Theory of Statistics - George Mason University

More documents

Recommendations

Info

496 6 Statistical Inference Based on Likelihood L(α;x) Exponential(α,1) α ] ( x(1) In some cases, it turns out that the estimation of a subset of the parameters does not depend on the value of some other subset. A good method of estimation of β in a linear model X = Zβ + ɛ where the residuals ɛ have a common variance σ 2 and zero correlation can be performed equally well no matter what the value of σ 2 is. (The Gauss-Markov theorem tells us that the least-squares method yields a good estimator.) If the residuals are independently distributed as normals with a common variance, we can formulate the problem as a problem in maximum likelihood estimation. The MLE for β (which just happens to be the same as the LSE) can be thought of in terms of a profile likelihood, because a particular value of σ 2 could be chosen a priori. (This is of course not necessary because the maximum or the likelihood with respect to β occurs at the same point regardless of the value of σ 2 .) Conditional Likelihood When there is a nuisance parameter for which we have a sufficient statistic, a simple approach is to use the PDF conditional on the sufficient statistic to form the likelihood function for the parameter of interest. After doing this, the MLE procedure continues as in the usual case. If the PDFs can be factored so that one factor includes θ2 and some function of the sample, S(x), and the other factor, given S(x), is free of θ2, then this factorization can be carried Theory of Statistics c○2000–2013 James E. Gentle
6.5 Variations on the Likelihood 497 into the likelihood. Such a likelihood is called a conditional likelihood of θ1 given S(x). Conditional likelihood methods often arise in applications in which the parameters of two different distributions are to be compared; that is, when only their relative values are of interest. Suppose µ = (µ1, µ2) and let θ1 = µ1/µ2. Although our interest is in θ1, we may not be able to write the likelihood as a function of θ1. If, however, we can find θ2 for which we have a sufficient statistic, T2(X), and we can factor the likelihood using the factorization theorem so that the factor corresponding to conditional distribution of X given T2(X) does not depend on θ2. This factor, as a function of θ1, is the conditional likelihood function. Sometimes a profile likelihood can be thought of as a particularly simple conditional likelihood. The linear model estimation problem referred to above could be formulated as a conditional likelihood. The actual form of the likelihood would be more complicated, but the solution is equivalent to the solution in which we think of the likelihood as a profile likelihood. Conditional Likelihood for the Exponential Class If X has a distribution in the exponential class with θ = (η1, η2), and its likelihood can be written in the form L(θ ; x) = exp(η T 1 T1(x) + η T 2 T2(x) − ζ(η1, η2))h(x), or, in the log-likelihood form, lL(θ ; x) = η T 1 T1(x) + η T 2 T2(x) − ζ(η1, η2) + c(x), we can easily write the conditional log-likelihood: lL(η1 ; x ; T2) = η T 1 T1(x) + ˜ζ(η1, T2) + c(x). Notice that this decomposition can be achieved iff η1 is a linear function of θ. If our interest is only in η1, we only determine the argument that maximizes the function η T 1 T1(x) + ˜ ζ(η1, T2), which is does not depend on η2. ********************* Partial Likelihood The idea of partial likelihood is somewhat similar to conditional likelihood. The most common area of application is in semiparametric models. These are models of the form f(x; θ) = g(x; θ)h(x), (6.67) Theory of Statistics c○2000–2013 James E. Gentle
Page 1 and 2:
James E. Gentle Theory of Statistic
Page 3 and 4:
Preface: Mathematical Statistics Af
Page 5 and 6:
Preface vii The objective in the di
Page 7 and 8:
x = ⎛ ⎜ ⎝ x1 . xd ⎞ ⎟ ⎠
Page 9 and 10:
• State and prove Fatou’s lemma
Page 11 and 12:
Contents Preface . . . . . . . . .
Page 13 and 14:
Contents xv 2.10 Multivariate Distr
Page 15 and 16:
Contents xvii 5.2.1 Expectation Fun
Page 17 and 18:
Contents xix 8.5.1 Nonparametric Pr
Page 19 and 20:
1 Probability Theory Probability th
Page 21 and 22:
1.1 Some Important Probability Fact
Page 23 and 24:
Page 25 and 26:
Definition 1.8 (exchangeability) Le
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
Theorem 1.5 (properties of a CDF) I
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
.5 PDF p X(x;θ) 0 x θ=1 θ=5 1.1
Page 41 and 42:
Joint and Marginal Distributions 1.
