Optimality

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222 W.-Y. Loh Pad= D4, D7,L4, L7,L8 Table 4 A Poisson loglinear model containing all main effects and all two-factor interactions involving Opening, Solder, and Mask. Regressor Coef t Regressor Coef t Constant -2.668 -9.25 maskA3 0.396 1.21 openmedium 0.921 2.95 maskB3 2.101 7.54 opensmall 2.919 11.63 maskB6 3.010 11.36 soldthin 2.495 11.44 padD6 -0.369 -5.17 maskA3:openmedium 0.816 2.44 padD7 -0.098 -1.49 maskB3:openmedium -0.447 -1.44 padL4 0.262 4.32 maskB6:openmedium -0.032 -0.11 padL6 -0.668 -8.53 maskA3:opensmall -0.087 -0.32 padL7 -0.490 -6.62 maskB3:opensmall -0.266 -1.12 padL8 -0.271 -3.91 maskB6:opensmall -0.610 -2.74 padL9 -0.636 -8.20 maskA3:soldthin -0.034 -0.16 padW4 -0.110 -1.66 maskB3:soldthin -0.805 -4.42 padW9 -1.438 -13.80 maskB6:soldthin -0.850 -4.85 panel2 0.334 7.93 openmedium:soldthin -0.833 -4.80 panel3 0.254 5.95 opensmall:soldthin -0.762 -5.13 Mask =B3 10 4 Pad= D4,D7, L4,L8 19 Solder =thick Pad= D6,L7, W4 15 7 Mask =B Pad= D,L4, L8,W4 24 Mask =B3 Pad= L6,L7, L9 Solder =thick 14 4 Open =small Pad= D,L4, W4 41 1 8 Solder =thick Pan =2,3 1 Pan =2,3 24 13 Mask =B3 3 1 Mask =B Pad= D4,D7, L4,L8, L9,W4 Pan =2,3 Open= large 11 6 4 Solder =thick 0.1 0.6 Mask =A1.5 1 Pad= D,L4, L9,W4 Fig 11. GUIDE piecewise constant Poisson regression tree for solder data. “Panel” is abbreviated as “Pan”. The sample mean yield is given beneath each leaf node. The leaf node with the lowest mean yield is painted black. 2 1
Solder =thick 2.5 Regression trees 223 Opening =small 16.4 3.0 Fig 12. GUIDE piecewise main effect Poisson regression tree for solder data. The number beneath each leaf node is the sample mean response. Table 5 Regression coefficients in leaf nodes of Figure 12 Solder thick Solder thin Opening small Opening not small Regressor Coef t Coef t Coef t Constant -2.43 -10.68 2.08 21.50 -0.37 -1.95 mask=A3 0.47 2.37 0.31 3.33 0.81 4.55 mask=B3 1.83 11.01 1.05 12.84 1.01 5.85 mask=B6 2.52 15.71 1.50 19.34 2.27 14.64 open=medium 0.86 5.57 aliased - 0.10 1.38 open=small 2.46 18.18 aliased - aliased - pad=D6 -0.32 -2.03 -0.25 -2.79 -0.80 -4.65 pad=D7 0.12 0.85 -0.15 -1.67 -0.19 -1.35 pad=L4 0.70 5.53 0.08 1.00 0.21 1.60 pad=L6 -0.40 -2.46 -0.72 -6.85 -0.82 -4.74 pad=L7 0.04 0.29 -0.65 -6.32 -0.76 -4.48 pad=L8 0.15 1.05 -0.43 -4.45 -0.36 -2.41 pad=L9 -0.59 -3.43 -0.64 -6.26 -0.67 -4.05 pad=W4 -0.05 -0.37 -0.09 -1.00 -0.23 -1.57 pad=W9 -1.32 -5.89 -1.38 -10.28 -1.75 -7.03 panel=2 0.22 2.72 0.31 5.47 0.58 5.73 panel=3 0.07 0.81 0.19 3.21 0.69 6.93 a main effects model to each node instead of a constant. This yields the much smaller piecewise main effect GUIDE tree in Figure 12. It has only two splits, first on Solder and then, if the latter is thin, on Opening. Table 5 gives the regression coefficients in the leaf nodes and Figure 13 graphs them for each level of Mask and Pad by leaf node. Because the regression coefficients in Table 5 pertain to conditional main effects only, they are simple to interpret. In particular, all the coefficients except for the constants and the coefficients forPad have positive values. Since negative coefficients are desirable for minimizing the response, the best levels for all variables except Pad are thus those not in the table (i.e, whose levels are set to zero). Further, W9 has the largest negative coefficient among Pad levels in every leaf node. Hence, irrespective of Solder, the best levels to minimize mean yield are A1.5 Mask, large Opening, W9 Pad, and Panel position 1. Finally, since the largest negative constant term occurs when Solder is thick, the latter is the best choice for minimizing mean yield. Conversely, it is similarly observed that the worst combination (i.e., one giving the highest predicted mean number of solder skips) is thin Solder, small Opening, B6 Mask, L4 Pad, and Panel position 2. Given that the tree has only two levels of splits, it is safe to conclude that
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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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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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