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| Ferguson T.S. — Mathematical Statistics. A Decision Theoretic Approach |
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| Предметный указатель |
 -distribut.ion 101 103
 -distribution 10:3
Absolute error loss 47
Acceptance region 258
Action space 1
Action space, terminal 309
Admissible decision rule 54 70
Admissible decision rule,  63
Admissible decision rule, almost 185
Agreeing preference pattern and utility 13
Alternative hypothesis 199
As good as 54
Basu's Theorem 350
Bayes deeision rule 31 37 44 31:3
Bayes deeision rule, 32 38
Bayes deeision rule, admissibility of 60 62
Bayes deeision rule, existence of 67
Bayes deeision rule, extended 50
Bayes deeision rule, generalized 50
Bayes deeision rule, limit of 49
Bayes deeision rule, nonrandomized 43 60
Bayes deeision rule, sequential 313 353
Bayes risk 31
Behavioral and randomized rule 24 25
Behavioral and randomized rule,equivalence of 25 26
Behavioral and randomized rule,relation between invariant 151 192
Behavioral strategy 25 26
Bernoulli trials 318 332 338 360 377
Bernoulli trials, dependent 328 340
Best invariant rule 154
Best invariant rule, not minimax 175
Best test 200
Beta distribution 101 223
Beta-binomial distribution 97 100 104 105 328
Better than 54
Binomial distribution 98
Binomial distribution, Bayes estimates for 52 53
Binomial distribution, Bayes sequential estimates for 318ff
Binomial distribution, minimax estimates for 93 95
Bio-assay 123
Bivariate normal distribution 112
Bivariate Poisson distribution 241
Bluffing 3 43 214
Boundary 67
Bounded from below 63
Boundedly complete sufficient statistic 132
Canonical form 266
Cauchy distribution 102 174 236 239 247 298
Chance move 3
Characteristic fundion 99 107
Classification problems 291
Closed from below 64
Closure 64
Compact 84
Complete class of decision rules 55
Complete class of decision rules, essentially 55
Complete class of decision rules, essentially minimal 212
Complete class of decision rules, minimal 55 69
Complete class of decision rules, minimal essentially 55
Complete class theorem 58 87
Complete sufficient statistic 132
Composite hypothesis 206
Confidence ellipsoids 276
Confidence sets 257
Confidence sets, invariant family of 261
Confidence sets, unbiased family of 260
Convex function 76
Convex function, strictly 80
Convex hull 36 65
Convex set 35
Convex set, strictly 75
Correlation coefficient 112
Cost function 310
Covariance matrix 105 106
Critical region 198
Decision function or rule 7
Decision function or rule, behavioral 25
Decision function or rule, best 28
Decision function or rule, equalizer 90
Decision function or rule, invariant 148 341
Decision function or rule, monotone multiple 286
Decision function or rule, nonrandomized 7
Decision function or rule, randomized 24
Decision function or rule, reasonable 29
Decision function or rule, sequential 312
Decision function or rule, terminal 311
Decision problem 7
Decision problem, sequential 315
Decision problemtruncated sequential 315
Diagonal matrix 105
Digamma function 179
Discrimination 295
Dominated family of distributions 116
Double exponential distribution 241 256 378
Dvoretsky — Wald — Wolfowitz theorem 79 142
Equalizer rule 90
Equivalence relation 11 153
Estimation 11 30 46
Estimation of a distribution function 192 348
Estimation of a median 197 350
Exponential distribution 101 180 255 327 349
Exponential family of distributions 126 139 209 217 227
Extended Bayes rule 50 59 87 91
Factorization theorem 115
Fibonacei numbers 339
Fisher — Yates test 252
Fundamental identity 373
Fundamental lemma 201
Fundamental problem of decision theory 28 34
Gambler's ruin 371 378
Game 1
Game, statistical 7
Gamma distribution 52 101 136 181
Gamma function 101
Generalized Bayes rule 50
Geometric distribution 95 99 180 234 327 339
Glivenko — Calltelli lemma 192
Group of trallsformations 143 147
Half-normal distribution 190
Hoelder's inequality 80 128
Hypergeometric distribution 97 99 328
Hypothesis, composite 206
Hypothesis, general linear 264
Hypothesis, null 199
Hypothesis, simple 199
Identifiability 144
Identity transformation 143
Inadmissible decision rule 54
Indicator function 100
Inner product 71
Interior 67
Interval estimate 184
Invariance 30 143
Invariance and convex loss 158
Invariance and sufficiency 157
Invariance in testing hypotheses 242
Invariance of cost 341
Invariance of decision problem 145 341
Invariance of distributions 144 341
Invariance of general linear hypothesis 268
Invariance of loss 145 341
Invariance of prior distribution 162 299
Invariant rules and prior information 152
Inverse probability methods 32
Inverse transformation 143
Jensen's inequality 76 122 331