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
Moment-Generating Functions 1.1 Som
Page 63 and 64:
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
A Taylor series expansion of this g
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
Page 83 and 84:
1.2 Series Expansions 65 X is the u
Page 85 and 86:
κ1 = E(Z) κ2 = E(Z 2 ) − (E(Z))
Page 87 and 88:
1.3 Sequences of Events and of Rand
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
We write 1.3 Sequences of Events an
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
1.3.6 Convergence of Functions 1.3
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
we have for fixed k, 1.3 Sequences
Page 119 and 120:
Multivariate Asymptotic Expectation
Page 121 and 122:
1.4 Limit Theorems 103 The bn can b
Page 123 and 124:
1.4 Limit Theorems 105 it applies t
Page 125 and 126:
lim n→∞ max σ j≤kn 2 nj σ2
Page 127 and 128:
1.5 Conditional Probability 109 The
Page 129 and 130:
Conditional Expectation over a Sub-
Page 131 and 132:
• monotone convergence: for 0 ≤
Page 133 and 134:
Fn(t) = ⌊nt⌋ + 1 n + 1 ; 1.5 Co
Page 135 and 136:
1.5 Conditional Probability 117 If
Page 137 and 138:
1.5 Conditional Probability 119 1.5
Page 139 and 140:
Conditional Entropy 1.6 Stochastic
Page 141 and 142:
1.6 Stochastic Processes 123 Stoppi
Page 143 and 144:
1.6 Stochastic Processes 125 (Exerc
Page 145 and 146:
1.6 Stochastic Processes 127 variab
Page 147 and 148:
1.6 Stochastic Processes 129 in a d
Page 149 and 150:
1.6 Stochastic Processes 131 We als
Page 151 and 152:
1.6 Stochastic Processes 133 X0 has
Page 153 and 154:
Convergence of Empirical Processes
Page 155 and 156:
and Fn(x, ω) ≥ Fn(xm,k−1, ω)
Page 157 and 158:
Notes and Further Reading 139 and f
Page 159 and 160:
Notes and Further Reading 141 we ca
Page 161 and 162:
Notes and Further Reading 143 After
Page 163 and 164:
Exercises 145 of betting system. Do
Page 165 and 166:
Exercises 147 1.22. Show that if th
Page 167 and 168:
1.36. Consider the the distribution
Page 169 and 170:
Exercises 151 1.59. A sufficient co
Page 171 and 172:
Exercises 153 1.85. Show that Doob
Page 173 and 174:
2 Distribution Theory and Statistic
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Example 2.1 complete and incomplete
Page 183 and 184:
2.2 Shapes of the Probability Densi
Page 185 and 186:
2.2 Shapes of the Probability Densi
Page 187 and 188:
2.3.2 The Le Cam Regularity Conditi
Page 189 and 190:
2.4 The Exponential Class of Famili
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
or or 2.5 Parametric-Support Famili
Page 197 and 198:
2.6 Transformation Group Families 1
Page 199 and 200:
2.6 Transformation Group Families 1
Page 201 and 202:
2.7 Truncated and Censored Distribu
Page 203 and 204:
2.9 Infinitely Divisible and Stable
Page 205 and 206:
Higher Dimensions 2.11 The Family o
Page 207 and 208:
2.11 The Family of Normal Distribut
Page 209 and 210:
Page 211 and 212:
Page 213 and 214:
Exercises 195 are considered at som
Page 215 and 216:
Exercises 197 This law has been use
Page 217 and 218:
Exercises 199 a) Show that for d
Page 219 and 220:
3 Basic Statistical Theory The fiel
Page 221 and 222:
How Does PH Differ from P? 3 Basic
Page 223 and 224:
3 Basic Statistical Theory 205 abov
Page 225 and 226:
3.1 Inferential Information in Stat
Page 227 and 228:
Page 229 and 230:
Page 231 and 232:
The Basic Paradigm of Point Estimat
Page 233 and 234:
Page 235 and 236:
Page 237 and 238:
Page 239 and 240:
Completeness 3.1 Inferential Inform
Page 241 and 242:
and so by completeness, 3.1 Inferen
Page 243 and 244:
Page 245 and 246:
and so I(µ, σ) = Eθ = E(µ,σ) =
Page 247 and 248:
