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Shorack G.R. — Probability for statisticians
Shorack G.R. — Probability for statisticians

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Название: Probability for statisticians

Автор: Shorack G.R.

Аннотация:

Probability for Statisticians is intended as a text for a one year graduate course aimed especially at students in statistics. The choice of examples illustrates this intention clearly. The material to be presented in the classroom constitutes a bit more than half the text, and the choices the author makes at the University of Washington in Seattle are spelled out. The rest of the text provides background, offers different routes that could be pursued in the classroom, ad offers additional material that is appropriate for self-study. Of particular interest is a presentation of the major central limit theorems via Stein's method either prior to or alternative to a characteristic funcion presentation. Additionally, there is considerable emphasis placed on the quantile function as well as the distribution function. The bootstrap and trimming are both presented. The martingale coverage includes coverage of censored data martingales. The text includes measure theoretic preliminaries, from which the authors own course typically includes selected coverage. The author is a professor of Statistics and adjunct professor of Mathematics at the University of Washington in Seattle. He served as chair of the Department of Statistics 1986 — 1989. He received his PhD in Statistics from Stanford University. He is a fellow of the Institute of Mathematical Statistics, and is a former associate editor of the Annals of Statistics.


Язык: en

Рубрика: Математика/Вероятность/Статистика и приложения/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 586

