Borovkov A.A. — Mathematical statistics :: Электронная библиотека попечительского совета мехмата МГУ

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Borovkov A.A. — Mathematical statistics

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Название: Mathematical statistics

Автор: Borovkov A.A.

Аннотация:

A wide-ranging, extensive overview of modern mathematical statistics, this work reflects the current state of the field while being succinct and easy to grasp. The mathematical presentation is coherent and rigorous throughout.
The author presents classical results and methods that form the basis of modern statistics, and examines the foundations of estimation theory, hypothesis testing theory and statistical game theory. He then considers statistical problems for two or more samples, and those in which observations are taken from different distributions. Methods of finding optimal and asymptotically optimal statistical procedures are given, along with treatments of homogeneity testing, regression, variance analysis and pattern recognition. The author also posits a number of methodological improvements that simplify proofs, and brings together a number of new results which have never before been published in a single monograph.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 570

Добавлена в каталог: 04.12.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

Actually attained level      265
Approximate computation of maximum-likelihood estimators      204
Asymptotic approach to comparison of estimators      82 88 94
Asymptotic confidence interval      227
Asymptotic confidence set      234
Bayes formula      110
Bayes principle      476
Bayesian approach complete      267 293
Bayesian approach partial      267 294
Behrens — Fisher problem      369 381
Beta distribution      46
Bias      80
Brownian bridge      20
Classification of particles      207
Close hypotheses      257 329
Complete class of strategies      471
Complete family of distributions      130
Condition       66
Condition       181
Condition $(A_\mu)$       67
Condition (r)      140
Condition (RR)      192 213
Conditional density      107
Conditional distribution      106
Conditional expectation      101
Conditional probability      106
Confidence bounds      226
Confidence interval      226
Confidence level      226
Confidence set      234 285
Confidence set, exact      290
Confidence set, invariant      291
Confidence set, unbiased      290
Continuity theorems      13
Correlation coefficient for a sample      13
Critical region      249
Data grouping      348
Decision function (decision rule, decision)      472
Decision function, asymptotically Bayesian      502
Decision function, asymptotically minimax      502
Decision function, invariant      485
Decision function, randomized      485
Decision function, unbiased      483
Density, a posteriori      110
Density, a priori      110
Density, conditional      107
Dispersion ellipsoid      89
Distance, $\chi^2$       177
Distance, Hellinger      177
Distance, Kullback — Leibler      177 253
distribution      1
Distribution polynomial      50
Distribution, $\chi^2$ (chi-square)      43
Distribution, a posteriori      110 243
Distribution, a priori      110 242
Distribution, Bernoulli      50
Distribution, Cauchy      49
Distribution, degenerate      49
Distribution, exponential      44
Distribution, Fisher      44
Distribution, least favorable or worst      112 247 276 295
Distribution, lognormal      49
Distribution, normal      41
Distribution, Poisson      50
Distribution, Student      45
Distribution, uniform      47
Empirical distribution      3 11
Empirical distribution function      5 12
Empirical distribution, smoothed      36
Empirical process      16
Equivariant estimation      165
Estimator      40 51
Estimator of density      39
Estimator of the scale parameter      157
Estimator of the shift parameter      157
Estimator, asymptotically Bayesian      115 174
Estimator, asymptotically efficient      94 194
Estimator, asymptotically minimax      115 174
Estimator, asymptotically normal      54
Estimator, asymptotically R-Bayesian      174
Estimator, asymptotically R-efficient      144
Estimator, Bayesian      110 115
Estimator, consistent      53
Estimator, efficient      92 100
Estimator, equivariant      166
Estimator, maximum-likelihood      69 188
Estimator, minimax      111 116
Estimator, Pitman      160
Estimator, R-efficient      144
Estimator, strongly consistent      53
Estimator, substitution      52
Estimator, sufficient      128
Estimator, superefficient      169
Exponential family of distributions      133
Fisher information      140 152 179
Functional limit theorem for empirical processes      21
Functional, continuously differentiable      26
Functionals of types I and II      8
Gamma distribution      42
Glivenko — Cantelli theorem      4
Hypothesis, composite      264
Hypothesis, simple      240
Interval estimation      225
Invariant family      165 281
Jensen inequality for conditional expectations      104

Kolmogorov — Smirnov test      383
Law of iterated logarithm for empirical distributions      12
Law of iterated logarithm for maximum-likelihood estimators      199
Likelihood ratio      189 211
Logarithmic likelihood function      70
Maximal invariant      284
Maximum-likelihood method      67
Mean square approach to comparison of estimators      80 87
Median      7
Minimal sufficient $\sigma$ -algebra      123
Minimum-distance method      65
Moments method      55 56
Monotone likelihood ratio      268
Neyman — Fisher theorem (factorization theorem)      117
Neyman — Pearson lemma      250
Neyman — Pearson lemma, generalized      274
Null hypothesis      249 264
Orbit      167
Order statistics      5
Poisson process      17
Power of a test      249
Principle of in variance      158
Principle of sufficiency      158
Principle of unbiasedness      158 277
Probability of errors of the first kind      264
Probability of errors of the ith kind      241 245
Quantile      7
Radiation source      163
Rao-Cramer inequality      139 146
Rao-Cramer inequality, difference      184
Rao-Cramer inequality, integral      167
Regression      390
Regression, linear      390
Regressor      391
Risk (risk function)      472
Robustness      357
Sample      1
Sample characteristics      7 12
Sample characteristics, median      7
Sample characteristics, moments      7 12
Sample characteristics, quantile      7
Schwarz inequality for conditional expectations      104
Sequential analysis      306
Sequential estimation      224
Sequential test      308
Sign test      319 384
Significance level      265
Size of a test      249 264
Statistic      8
Statistic, $\chi^2$       23 27
Statistic, complete      130
Statistic, minimal sufficient      122
Statistic, of types I and II      8
Statistic, sufficient      116
Statistic, trivial sufficient      123
Strategy      462
Strategy randomized (mixed)      463
Strategy, Bayesian      463
Strategy, equalizing      468
Strategy, minimax      465
Strategy, pure      463 474
Strategy, uniformly optimal      462
Substitution method      51
Sufficient $\sigma$ -algebra      123
Sufficient estimator      128
Support of a distribution      8
test      241
Test for testing homogeneity      382
Test for testing homogeneity, $\chi^2$       389
Test for testing homogeneity, asymptotically minimax      370
Test for testing homogeneity, consistent      382
Test for testing homogeneity, nonparametric      383
Test nonparametric      319
Test of asymptotic level      253 325
Test unbiased      277
Test uniformly most powerful      266
Test, $\chi^2$       345
Test, $\omega^2$ (von Mises — Smirnov)      318
Test, asymptotically Bayesian      325 331
Test, asymptotically Bayesian unbiased      328
Test, asymptotically Bayesian, equivalent      258 332
Test, asymptotically Bayesian, minimax      331
Test, asymptotically Bayesian, most powerful      258
Test, asymptotically Bayesian, uniformly most powerful      330
Test, Bayesian      242 266 293
Test, consistent      316
Test, invariant      281
Test, Kolmogorov      317
Test, likelihood ratio      303
Test, minimax      246 267
Test, Moran      320
Test, most powerful      242 247
Testing homogeneity      368
Total probability formula      105
Two-person game      462
Two-person game randomized (mixed)      463 474
Two-person game statistical      472
Uniform central limit theorem      219
Uniform convergence in probability      220
Uniform convergence of distributions      220
uniform convergence of integrals      193
Uniform law of large numbers      219
Value of the game      465
Wiener process      20
Wilcoxon test      385

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