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Gruber M.H.J. Ч Regression Estimators: A Comparative Study
Gruber M.H.J. Ч Regression Estimators: A Comparative Study

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Ќазвание: Regression Estimators: A Comparative Study

јвтор: Gruber M.H.J.

јннотаци€:

An examination of mathematical formulations of ridge-regression-type estimators points to a curious observation: estimators can be derived by both Bayesian and Frequentist methods. In this updated and expanded edition of his 1990 treatise on the subject, Marvin H. J. Gruber presents, compares, and contrasts the development and properties of ridge-type estimators from these two philosophically different points of view. The book is organized into five sections. Part I gives a historical survey of the literature and summarizes basic ideas in matrix theory and statistical decision theory. Part II explores the mathematical relationships between estimators from both Bayesian and Frequentist points of view. Part III considers the efficiency of estimators with and without averaging over a prior distribution. Part IV applies the methods and results discussed in the previous two sections to the Kalman Filter, analysis of variance models, and penalized splines. Part V surveys recent developments in the field. These include efficiencies of ridge-type estimators for loss functions other than squared error loss functions and applications to information geometry. Gruber also includes an updated historical survey and bibliography. With more than 150 exercises, Regression Estimators is a valuable resource for graduate students and professional statisticians.


язык: en

–убрика: ћатематика/

—ери€: —делано в холле

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

»здание: 2-nd edition

√од издани€: 2010

 оличество страниц: 424

ƒобавлена в каталог: 02.10.2012

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
"half Kalman Filter"      224
Additional observations      10 69 79 82 106 120 171 296
Admissible estimator      17 21 50 53Ч55 145 146
Akdeniz, E.      26
Akdeniz, F.      23 25 26 28
Albert, A.      48
Alheety, M.I.      28
Ali, M.A.      17
Alternative forms of the Bayes estimator      95Ч100 103 116 117Ч129 130 131 133
Alternative forms of the mixed estimator      103Ч109
Alternative forms of the MSE      153Ч157 167
Amari, S.      19 355
Analysis of variance (Anova)      10 31 35 252Ч262 288 298
Analysis of variance (ANOVA), one-way classification      252 253 253Ч263 278
Analysis of variance (ANOVA), two-way classification      252 267Ч273 277 278
Andrews, A.P.      19
Applications, aerospace tracking      223 224
Applications, analysis of life lengths      223
Applications, design of experiments      252 270 272 277
Applications, dose-response experiment      223
Applications, Earth's magnetic field      245
Applications, gross national product      6
Applications, measurement process      246
Applications, navigation      19 244Ч248
Applications, satellite tracking      244 245
Applications, ship's motion      245Ч248
Applications, underwater sonar      223
Approximate Bayes estimator      253
Arsenin, V.Y.      11
Arslan, O.      23
Askin, R.G.      22 23
Asters, R.C.      11
Atkinson, C.      19 366 367
Augmented linear model      11 69 111 117 121 130 187 234 308 341 384
Average MSE      13 32 50 52 136 138
Average MSE of Bayes estimator      153Ч155
Average MSE of contraction estimator      162Ч164
Average MSE of empirical Bayes estimator      265 276 278Ч282
Average MSE of Kalman Filter      239Ч242
Average MSE of Liu estimator      165
