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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
Inequality, Bonferroni      50
Inequality, bounds on $(1-x/n)^{n}$      354 550
Inequality, Cantelli      221
Inequality, Cauchy — Schwarz      48 197
Inequality, Chang      336
Inequality, Chebyshev      49 218
Inequality, chf bound on tails      350
Inequality, chf key inequality      372
Inequality, convexity      48
Inequality, correlation      48
Inequality, Daniels' equality      214 230 335
Inequality, dispersion      167 209
Inequality, Doob      247 249 335
Inequality, Etemadi      212
Inequality, geometric mean      50
Inequality, Gine — Zinn symmetrization      213 273
Inequality, Hajek — Renyi      221 248 333 452 453
Inequality, Hardy      50
Inequality, Hilbert space inequalities      104 106
Inequality, Hoeffding — Frechet      85
Inequality, Hoelder      47 48 163 209
Inequality, Hoffman — Jorgensen      250
Inequality, Jensen      49 163 166 167 209 248 468
Inequality, Khinchin      213 273
Inequality, Kolmogorov      210 217 218 247 248 253 285 321
Inequality, Kolmogorov's other      243
Inequality, Levy      211 220 235
Inequality, Liapunov      48 163 209 371
Inequality, Littlewood      48
Inequality, Markov      49 209
Inequality, Mills' ratio      235 237 321 322
Inequality, Minkowski      49 53 163
Inequality, moment expansions of chfs      352 354 371 374
Inequality, monotone      205 221 247 248 321
Inequality, Ottavani — Skorokhod      212 242
Inequality, Paley–Zygmund      49 273 324
Inequality, Pyke–Shorack      328 334 336
Inequality, sandwiching the mean      206 207 215 217 218
Inequality, Shorack      141 143 147 214 256 458
Inequality, Shorack–Smythe      247
Inequality, symmetrization      210 219
Inequality, truncation      219 227–229 232
Inequality, upcrossing      473
Inequality, Wellner      50
Inequality, Winsorized variance      141 143 147
Inequality, Young      47
Infinitely divisible      400
Infinitely divisible, limits of      400
Infinitely divisible, log chf is never zero      400
Infinitely divisible, subclass $\mathcal{I}_{2}$      400
information      35
Integrable      38
Integrable collection      54
Integrable, $\mathcal{L}_{1}$      37
Integrable, $\mathcal{L}_{2}$      78
Integrable, $\mathcal{L}_{r}$      37 47
Integrable, $\mathcal{L}_{r}^{+}$      37
Integrable, product      82
Integrable, uniformly      54—56 221 252 473 477 478 488 492 511 516
integral      38
Integral, improper      44
Integral, Lebesgue      37
Integral, Lebesgue — Stieltjes      44
Integral, linearity      38 163
Integral, Riemann      1 75
Integral, Riemann — Stieltjes      44
Integration by parts formulas      115 118
Inverse image      21 22
Inverse image of $\sigma$-fields      22
Inverse transformation      111
Jacobian      78 190
Khinchin equivalent rvs      206 207 215 244
Kolmogorov consistency theorem      92 302
Kolmogorov Extension Theorem      87
Kolmogorov inequality (see also)      210
Kolmogorov representation theorem for $\mathcal{I}$      402
Kolmogorov zero-one law      155 204 482
Kolmogorov, Gnedenko — Kolmogorov theorem      123 232
Kolmogorov, Kolmogorov — Smirnov      316 331
Kolmogorov, SLLN (see also)      215 275
Kullback — Leibler information      564 567
Large deviations      395
Lebesgue decomposition theorem      64 66 108
Lebesgue integral      37
Lebesgue measure      6 23 38 67
Lebesgue measure $\lambda_{n}$      80
Lebesgue sets      15 17
Lebesgue singular df      71 108
Lebesgue sums      1
Lebesgue theorem re derivatives      70 71 78
Lebesgue, Lebesgue — Stieltjes measure      4 18 20 77 78 80
Likelihood ratios      470
LIL      235 238 321 323 303 see
LIL, U-statistics      452
Limit determining class      292
Limit theorem, general uan terms      405
Limit theorem, uan terms with negligible variances      403
Lindeberg's $LF_{n}^{\in}$      260 266 267 278 283 373 375 376 436
