Optimality

More documents

Recommendations

Info

226 W.-Y. Loh germination temperature =11 o C moisture level =high or medium moisture level =high 134/300 122/300 343/600 germ. temp. =11 o C 256/300 252/300 Fig 15. Piecewise simple linear LOTUS logistic regression tree for seed germination experiment. The fraction beneath each leaf node is the sample proportion of germinated seeds. Probability of germination 0.0 0.2 0.4 0.6 0.8 1.0 Moisture level = high 20 30 40 50 60 Storage temperature Probability of germination 0.0 0.2 0.4 0.6 0.8 1.0 Moisture level = medium 20 30 40 50 60 Storage temperature Probability of germination 0.0 0.2 0.4 0.6 0.8 1.0 Moisture level = low 20 30 40 50 60 Storage temperature Fig 16. Fitted probability functions for seed germination data. The solid and dashed lines pertain to fits at germination temperatures of 11 and 21 degrees, respectively. The two lines coincide in the middle graph. which extends the GUIDE algorithm to logistic regression. It yields the logistic regression tree in Figure 15. Since there is only one linear predictor in each node of the tree, the LOTUS model can be visualized through the fitted probability functions shown in Figure 16. Note that although the tree has five leaf nodes, and hence five fitted probability functions, we can display the five functions in three graphs, using solid and dashed lines to differentiate between the two germination temperature levels. Note also that the solid and dashed lines coincide in the middle graph because the fitted probabilities there are independent of germination temperature. The graphs show clearly the large negative effect of storage temperature, especially when moisture level is medium or high. Further, the shapes of the fitted functions for low moisture level are quite different from those for medium and high moisture levels. This explains the strong interaction between storage temperature and moisture level found by Collett [9]. 7. Conclusion We have shown by means of examples that a regression tree model can be a useful supplement to a traditional analysis. At a minimum, the former can serve as a check on the latter. If the results agree, the tree offers another way to interpret the main
Regression trees 227 effects and interactions beyond their representations as single degree of freedom contrasts. This is especially important when variables have more than two levels because their interactions cannot be fully represented by low-order contrasts. On the other hand, if the results disagree, the experimenter may be advised to reconsider the assumptions of the traditional analysis. Following are some problems for future study. 1. A tree structure is good for uncovering interactions. If interactions exist, we can expect the tree to have multiple levels of splits. What if there are no interactions? In order for a tree structure to represent main effects, it needs one level of splits for each variable. Hence the complexity of a tree is a sufficient but not necessary condition for the presence of interactions. One way to distinguish between the two situations is to examine the algebraic equation associated with the tree. If there are no interaction effects, the coefficients of the cross-product terms can be expected to be small relative to the main effect terms. A way to formalize this idea would be useful. 2. Instead of using empirical principles to exclude all high-order effects from the start, a tree model can tell us which effects might be important and which unimportant. Here “importance” is in terms of prediction error, which is a more meaningful criterion than statistical significance in many applications. High-order effects that are found this way can be included in a traditional stepwise regression analysis. 3. How well do the tree models estimate the true response surface? The only way to find out is through computer simulation where the true response function is known. We have given some simulation results to demonstrate that the tree models can be competitive in terms of prediction mean squared error, but more results are needed. 4. Data analysis techniques for designed experiments have traditionally focused on normally distributed response variables. If the data are not normally distributed, many methods are either inapplicable or become poor approximations. Wu and Hamada [22, Chap. 13] suggest using generalized linear models for count and ordinal data. The same ideas can be extended to tree models. GUIDE can fit piecewise normal or Poisson regression models and LOTUS can fit piecewise simple or multiple linear logistic models. But what if the response variable takes unordered nominal values? There is very little statistics literature on this topic. Classification tree methods such as CRUISE [15] and QUEST [19] may provide solutions here. 5. Being applicable to balanced as well as unbalanced designs, tree methods can be useful in experiments where it is impossible or impractical to obtain observations from particular combinations of variable levels. For the same reason, they are also useful in response surface experiments where observations are taken sequentially at locations prescribed by the shape of the surface fitted up to that time. Since a tree algorithm fits the data piecewise and hence locally, all the observations can be used for model fitting even if the experimenter is most interested in modeling the surface in a particular region of the design space. References [1] Box, G. E. P., Hunter, W. G. and Hunter, J. S. (2005). Statistics for Experimenters, 2nd ed. Wiley, New York.