Average risk averaging over LINEX loss function      333Ч340
Average risk averaging over squared error loss function      13 32 50 52 136 138 153Ч155 162Ч164 165 239Ч242 265 276 278Ч282
Average risk averaging over Zellner's balanced loss function      312Ч315
Average variance of BE      147 151Ч152
Baksalary, J.K.      18
Baldwin, K.F.      9 12 13 20 23 24 27
Bannerjee, K.S.      11
Basis for a vector space      15 273 278 357 358
Basis for spline model      285Ч287 289
Batah, F.M.      30
Baye, M.R.      21 30 110 116
Bayes estimator, admissibility of      53 145
Bayes estimator, alternative forms of      12 22 95Ч100
Bayes estimator, average MSE of      138 147Ч156 157Ч160 162Ч165 336Ч340
Bayes estimator, comparison with least square estimator      174Ч187
Bayes estimator, comparison with minimax estimator      56 296
Bayes estimator, comparison with mixed estimator      233Ч234 308
Bayes estimator, comparison with ridge-type estimator      101Ч111
Bayes estimator, conditional MSE of      168Ч171 174Ч187 192Ч193 201 315Ч318 340Ч352
Bayes estimator, distance between      354 369Ч384
Bayes estimator, extensions of BLUR      24
Bayes estimator, formulation of      12 26 27 31 34 48Ч53 88Ч92 263Ч264 273Ч275 306Ч307 333Ч336
Bayes estimator, iterated      225Ч228
Bayes estimator, linear      18 70 88Ч92 145 146
Bayes estimator, robustness of      203Ч208 214Ч240
Bayes estimator, special cases of      3 12 14 94 117Ч129 291Ч299
Bayes risk      26 56 138 143 144 146 174 333Ч339 352
Bayes theorem      34 48 49 227
Berger, J.O.      14 15 203
Berliner, M.      14 203
Best linear unbiased estimator (BLUE)      74 76
Best linear unbiased predictor (BLUP)      18 24 284 290Ч299
Bias      9 13 17 18 20 27Ч30 52 111 135 137Ч138 148 168Ч170 196Ч198 336 337
Biased estimator      13 26 30 111
Biased prior information      203 211
Biasing parameter      13 20 21 22 24 25 27 28 29 30
Billor, N.      23
Birkes, D.      19
Bock, M.E.      203
Borchers, B.      11
Bozdogan, H.      29
Brown, K.G.      12
Brown, R.G.      227
Bucy, R.S.      223
Bulmer, M.G.      292
Bunke, H.      15
Bunke, O.      15 20
Burbea, J.      19
Burr, T.L.      26
Camara, V.A.P.      27 325 326
Carr, R.N.      11
Carroll, R.J.      24 283 289 298
Cassella, G.      24
Catlin, D.      223
Cauchy Ч Schwarz inequality      60
Chang, X.      31
Change of coordinate      177
Characteristic equation      62 63
Chaubey, Y.P.      17 30
Chawla, J.S.      17
Clark, A.E.      29
Column space      41 104 121 122 124 125 273
Comparison of conditional and average MSE      179Ч181
Comparison of conditional MSE for a BE and an LS      174Ч179 181Ч187
Comparison of conditional MSE for ridge-type estimator and an LS      12 18 178 181
Comparison of conditional MSE for two BE      192Ч194
Comparison of ridge-type estimators and LS estimators under incorrect prior assumptions      202Ч220
Completely randomized design      252
Conditional expectation      52 336
Conditional MSE      32 135Ч137 167Ч201 202 209
Conditional MSE of Bayes estimator      52 139 214 215 218
Conditional MSE of least square estimator      148 218
Conditional MSE of ridge-type estimator      60 139
Conklin, W.M.      29
Constraint, exact      24Ч26 254 267
Constraint, linear      24 74
Constraint, quadratic      25 29 289 295
Constraint, stochastic prior      22 25 26 121 211 307 385Ч387
Contaminated Bayes estimators      217Ч228
Contaminated priors      213Ч217
Contraction estimators      12 162Ч164 179 194 195 197
contrast      254 256 261 269 278
Convex combinations      303 305 310 312
Convex loss function      333