Linear algebra      191
Lipschitz condition      74 540
LLN      126 148 218 222 224—226
LLN, Glivenko — Cantelli (see also)      223
LLN, negligibility      215 226
LLN, random sample size      220
LLN, SLLN of Kolmogorov      208 215 220 275 481
LLN, strong      221 224 238 245 325
LLN, U-statistics      452 453 481
LLN, uniform SLLN of Chung      252
LLN, weak      208 224 226 228 245 366 377 406 425
LLN, WLLN of Feller      208 215 220
LLN, WLLN of Khinchin      208
Local limit theorem      380
Mann — Whitney statistic      456
Martingale      246 311 317 467
Martingale, $\geqq$ notation      246
Martingale, closes      473 477 478 493
Martingale, CLT      530
Martingale, convergence theorem      473 477 478
Martingale, counting process      470
Martingale, decomposition      487 488 502 511 512
Martingale, equivalence      246 467
Martingale, examples re empiricals      333
Martingale, exponential      470 499 500
Martingale, integrable      468
Martingale, Kakutani      470 482
Martingale, local      511
Martingale, optional sampling theorem      312 472 492 493
Martingale, reversed      335 453 478 481 498
Martingale, s-mg      246 467
Martingale, square-integrable      468
Martingale, sub mg      246 247 467
Martingale, transform      489 490 502 513 516
Martingale, U-statistic      452
Martingale, Wald      469
Mason theorem      329 332 336
Maximum likelihood      551 563
Measurability criterion      24 25
Measurable      24
Measurable as a limit      25 29
Measurable function spaces      90
Measurable function spaces, $(C, \mathcal{C} )$      90 295 298
Measurable function spaces, $(C_{[0,\infty)}, \mathcal{C}_{[0,\infty)})$      297
Measurable function spaces, $(D, \mathcal{D})$      295 298 469
Measurable function spaces, $(M, d)$ with $\mathcal{M}_{d} or \mathcal{M}_{d}^{B}$      295
Measurable function spaces, $(R_{T}, \mathcal{B}_{T})$      91
Measurable function spaces, $(R_{[0,1]}, \mathcal{B}_{[0,1]})$      90
Measurable function spaces, general space $(M_{T}, \mathcal{M}_{T})$      90
Measurable partition      37
Measurable set      26 29
Measurable space      3
Measurable, $\mathcal{A'-A}$-measurable      24
Measurable, $\mathcal{A}_{\tau}$-measurable      306
Measurable, $\mathcal{B}_{n}$-measurable      84
Measurable, $\mathcal{B}_{\infty}$-measurable      86
Measurable, $\mathcal{C}$-measurable      298
Measurable, $\mathcal{D}$-measurable      301 320
Measurable, $\mathcal{F}(X)$-measurable      24 28 86 158
Measurable, $\mathcal{F}(X_{1},X_{2},\dots)$-measurable      35 86 91
Measurable, $\mathcal{F}(X_{1},\dots,X_{n})$-measurable      35
Measurable, $\mathcal{F}(X_{s} : s\leq t)$-measurable      36 91
Measurable, $\mathcal{F}(\tilde{S})$-measurable      239
Measurable, $\mathcal{F}_{t}$-measurable      36
Measurable, $\sigma[\mathcal{C}]$-measurable      24 28
Measurable, common functions are      25
Measurable, measurability criterion      28
Measurable, non      16 92
Measurable, progressively      306
Measure      4
Measure, $\sigma$-finite      12 14 61 64 66 80
Measure, absolute continuity      63 64 66—68 78 108
Measure, Borel      531
Measure, complete      15 18 29
Measure, continuous      7 62 88
Measure, counting      6 67
Measure, finite      61
Measure, induced      23 24 27 33 42 52 92
Measure, Lebesgue      4 6 23 38 67
Measure, Lebesgue $\lambda_{n}$      80
Measure, Lebesgue — Stieltjes      4 18 20 77 78 80
Measure, monotone property      6 62
Measure, motivation      4
Measure, outer      4
Measure, outer extension      12
Measure, positive part      62 64
Measure, positive set      62
Measure, probability      20
Measure, product      80 81
Measure, regular      16 20
Measure, signed      61 64 66 76
Measure, singular      63 64 71 108
Measure, space      4
Measure, total variation      62 513
Measure, uniform convergence      41 54 60
mesh      44