Laplace transformation 134
Latin square 274
Least favorable distribution 34 39 57 81 90
Least squares 269
| Lebesgue integral 7 138
Lebesgue measure 50
Locally best test 235 249 255
Locally best test, one-sided 235
Locally best test, unbiased 237
Location parameter 164
Location parameter, best invariant estimate of 168 186 345
Location parameter, best invariant test for 245
Location parameter, minimax estimate of 171
Location-scale parameter 166 184
Logistic distribution 102 123 215 241 256 200
logit 121
Loss function 1 309
Lower boundary point 64
Lower quantant 63
Lower semicontinuity 84 86
Mann — Whitney test 251
Maximal invariant 243 247 343
Maximum likelihood estimate 123
Mean squared error 11
Minimax decision rule 83 39 57 90
Minimax decision rule, 33 39
Minimax theorem 57 82 85
Minimum estimate 123
Minimum Bayes risk 31 39
Minimum logit- estimate 123
Mixed strategy 23 26
Modal interval 51
Moment generating function 37:3
Monotone likelihood ratio 208 259 286 218 307
Move 2
Multinomial distribution 205
Multiple decision problem 10 284
Multiple decision problem, monotone 285
Natural ordering 54
Natural parameter space 128
Necessary and sufficient statistic 132
Negative binomial distribution 96 99
Neyman structure 227
Neyman — Pearson lemma 201
Neyman — Pearson lemma, generalized 204 238
Noneentral , , and t-distributions 102 103
Nonnegative definite matrix 80 106
Nonregular families 130 137 259
Nonsingular normal distribution 109
Normal distribution 101
Normal distribution, admissibility of sample mean 141 142 185
Normal distribution, inadmissibility of sample mean 169
Normal distribution, independence of sample mean and variance 115
Normal distribution, invariant estimates for 135 182ff
Normal distribution, minimax estimates for 94 176ff
Normal distribution, multiple decisions for 288 290 294 303ff
Normal distribution, multivariate 107 119 153 169 294
Normal distribution, tests for 207 234 246 249 265ff
Normal distribution, unbiased estimates for 48 122 135
Normal form of game 1
Null hypothesis 199
Objectivists 32
Odd or even 2 8 40 42
One-sided t.est 212 286
One-to-one transformation 143
One-way classification 265 273 280 282
Onto transformation 143
Orbit 149
Order statistics 119 153 192 249 252
Orthogonal group 148 153 244 268
Orthogonal matrix 105
Parameter space 1 309
Payoff 11
Personal probability 17
Pitman estimate 188 247
Poisson distribution 99
Poisson distribution, Bayes estimates for 52 326
Poisson distribution, minimax estimates for 96 327
Poisson distribution, unbiased estimates for 136
Poisson distribution, unbiased tests for 233
Positive definite matrix 106
Power function 198 199 235
Preference pattern 12
Prior distribution 31 37 60
Prior distribution, invariant 162 299
Probability of error 10 200
Pseudo-game 214
Quantile 51
Random stopping time 311
Randomized decisions 22
Randomized decisions, space of 22
Rank test 251
Ranking problems 294
Rao — Blackwell theorem 121 125 331
Rao-Blackwellization 124
Regression 265 273 283
Riemann integral 8 133
Riemann — Stieltjes integral, S 170
Risk function 7 312
Risk set 34 63
Risk set, nonrandomized 36
Rule see Decision function
Sample distribution function 192 196 348
Sample space 6 310
Scale parameter 164 178 190
Semicontinuity 84
Separating hyperplane theorem 73 82 87
Sequential probability ratio test 361
Sequential probability ratio test, extended 362
Sequential probability ratio test, optimal property of 365 369
Set estimate 283
Similar test 266
Simple function 8
Simple hypothesis 199
Singular normal distribution 109
Size of a test 200 206
Slippage problems 299ff
Spectral theorem 106
Squared error loss 46
Statistical decision problem or game 7
Stopping rule 310
Stopping rule, invariant 341
Strategy 2
Strategy, mixed 23
Subjective probability 17
Subjectivists 32
Sufficiency and invariance 157
Sufficient, sequence 330
Sufficient, statistic 113
Sufficient, statistic, complete class of rules based on 120
Sufficient, statistic, minimal 132
Support of a distribution 61
Supporting hyperplane theorem 73 77
t-distribution 103
t-test 230 232 234 246 263
Terminal action space 309
Terminal decision rule 311
test 198 199
Testing hypotheses 10 198
Three-way classification 274
Tmnsitive group 174
Tramformation 143
Transitive sufficient, sequence 334
Transpose of a vector 71
Two-sided problems 222
Two-sided tests 216
Two-way classification 266 271 281 283
Unbiased estimates 29 48 122 134
Unbiased estimates, sequential 332
Unbiased tests 224ff
Uniform distribution 100
Uniform distribution, Bayes estimates for 45
Uniform distribution, best invariant estimate for 181 185 190 191
Uniform distribution, minimax sequential estimate for 346
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