Page 1 and 2:
Institute of Mathematical Statistic
Page 3 and 4:
Institute of Mathematical Statistic
Page 5 and 6:
iv Contents Copulas and Decoupling
Page 7 and 8:
vi Finally, thanks go the Statistic
Page 9 and 10:
SCIENTIFIC PROGRAM The Second Erich
Page 11 and 12:
x Recent Advances in Longitudinal D
Page 13 and 14:
The Second Lehmann Symposium—Opti
Page 15 and 16:
xiv Richard Davis Colorado State Un
Page 17 and 18:
xvi Matthias Matheas Rice Universit
Page 19 and 20:
xviii Hui Zhao University of Texas
Page 21 and 22:
IMS Lecture Notes-Monograph Series
Page 23 and 24:
Proof. The maximum LR test rejects
Page 25 and 26:
4. Location-scale families On likel
Page 27 and 28:
6. Conclusions On likelihood ratio
Page 29 and 30:
Page 31 and 32:
Student’s t-test for scale mixtur
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Optimality in multiple testing 17 w
Page 39 and 40:
Optimality in multiple testing 19 I
Page 41 and 42:
power 0.02 0.03 0.04 0.05 0.06 0.07
Page 43 and 44:
Optimality in multiple testing 23 i
Page 45 and 46:
Optimality in multiple testing 25 (
Page 47 and 48:
Optimality in multiple testing 27 M
Page 49 and 50:
6. Summary Optimality in multiple t
Page 51 and 52:
Optimality in multiple testing 31 [
Page 53 and 54:
Page 55 and 56:
On the false discovery proportion 3
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
≤ N� � N� P m=1 On the fals
Page 63 and 64:
Since j≤ s, it follows that On th
Page 65 and 66:
0.025 0.02 0.015 0.01 0.005 On the
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Massive multiple hypotheses testing
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
P value EDF qdf 0.0 0.2 0.4 0.6 0.8
Page 83 and 84:
Page 85 and 86:
Proof. See Appendix. Massive multip
Page 87 and 88:
X1 Massive multiple hypotheses test
Page 89 and 90:
Page 91 and 92:
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4
Page 93 and 94:
Then π0α ∗ cal π0α ∗ cal +
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Frequentist statistics: theory of i
Page 101 and 102:
Page 103 and 104:
2.3. Failure and confirmation Frequ
Page 105 and 106:
3.1. Types of null hypothesis Frequ
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
together with the probabilistic ass
Page 121 and 122:
Statistical models: problem of spec
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Modeling inequality, spread in mult
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Semiparametric transformation model
Page 155 and 156:
2.1. The model Semiparametric trans
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
6.3. Part (ii) Put (6.1) (6.2) ˙Γ
Page 181 and 182:
Page 183 and 184:
8. Proof of Proposition 2.3 Semipar
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Bayesian transformation hazard mode
Page 193 and 194:
Bayesian transformation hazard mode
Page 195 and 196: Bayesian transformation hazard mode
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
Page 201 and 202: Bayesian transformation hazard mode
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
Page 233 and 234: Abs(effects) 0.00 0.05 0.10 0.15 0.
Page 235 and 236: Regression trees 215 of the tree. T
Page 237 and 238: Regression trees 217 GUIDE methods
Page 239 and 240: Regression trees 219 and E appear a
Page 241 and 242: Regression trees 221 Table 3 Result
Page 243 and 244: Solder =thick 2.5 Regression trees
Page 245: Observed values 0 10 20 30 40 50 60
Page 251 and 252: On Competing risk and degradation p
Page 263 and 264: Restricted estimation in competing
Page 271 and 272: 9. Conclusion 0.0 0.1 0.2 0.3 0.4 0
Page 275 and 276: 2.1. Maximin efficiency robust test
Page 277 and 278: Robust tests for genetic associatio
Page 287 and 288: 1 . . . 1 i . . . . . . R j . . . O
Page 289 and 290: Optimal sampling strategies 269 as
Page 291 and 292: Optimal sampling strategies 271 γ1
Page 293 and 294: Optimal sampling strategies 273 Def
Page 295 and 296: Optimal sampling strategies 275 Usi
Page 297 and 298:
Optimal sampling strategies 277 Def
Page 299 and 300:
optimal 1 2 3 4 leaf sets 5 6 (leaf
Page 301 and 302:
6. Related work Optimal sampling st
Page 303 and 304:
Optimal sampling strategies 283 Cla
Page 305 and 306:
Optimal sampling strategies 285 Sin
Page 307 and 308:
Optimal sampling strategies 287 µ
Page 309 and 310:
Optimal sampling strategies 289 on
Page 311 and 312:
Page 313 and 314:
Linear prediction after model selec
Page 315 and 316:
y the P× 1 vector (3.1) ηn(p) = L
Page 317 and 318:
Page 319 and 320:
Page 321 and 322:
Page 323 and 324:
Page 325 and 326:
Page 327 and 328:
Page 329 and 330:
Appendix B: Proofs for Section 5 Li
Page 331 and 332:
Page 333 and 334:
Local asymptotic minimax risk/asymm
Page 335 and 336:
Page 337 and 338:
Then (3.5) Local asymptotic minimax
Page 339 and 340:
Page 341 and 342:
Page 343 and 344:
Moment-density estimation 323 may n
Page 345 and 346:
3. The bias and MSE of ˆf ∗ α M
Page 347 and 348:
Moment-density estimation 327 Corol
Page 349 and 350:
Moment-density estimation 329 Suppo
Page 351 and 352:
6. Simulations Moment-density estim
Page 353 and 354:
Moment-density estimation 333 [7] M
Page 355 and 356:
for positive α, and for α = 0 as
Page 357 and 358:
Asymptotics of the MDPDE 337 Lemma
Page 359:
Asymptotics of the MDPDE 339 asympt
show all

Optimality

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?