Metric space      101 295
Metric space, Arzela theorem      103 539 541
Metric space, Ascoli theorem      103
Metric space, compact      102
Metric space, compactness equivalents      102
Metric space, complete      102 537
Metric space, covering numbers      103
Metric space, discontinuity set      26
Metric space, equicontinuity      103
Metric space, equivalent metrics      101
Metric space, properties      101
Metric space, regular measure      16
Metric space, relatively compact      537
Metric space, separable      295 537
Metric space, sup norm      103
Metric space, totally bounded      102
Metric space, uniform continuity      103
Metrics d on $(D, \mathcal{D})$      297
Metrics, Dudley      540
Metrics, Hellinger      68 543
Metrics, Kolmogorov      543
Metrics, Levy      290
Metrics, Prohorov      540
Metrics, total variation      68 380 543 544
Modulus of continuity      539
moment      46 47 117
Moment, conditional      195
Moment, consistent estimation      226 228 231 270 272 275
Moment, convergence of      53 244 289 293 376
Moment, correlation      48 157 463
Moment, covariance      46 117 157 199
Moment, cumulant      357 388
Moment, generating function      395
Moment, mean      46 116 117 119
Moment, partial      127 128 134 226—228 231 269
Moment, partial variance      145
Moment, skewness $\gamma_{1}$ and tail heaviness $\gamma_{2}$      279 357 383 386—388 391 392 554
Moment, standard deviation      46
Moment, variance      46 116 117 119 233
Moments determine the normal      293
Moments of stable laws      409
Monte Carlo      225
Natural parameters      272
Negligibility      213—215 217—221 226 228 231 233 264 274 275 367 427 430
Negligibility, uan      260 264 283 326 377 399
Nonnegative definite      193 202 363
Norm, $\mathcal{L}_{2}$-norm      104
Norm, q-norm      329
Norm, rth mean      46
Norm, sup      22 103 295 330
Null set      15 83
Oh, big $O_{p}$      209
Oh, big O      9
Oh, little $o_{p}$      209
Oh, little o      9
Oh, o-plus $\bigoplus$, or “at most”      9
Optional sampling      472 492 499
Order statistics      120 172 173 184 281 325
Orthogonal      104 191 338 339 369 379 512
Partial sum process $\mathbb{S}_{n}$      318 319 368 536 539
Partition      37
PLT      367 404
PLT, negligibility      367
PLT, Poisson limit theorem      367 404
Poisson approximation      386 389
Poisson process      181 470 500
Poisson, compound      402
Poisson, generalized      402
Positive definite      191
Positive part      21 119 468
Predictable      487 489 502 511 513 514 516
Predictable $\sigma$-field      513 514
Predictable covariation      512 517
Predictable variation      489 490 501 502 512 516
Probability integral transformation      113
Probability integral transformation, inverse      111
Process      90
Process counting      470
Process realizations (equivalent)      90
Process realizations (smoother)      93
Process, $\mathcal{D}$-class      511
Process, convergence on $(D, \mathcal{D})$      538
Process, empirical process (see also)      121
Process, existence of on $(C, \mathcal{C})$      298
Process, existence of on $(D, \mathcal{D})$      301
Process, general      91
Process, increasing      511
Process, independent increments      297
Process, normal      90 202
Process, predicatable (see also)      487
Process, stationary      297
Process, stationary increments      297
Process, versions      90
Product $\sigma$-field      79
Product lemma      353 366 373
Product limit estimator      507
Product measure      80 81
Product null sets      83
Product sections      81 82
Product space      79
Product topology      99
Product, $\mathcal{F}_{0}$, $\mathcal{F}$, $\mathcal{A\times A'} = \sigma[F]$, $\mu\times\nu$      79
Product, countable      86
Product, cylinder set      86
Product, Fubini theorem      82
Product, integrable      82
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