Добавлена в каталог: 10.06.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$P_{\infty}$      36
$\sigma$-field      3
$\sigma$-field, $\mathcal{A} = \sigma[C]$      12
$\sigma$-field, $\mathcal{\hat{A}}_{\mu}$, the completed $\sigma$-field      15
$\sigma$-field, $\mu^{*}$-measurable sets $\mathcal{A}^{*}$      4
$\sigma$-field, histories      305 469
$\sigma$-field, induced $\mathcal{F}(X)\equivX^{-1}(\mathcal{\bar{B}})$      24
$\sigma$-field, predictable      513
$\sigma$-field, preservation of      22
$\sigma$-field, symmetric      156
$\sigma$-field, tail      155
$\Sigma^{-}$ and $\Sigma^{-1/2}$      192 379
A.b.f.      8
Absolute continuity      255 258
Absolute continuity of functions      74 76 78 323
Absolute continuity of log f      78 138
Absolute continuity of measures      60 63 64 66—68 78 108
Absolute continuity of the integral      42 54 66 76 480
Absolute continuity Radon — Nikodym      66
Absolute continuity, fundamental theorem of calculus      76 78 137
Adapted      35 36 305
Added touch      124 268
Ancillary statistic      173
Approximation by continuous functions      59 60
Approximation by elementary functions      26
Approximation by simple functions      26
Approximation by step functions      59 78
Approximation for $\mathcal{L}(h(\bar X_{n}))$      396
Approximation lemma, for sets      16 80 155 156
Associated rvs      264
Best linear predictor      194
Bootstrap      274 277 388 432
Bootstrap Bayesian      434
Borel sets      5 6 17 18 23
Borel sets $\mathcal{B}$, $\mathcal{\hat{B_{\mu}}}$, $\mathcal{\bar B}$      18
Borel sets, countable base $\mathcal{B_{C}}$ of $\mathcal{B_{T}}$      91
Borel sets, generators of $\mathcal{B_{\infty}}$      36 86
Borel sets, generators of $\mathcal{B}$      24
Borel sets, generators of $\mathcal{B}_{n}$      23 35 80
Borel–Cantelli      204 217 218 236 237 244 321 376
Bounded variation      74—76
Bounded variation, total variation measure      62
Branching processes      484
Brownian bridge      202 302 326
Brownian motion      202 235 302 311 318 469 500 536
Brownian motion integrals of      303
Brownian motion LIL      303 320
Brownian motion path properties      323
Brownian motion reflection principle      314 315 321 322 331 409
Brownian motion transformations      303 318
Cantor set      108 114
Caratheodory covering      12 17 77
Caratheodory extension theorem      5 10 12 80 88 480
Censored data      523 527
Chain rule      67 78
Change of variable      78 113 134
Change of variable, Radon–Nikodym      66—68
Change of variable, unconscious statistician      42 67 113
CHF      242 294 341 351
Chf continuity theorem      342 350 351 366 381
Chf distributions on grids      361
Chf examples      343—345 349
Chf expansion for uan array      400
Chf Fourier, Laplace, etc.      341 356
Chf moment expansions      352—354 371 374
Chf tail behavior      355
Chf uniqueness theorem      342 346 347 350 351 366
Chf, complementary pair      346
Chf, empirical      348
Chf, Esseen's lemma      358 371 373 385 391 394
Chf, inversion formula      346—348 359 362 380 381 385 386 393
Chf, multivariate normal      200
Chisquare tests      369 379 389
CLT      125 126 147
CLT for general iid uan arrays      406
CLT, asymptotic normality condition      265 406 529
CLT, basic dependent      529
CLT, Berry–Esseen      259 261 263 267 270 276 278 371 376 450
CLT, Berry–Esseen for U-statistics      454
CLT, bootstrap      274 434
CLT, classical      183 238 342 366
CLT, connection with variance estimation      125 233 270 271
CLT, creating normality      428
CLT, delta method      279
CLT, Doeblin      285 370
CLT, Edgeworth (see also)      391
CLT, examples      279 280 282 283 286 287 368 369
CLT, finite sampling      388 430 464 465
CLT, Frechet — Shohat via moments      293
CLT, gamma approximation      383 388 389 459
CLT, Hoeffding's combinatorial      458
CLT, L-statistics      441 442
CLT, Liapunov      371 377
CLT, Lindeberg — Feller      260 266 267 373 404 529
CLT, local limit (see also)      380
CLT, martingale      530
CLT, multivariate      201 368
CLT, negligibility      215 233 234 367
CLT, Poisson approximation      386
CLT, R-statistics      427
CLT, random sample size      285
CLT, Rebolledo      503 505
CLT, slowly varying $\tilde{\sigma}(\cdot)$      233 268 270—272
CLT, Stein's approach      255 262 263 459
CLT, Studentized CLT for $\tilde{\sigma}^{2}(\cdot) \in \mathcal{L}$      270 272
CLT, trimmed means      416 417 420
CLT, U-statistics      450 453
CLT, universal bootstrap CLT      277
CLT, universal studentized CLT      276
CLT, Winsorized      264 270
Compensator      487 488 501 511
Complete measure      15 18 29
Complete space $\mathcal{L}_{r}$      51
Completeness assumption      469
Conditional expectation      158 159 168 171 246 248
Conditional expectation, properties of      163
Conditional probability      159 160 186 201
Conditional probability, regular      10 168—170
Contiguity      567
Convergence a.e.      29—31 51
Convergence a.s.      33 78 89 114 241
Convergence almost uniform      60
Convergence in $\mathcal{L}_{2}$      78
Convergence in distribution      33 52 89 112 114 241 288 290 291 293 342 531
Convergence in measure      30 31 51
Convergence in probability      33 51 89 241
Convergence in quantile      112
Convergence in rth mean      33 51 89
Convergence in sub-df      288
Convergence modes      57
Convergence set      29 217 242 252
Convergence uniform      60
Convergence, Cesaro summability      205 217
Convergence, DCT      41
Convergence, fd convergence $\rightarrow_{fg}$      91
Convergence, MCT      40
Convergence, mutual      29—31 51
Convex      46 48 49 56 468
Convolution formula      187 347–349
Correlation coefficient      286 463
Correlation coefficient, multiple      194
Correlation coefficient, Spearman      462
Counting process      470 501
Coupling      262 389 542
Cramer — Levy continuity theorem      350
Cramer — von Mises      304 331 338 339
Cramer — Wold device      351 368 378
Cumulant gf      356 371 384 387 388 392
Cumulative hazard function      470 506 522 527
Decomposition of Normals      405
Decomposition of Poissons      405
Dense in Hilbert space      106
