Optimality

More documents

Recommendations

Info

IMS Lecture Notes–Monograph Series 2nd Lehmann Symposium – <strong>Optimality</strong> Vol. 49 (2006) 312–321 c○ Institute of Mathematical Statistics, 2006 DOI: 10.1214/074921706000000527 Local asymptotic minimax risk bounds in a locally asymptotically mixture of normal experiments under asymmetric loss Debasis Bhattacharya 1 and A. K. Basu 2 Visva-Bharati University and Calcutta University Abstract: Local asymptotic minimax risk bounds in a locally asymptotically mixture of normal family of distributions have been investigated under asymmetric loss functions and the asymptotic distribution of the optimal estimator that attains the bound has been obtained. 1. Introduction There are two broad issues in the asymptotic theory of inference: (i) the problem of finding the limiting distributions of various statistics to be used for the purpose of estimation, tests of hypotheses, construction of confidence regions etc., and (ii) problems associated with questions such as: how good are the estimation and testing procedures based on the statistics under consideration and how to define ‘optimality’, etc. Le Cam [12] observed that the satisfactory answers to the above questions involve the study of the asymptotic behavior of the likelihood ratios. Le Cam [12] introduced the concept of ‘Limit Experiment’, which states that if one is interested in studying asymptotic properties such as local asymptotic minimaxity and admissibility for a given sequence of experiments, it is enough to prove the result for the limit of the experiment. Then the corresponding limiting result for the sequence of experiments will follow. One of the many approaches which are used in asymptotic theory to judge the performance of an estimator is to measure the risk of estimation under an appropriate loss function. The idea of comparing estimators by comparing the associated risks was considered by Wald [19, 20]. Later this idea has been discussed by Hájek [8], Ibragimov and Has’minskii [9] and others. The concept of studying asymptotic efficiency based on large deviations has been recommended by Basu [4] and Bahadur [1, 2]. In the above context it is an interesting problem to obtain a lower bound for the risk in a wide class of competing estimators and then find an estimator which attains the bound. Le Cam [11] obtained several basic results concerning asymptotic properties of risk functions for LAN family of distributions. Jeganathan [10], Basawa and Scott [5], and Le Cam and Yang [13] have extended the results of Le Cam for Locally Asymptotically Mixture of Normal (LAMN) experiments. Basu and Bhattacharya [3] further extended the result for Locally Asymptotically Quadratic (LAQ) family of distributions. A symmetric loss structure (for example, 1Division of Statistics, Institute of Agriculture, Visva-Bharati University, Santiniketan, India, Pin 731236 2Department of Statistics, Calcutta University, 35 B. C. Road, Calcutta, India, Pin 700019 AMS 2000 subject classifications: primary 62C20, 62F12; secondary 62E20, 62C99. Keywords and phrases: locally asymptotically mixture of normal experiment, local asymptotic minimax risk bound, asymmetric loss function 312
Local asymptotic minimax risk/asymmetric loss 313 squared error loss) has been used to derive the results in the above mentioned references. But there are situations where the loss can be different for equal amounts of over-estimation and under-estimation, e. g., there exists a natural imbalance in the economic results of estimation errors of the same magnitude and of opposite signs. In such cases symmetric losses may not be appropriate. In this context Bhattacharya et al. [7], Levine and Bhattacharya [15], Rojo [16], Zellner [21] and Varian [18] may be referred to. In these works the authors have used an asymmetric loss, known as the LINEX loss function. Let ∆ = ˆ θ− θ, a�= 0 and b > 0. The LINEX loss is then defined as: (1.1) l(∆)=b[exp(a∆)−a∆−1]. Other types of asymmetric loss functions that can be found in the literature are as follows: � C1∆, for ∆≥0 l(∆) = −C2∆, for ∆ < 0, C1, C2 are constants, or l(∆) = � λw(θ)L(∆), for ∆≥0, (over-estimation) w(θ)L(∆), for ∆ < 0, (under-estimation), where ‘L’ is typically a symmetric loss function, λ is an additional loss (in percentage) due to over-estimation, and w(θ) is a weight function. The problem of finding the lower bound for the risk with asymmetric loss functions under the assumption of LAN was discussed by Lepskii [14] and Takagi [17]. In the present work we consider an asymmetric loss function and obtain the local asymptotic minimax risk bounds in a LAMN family of distributions. The paper is organized as follows: Section 2 introduces the preliminaries and the relevant assumptions required to develop the main result. Section 3 is dedicated to the derivation of the main result. Section 4 contains the concluding remarks and directions for future research. 2. Preliminaries Let X1, . . . , Xn be n random variables defined on the probability space (X,A, Pθ) and taking values in (S,S), where S is the Borel subset of a Euclidean space and S is the σ-field of Borel subsets of S. Let the parameter space be Θ, where Θ is an open subset of R 1 . It is assumed that the joint probability law of any finite set of such random variables has some known functional form except for the unknown parameter θ involved in the distribution. LetAn be the σ-field generated by X1, . . . , Xn and let Pθ,n be the restriction of Pθ toAn. Let θ0 be the true value of θ and let θn = θ0 + δnh (h∈R 1 ), where δn→ 0 as n→∞. The sequence δn may depend on θ but is independent of the observations. It is further assumed that, for each n≥1, the probability measures Pθo,n and Pθn,n are mutually absolutely continuous for all θ0 and θn. Then the sequence of likelihood ratios is defined as Ln(Xn;θ0, θn) = Ln(θ0, θn) = dPθn,n , dPθ0,n where Xn = (X1, . . . , Xn) and the corresponding log-likelihood ratios are defined as Λn(θ0, θn) = log Ln(θ0, θn) = log dPθn,n . dPθ0,n
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:
Page 195 and 196:
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:
Page 203 and 204:
Page 205 and 206:
Copulas, information, dependence an
Page 207 and 208:
Page 209 and 210:
Page 211 and 212:
Page 213 and 214:
Page 215 and 216:
Page 217 and 218:
Page 219 and 220:
Page 221 and 222:
Page 223 and 224:
Page 225 and 226:
Page 227 and 228:
Page 229 and 230:
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 and 246:
Observed values 0 10 20 30 40 50 60
Page 247 and 248:
Regression trees 227 effects and in
Page 249 and 250:
Page 251 and 252:
On Competing risk and degradation p
Page 253 and 254:
Page 255 and 256:
Page 257 and 258:
Page 259 and 260:
Page 261 and 262:
Page 263 and 264:
Restricted estimation in competing
Page 265 and 266:
Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
9. Conclusion 0.0 0.1 0.2 0.3 0.4 0
Page 273 and 274:
Page 275 and 276:
2.1. Maximin efficiency robust test
Page 277 and 278:
Robust tests for genetic associatio
Page 279 and 280:
Robust tests for genetic associatio
Page 281 and 282: 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: IMS Lecture Notes-Monograph Series
Page 313 and 314: Linear prediction after model selec
Page 315 and 316: y the P× 1 vector (3.1) ηn(p) = L
Page 329 and 330: Appendix B: Proofs for Section 5 Li
Page 331: Linear prediction after model selec
Page 335 and 336: Local asymptotic minimax risk/asymm
Page 337 and 338: Then (3.5) Local asymptotic minimax
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

Create successful ePaper yourself

Delete template?

Save as template?