Optimality

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224 W.-Y. Loh Regression coefficient for Mask Regression coefficients for Pad 0.5 1.0 1.5 2.0 2.5 0.0 0.5 Solder thick Solder thick Mask B6 Mask B3 Mask A3 Solder thin Opening small Pad D6 Pad D7 Pad L4 Solder thin Opening small Pad L6 Pad L7 Pad L8 Pad L9 Pad W4 Pad W9 Fig 13. Plots of regression coefficients for Mask and Pad from Table 5. Solder thin Opening not small Solder thin Opening not small four-factor and higher interactions are negligible. On the other hand, the graphs in Figure 13 suggest that there may exist some weak three-factor interactions, such as between Solder, Opening, and Pad. Figure 14, which compares the fits of this model with those of the Chambers-Hastie model, shows that the former fits slightly better. 6. Logistic regression The same ideas can be applied to fit logistic regression models when the response variable is a sample proportion. For example, Table 6 shows data reported in Collett [9, p. 127] on the number of seeds germinating, out of 100, at two germination temperatures. The seeds had been stored at three moisture levels and three storage temperatures. Thus the experiment is a 2×3×3 design. Treating all the factors as nominal, Collett [9, p. 128] finds that a linear logistic
Observed values 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Regression trees 225 Observed values 0 10 20 30 40 50 60 0 10 20 30 40 50 60 GUIDE fits Fig 14. Plots of observed versus fitted values for the Chambers–Hastie model in Table 4 (left) and the GUIDE piecewise main effects model in Table 5 (right). Table 6 Number of seeds, out of 100, that germinate Germination Moisture Storage temp. ( o C) temp. ( o C) level 21 42 62 11 low 98 96 62 11 medium 94 79 3 11 high 92 41 1 21 low 94 93 65 21 medium 94 71 2 21 high 91 30 1 Table 7 Logistic regression fit to seed germination data using set-to-zero constraints Coef SE z Pr(> |z|) (Intercept) 2.5224 0.2670 9.447 < 2e-16 germ21 -0.2765 0.1492 -1.853 0.06385 store42 -2.9841 0.2940 -10.149 < 2e-16 store62 -6.9886 0.7549 -9.258 < 2e-16 moistlow 0.8026 0.4412 1.819 0.06890 moistmed 0.3757 0.3913 0.960 0.33696 store42:moistlow 2.6496 0.5595 4.736 2.18e-06 store62:moistlow 4.3581 0.8495 5.130 2.89e-07 store42:moistmed 1.3276 0.4493 2.955 0.00313 store62:moistmed 0.5561 0.9292 0.598 0.54954 regression model with all three main effects and the interaction between moisture level and storage temperature fits the sample proportions reasonably well. The parameter estimates in Table 7 show that only the main effect of storage temperature and its interaction with moisture level are significant at the 0.05 level. Since the storage temperature main effect has two terms and the interaction has four, it takes some effort to fully understand the model. A simple linear logistic regression model, on the other hand, is completely and intuitively explained by its graph. Therefore we will fit a piecewise simple linear logistic model to the data, treating the three-valued storage temperature variable as a continuous linear predictor. We accomplish this with the LOTUS [5] algorithm,
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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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Page 199 and 200: Hazard function 0.0 0.5 1.0 1.5 2.0
Page 203 and 204: IMS Lecture Notes-Monograph Series
Page 205 and 206: Copulas, information, dependence an
Page 231 and 232: Regression trees 211 right model in
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Page 241 and 242: Regression trees 221 Table 3 Result
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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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