Dense, Bernoulli polynomials      223
Dense, continuous functions      59 78
Dense, step functions      59 78 482
Density estimation      348
Derivative series      71
Derivative under integral sign      45 317
Derivative, differentiable      70 75 76
Derivative, Dini derivates      70
Derivative, Lebesgue theorem      70 71 78
Derivative, Radon — Nikodym      66—68 74 76—78 158
Derivative, Taylor's theorem      72 352
Determining class      347
Determining class limit      53
Determining class moments      293 294 354
Determining class, $C$, $C_{0}$, $C_{b}$, $C_{bu}$, $C_{b}^{(k)}$, $C_{b}^{(\infty)}$      292
Determining class, $\mathcal{G}$, $\mathcal{G}_{0}$, $\mathcal{G}_{1}$, $\mathcal{G}_{2}$      292
Determining class, chf      294
df      20 33 107
Df properties      113
Df support      110
Df, absolutely continuous      108
Df, decomposition of      107
Df, generalized      18 19 27
Df, joint      84 85 154
Df, jumps and flat spots      110 134
Df, Lebesgue singular      71 108
Df, singular      108
Df, sub      33 107 288
Diagonalization      289
Discontinuity set      26
Distribution sampling without replacement      180
Distribution, Bernoulli      179
Distribution, Beta      184 549
Distribution, binomial      179 555
Distribution, Cauchy      184 187 242 284 342—344 412
Distribution, chisquare      183
Distribution, compound Poisson      402
Distribution, de la Vallee Poussin      343 346 348 363
Distribution, discrete uniform      555
Distribution, double exponential      184 283 345 558
Distribution, exponential      181 287
Distribution, extreme value      286 559
Distribution, gamma      182 383 392 406 556
Distribution, geometric      179
Distribution, hitting time      409
Distribution, hypergeometric      180 555
Distribution, log gamma      559
Distribution, logistic      184 345 558
Distribution, multinomial      185 369 402
Distribution, multivariate normal      199 286
Distribution, NegBiT      179 387 555
Distribution, noncentral      378 379 389
Distribution, normal      183 287 405 549 556
Distribution, Pareto      562
Distribution, Poisson      181 284 287 386 405 555
Distribution, Snedecor's $F_{m,n}$      188 557
Distribution, stable laws (see also)      407
Distribution, Student's $t_{m}$      188 557
Distribution, triangular      343 346 348
Distribution, uniform      184 558
Distribution, Weibull      561
Divergence set      29
Domain of attraction      145
Domain of attraction, $\mathcal{D}$(Normal)      267 268 271 272 277
Domain of attraction, $\mathcal{D}(G)$ of the stable law $G$      407 412
Domain of attraction, $\mathcal{D}_{N}$(Normal)      413
Domain of attraction, $\mathcal{D}_{N}(G)$ of the stable law $G$      407 412
Domain of attraction, statistical $\tilde{\mathcal{D}}$      145 417 442
Domain of attraction, total $\tilde{\mathcal{D}}$      145
Edgeworth expansions      386 391 392
Eigenvalues      191 196 340
Embedding, Csoergo, Csoergo, Horvath, Mason      328
Embedding, Shorack      328
Embedding, Skorokhod      235 312 318 329
Empirical chf      348
Empirical df $\mathbb{F}_{n}$      121 223 325
Empirical df $\mathbb{G}_{n}$      121 223 325 417
Empirical finite sampling process $\mathbb{R}_{n}$      326
Empirical process $\mathbb{E}_{n}$      121
Empirical process $\mathbb{U}_{n}$      121 325 368 417
Empirical qf $\mathbb{K}_{n}$      120
Empirical quantile process $\mathbb{V}_{n}$      325
Empirical two-sample process $\mathbb{W}_{m,n}$      331
Empirical weighted process $\mathbb{W}_{m}$      326
Equivalent summands      267
Euclidean space      23
Euler's constant      186 546 550
Expectation      38
Expectation of products      152
Extremes      370
Filtration      469 489 493 511
Finite-dimensional convergence projection mappings $\pi_{t_{1},\dots,t_{k}}$      90
Finite-dimensional convergence rectangle      86 89
Finite-dimensional convergence subsets      90
Finite-dimensional convergence, $\rightarrow_{fd}$      91
Finite-dimensional convergence, $\sigma$-field      90
Finite-dimensional convergence, distributions      90
Function, $\psi$-function      59
Function, elementary      24—26
Function, indicator      3 24
Function, measurable      24
Function, simple      24—26 28 37 166
Function, step      59
Gambler's ruin      499 500
Gamma approximation; the GLT      383
Gamma function      546
Gamma function, digamma      546
Generalized inverse      193
Generators      4
Generators of the induced $\mathcal{F}(X)\equivX^{-1}(\mathcal{\bar{B}})$      24
Generators of the induced $\mathcal{F}(X_{1},X_{2},\dots)$      35
Generators of the induced $\mathcal{F}(X_{1},\dots,X_{n})$      35
Generators regarding $X^{-1}$      22
Generators regarding independence      9
Generators, $\sigma$-field      3 10 24
Generators, Glivenko — Cantelli      223 325 326
Generators, various for Borel sets $\mathcal{B}$      24
Generators, various for Borel sets $\mathcal{B}_{n}$      23 35 80
Generators, various for Borel sets $\mathcal{B}_{\infty}$      36 86
Hermite polynomials      317 390 391 470
Hewitt — Savage zero-one law      156
Hilbert space      104 174
Hilbert space, $\ell_{2}$-isomorphism      106
Hilbert space, $\mathcal{L}_{2}$ is a Hilbert space      104
Hilbert space, alternating projections (ACE)      175
Hilbert space, Fourier coefficients      105 106 340
Hilbert space, Gram–Schmidt      106
Hilbert space, idempotent operator      174
Hilbert space, inequalities      104 106
Hilbert space, orthogonal projections      105 174
Hilbert space, orthonormal basis      105 106
Hilbert space, projection operator      176
Hilbert space, self-adjoint operator      174
Hilbert space, subspaces      105
Hitting time      311 314 409
Hodges — Lehmann estimator      455 456
I.o.      8 204
Increment function      18
Independence tests      369
Independent      163
Independent $\sigma$-fields      151 153
Independent rvs      151 153 154 157 189 200 201 351
Indicator function proof      28 38 42 66 81 83 164 166
Induced distribution      23 24 27 33 42 52 92
Induced distribution consistency      87
Inequality, $C_{r}$-inequality      47 163 209 313 372
Inequality, basic      49
Inequality, Birnbaum–Marshall      249 333
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