Correct prior assumptions      137 139 202Ч205 207 209 211 212 242 243
Costa, S.I.      19
Craig, A.      143
Crivelli, A.      19
Crouse, R.H.      20 21 29
Dagnelie, P.      20
Daniel, C.      17
De Boer, P.M.      27
Decision function      56
Dempster, A.P.      9 12 13 154
Design matrix      10 11 20 22 34 94 129 174 245 252
Determinant      35 36 196 372
Dey, D.      26
Diderrich, G.T.      224
Differential equation      83
Differential equation for geodesic      362Ч363
Differential geometry      354 355Ч369 389
Differential geometry, affine connection      355 361
Differential geometry, Christoffel symbols      361
Differential geometry, covariant derivative      361Ч362
Differential geometry, geodesic      354 355 361 362 364Ч367 385 387Ч389
Differential geometry, manifold      354Ч357 359Ч362 389
Differential geometry, Riemannian metric      354 355 358Ч360 389
Differential geometry, tangent space      355 357Ч359 361 389
Discrete Kalman Filter      32 224
dispersion      4 30 143 155 161 205 214 251 340 354 367 376 381Ч389
Dispersion of estimator      5 36
Dispersion of prior      14 101Ч103 117Ч120 157Ч159 163 165 174 179 185 193Ч195 197 248
Dispersion, conditional      196
Dispersion, initial      248
Dispersion, misspecified      202 203 242
Do Carmo, M.P.      355 362
Dodge, Y.      19
Druilhet, P.      30
Drygas, H.      21
Duncan, D.B.      11 95
Ebegil, M.      29
Efficiency of estimators      22 25 32 33 135 147Ч148 153 165 196 209 217 218 253 292Ч294
Efron, B.      253 267
Eigenvalue      26 61
Eigenvalue in distance functions      355 367 369 371 372 383 385
Eigenvalue in SVD      38Ч40 255
Eigenvalue of design matrix      5Ч7 10 107 108 268 343
Eigenvalue of positive semi-definite matrix      35Ч37 58
Eigenvalue of prior      209
Eigenvalue of shrinkage matrix      28
Eigenvector      58 61 255 268 367 369 376 379 389
Ekni, M.      29
Ellipsoid for comparing efficiency of estimators      167 174Ч193 201
Ellipsoid for incorrect prior assumptions      137 202Ч220
Ellipsoid for minimax estimators      14Ч16 20 22 26 30 35 56 86 136 147 208Ч210
Ellipsoid for ridge regression      6 10 76
Empirical Bayes estimator (EBE)      12 15 26 31 94 253 259 263Ч267 273Ч278
Epsilon contaminated prior distribution      14 214Ч217
Error of measurement      246
Estimable parametric functions for estimators      98Ч101 119 153Ч154 159 170 175 188 190Ч192 217 264 275
Estimable parametric functions for studying the efficiency of estimators      167 174Ч193 201
Estimable parametric functions, (X, R) estimable      81 84 105 136 149 151
Estimable parametric functions, definition and properties of      70Ч74 92 148 253Ч257 261 268Ч269 303 305 306 309 319
Estimable parametric functions, R estimable      81 104 149 157
Estimable parametric functions, X estimable      81 84 104 105 151 162
Eubank, R.L.      25 223
Exchangeability of sample and prior information      131
Experimental design models      4 32 252Ч282
Farebrother, R.L.      12 13 14 18 60 71 181
Ferguson, T.S.      56
Feynman, R.P.      135
Firinguetti, L.      17 19
Fixed effects model      253 271
Flack, V.F.      17
Foucart, T.      28
Frequentist argument      101 318
Fry, H.      26
Fu, W.J.      22
Gaffke, N.      18
Galarneau, D.I.      13 22
Gamma distribution      368
Gana, R.      20
Gauss Ч Markov theorem (GM)      11 15 69 74Ч76 81 89 94 116Ч117 130
Gauss Ч Markov theorem (GM), extended GM      96 98 116Ч117 130
Gauss Ч Markov theorem (GM), statement and proof of GM      74Ч75 90 116Ч117 130
Generalized inverse and least square estimators      70Ч76 80 81 256Ч268
