(s) is an m×m matrix with entries ¯ dm(s) = [k(si, sj)], and ¯ Dm(t, u;s) is an (m + 1)×(m + 1) matrix ¯Dm(t, u;s) = � k(t, u), Um(t;s) Vm(s;u), ¯ dm(s) where Um(t;s) = [k(t, s1), . . . , k(t, sm)], Vm(s;u) = [k(s1, u), . . . , k(sm, u)] T . By Fredholm determinant formula [25], the resolvent of the kernel k is given by ˜∆(t, u, λ) = D(t, u, λ)/D(λ), for all λ such that D(λ)�= 0, so that � � dm = . . . det s1 ,...,sm∈(0,τ] distinct ¯ dm(s)b(ds1)·. . .·b(dsm), because the determinant is zero whenever two or more points si, i = 1, . . . , m are equal. By Fubini theorem, the right-hand side of the above expression is equal to � � � . . . det ¯ dm(sπ(1), . . . , sπ(m))b(ds1)·. . .·b(dsm) π � = m! 0
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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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Page 131 and 132: Statistical models: problem of spec
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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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