Generalized inverse and parametric functions      254Ч255 305
Generalized inverse and the singular value decomposition      34 38
Generalized inverse, definition and properties of      5 32 41Ч48
Generalized ridge estimator      15 32 343
Generalized ridge estimator as a special case of a broader class of estimators      109Ч116 130
Generalized ridge estimator for LINEX loss function      342Ч351
Generalized ridge estimator for Zellner's balanced loss function      310Ч312 322
Generalized ridge estimator, comparison with least square estimator      14 17 20 22 178 342Ч351
Generalized ridge estimator, comparison with other ridge-type estimators      17 27 30 101Ч103 162 195
Generalized ridge estimator, derivation of      76Ч77 129Ч130 237Ч239
Generalized ridge estimator, examples of      77Ч79 102Ч103 113Ч115 180
Ghosh, H.      31
Gibbons, D.G.      13 28
Gokpinar, E.      29
Goldberger, A.S.      10 14 18
Goldstein, M.      11
Golub, G.      13
Gore, S.D.      28 30
Graybill, F.A.      48 64
Grewal, M.S.      19
Gross, J.      20 21 22 24 146
Gruber, M.H.J.      9 12 13 14 18 22 23 25 30 55 96 111 224 317 324
Gunst, R.F.      20
Hafner, C.M.      27
Haitovsky, Y.      13
Hald, A.      22 23
Hampel, F.R.      23
Hanumara, R.C.      20 29
Haq, W.S.      21
Harrison, C.F.      223
Harville, D.H.      11 95
Haughton, D.M.A.      29
Heath, M.      13
Heffes, H.      224
Heiligers, M.R.      18
Hemmerle, W.J.      27
Henderson, C.R.      18 24
Herring, F.      16
Heumann, B.      25
Higgens Ч Tsoskos loss function      340 351Ч353
Hocking, R.R.      24
Hoerl, A.E.      4 6 9 10 11 12 13 18 20 21 23 24 25 27 28 30 69 77 101 110 112 129 181 208 310 317 322 343 376 377 382 383
Hoerl, R.W.      11
Hogg, R.V.      143
Holzworth, R.J.      21
Horn, S.D.      11 95
Hu, H.      27
Huber, P.J.      23 29
Hubert, M.H.      28
Humak, K.M.S.      15
Ibrahim, M.      28
Iemma, A.F.      20
Inadmissible estimator      10 15 34 53Ч55 111 144 330
Incorrect, ellipsoid      137 220
Incorrect, prior information      14 28 32 136Ч140 201Ч220 242Ч244
Incorrect, prior mean      201Ч205 210 242
Increasing function      163 195 328 329 378 384
Independent linear models      140 303 304 324
Inference procedure      225 251
Information geometry      4 19 33 354Ч389
Information geometry, Fisher information      19 354 355 358Ч361 389
Information geometry, Rao distance      19 354 355 361 363Ч369 372Ч374 386 389
Initial prior dispersion      248
Initial prior information      234 239
Initial prior mean      225
Inner product      358 359
Inoue, T.      23
Iterative procedure      225 227
Iyengar, N.S.      14
Jack-knife estimator      16Ч18 30 171
James Ч Stein estimator      10 12 13 15 20 22 23 25 27 54 55 94 110
James, W.      10 12 13 15 20 22 23 25 27 54 55 94 110 111 140 219 251 259 265 275 277 278 322
Jin, C.      20 29
Judge, G.      203
Kaciranlar, M.R.      29 30 111
Kadiyala, K.      18 23
Kagan, L.M.      91
Kalman filter      4 18 19 32 95 223Ч251 354 355 387Ч389
Kalman gain      227 237
Kalman, R.E.      4 18 32 95 223 224 225 227 228 229 230 235 236 237 239 242 244 245 248 251 354 387 388
Kauermann, G.      289
Kennard, R.W.      4 6 9 10 11 12 13 18 20 21 23 24 25 27 28 30 69 77 101 110 112 129 181 208 310 317 322 343 376 377 382 383
Khalaf, G.      27
Kibria, B.M.G.      21 24 25 26 29
Kim, D.      31
Kim, D.H.      31
Kim, M.      31
Klintworth, M.      26
Knots      283 286Ч289 291 294
Kozak, J.      14
Kronecker product      33 63Ч65
Krzanowski, W.J.      19 368 376
Kubokawa, T.      26 27
Kuks, J.      20 22
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