Page 249 and 250:
Page 251 and 252:
Robustness 3.1 Inferential Informat
Page 253 and 254:
3.2 Statistical Inference: Approach
Page 255 and 256:
Page 257 and 258:
Page 259 and 260:
Page 261 and 262:
Page 263 and 264:
Page 265 and 266:
Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
Page 273 and 274:
3.3 The Decision Theory Approach to
Page 275 and 276:
Page 277 and 278:
Page 279 and 280:
Page 281 and 282:
Page 283 and 284:
Page 285 and 286:
Page 287 and 288:
Page 289 and 290:
Page 291 and 292:
Risk 0.005 0.010 0.015 0.020 0.025
Page 293 and 294:
3.4 Invariant and Equivariant Stati
Page 295 and 296:
Page 297 and 298:
Page 299 and 300:
Equivariant Point Estimation 3.4 In
Page 301 and 302:
Page 303 and 304:
Page 305 and 306:
3.5 Probability Statements in Stati
Page 307 and 308:
Test Rules 3.5 Probability Statemen
Page 309 and 310:
Page 311 and 312:
Page 313 and 314:
Page 315 and 316:
if 3.6 Variance Estimation 297 Give
Page 317 and 318:
3.6 Variance Estimation 299 J(T) =
Page 319 and 320:
3.7 Applications 3.7.1 Inference in
Page 321 and 322:
3.8 Asymptotic Inference 303 The ca
Page 323 and 324:
3.8 Asymptotic Inference 305 The mo
Page 325 and 326:
3.8 Asymptotic Inference 307 wherea
Page 327 and 328:
3.8 Asymptotic Inference 309 tendin
Page 329 and 330:
Definition 3.20 (asymptotic signifi
Page 331 and 332:
Proof. *************** Notes and Fu
Page 333 and 334:
Notes and Further Reading 315 Least
Page 335 and 336:
Approximations and Asymptotic Infer
Page 337 and 338:
Exercises 319 b) the expectation of
Page 339 and 340:
4 Bayesian Inference We have used a
Page 341 and 342:
4.1 The Bayesian Paradigm 323 For m
Page 343 and 344:
4.1 The Bayesian Paradigm 325 A qua
Page 345 and 346:
4.2 Bayesian Analysis 327 even if t
Page 347 and 348:
4.2 Bayesian Analysis 329 Hence P(B
Page 349 and 350:
4.2 Bayesian Analysis 331 B1. ∀θ
Page 351 and 352:
4.2 Bayesian Analysis 333 This mean
Page 353 and 354:
Prior Prior 0.0 0.5 1.0 1.5 2.0 0 2
Page 355 and 356:
4.2 Bayesian Analysis 337 We go thr
Page 357 and 358:
4.2 Bayesian Analysis 339 We constr
Page 359 and 360:
4.2 Bayesian Analysis 341 as the mo
Page 361 and 362:
Assessing the Problem Formulation 4
Page 363 and 364:
Posterior Posterior 0 5 10 15 0 1 2
Page 365 and 366:
4.2 Bayesian Analysis 347 distribut
Page 367 and 368:
4.3 Bayes Rules 349 • the nature
Page 369 and 370:
Then we have E(g(θ)T(X)) = E(T(X)E
Page 371 and 372:
4.3 Bayes Rules 353 equivariance Fo
Page 373 and 374:
4.3 Bayes Rules 355 Example 4.9 (Co
Page 375 and 376:
Page 377 and 378:
p0 = Pr(Θ ∈ Θ0) and p1 = Pr(Θ
Page 379 and 380:
The Weighted 0-1 or α0-α1 Loss Fu
Page 381 and 382:
The term ˆp0 ˆp1 = p0 p1 BF(x) =
Page 383 and 384:
4.5 Bayesian Testing 365 • Bayes
Page 385 and 386:
where fΘ|x(θ|x) = p1 if θ = θ0
Page 387 and 388:
4.6 Bayesian Confidence Sets 369 as
Page 389 and 390:
Posterior 0 1 2 3 95% Credible Regi
Page 391 and 392:
Markov Chain Monte Carlo 4.7 Comput
Page 393 and 394:
4.7 Computational Methods 375 β/(t
Page 395 and 396:
Notes and Further Reading 377 An al
Page 397 and 398:
4.4. Given the conditional PDF a) U
Page 399 and 400:
Exercises 381 4.14. Consider again
Page 401 and 402:
Exercises 383 4.26. As in Exercise
Page 403 and 404:
5 Unbiased Point Estimation In a de
Page 405 and 406:
Estimability 5 Unbiased Point Estim
Page 407 and 408:
s1π + s0(1 − π) = π, 5.1 UMVUE
Page 409 and 410:
5.1 UMVUE 391 Let T be UMVUE for g(
Page 411 and 412:
5.1 UMVUE 393 Example 5.7 UMVUE of
Page 413 and 414:
5.1 UMVUE 395 The risk of T ∗ is
Page 415 and 416:
5.1 UMVUE 397 estimator of θ that
Page 417 and 418:
(Compare this with Tfdx = g(θ).)
Page 419 and 420:
¯h(X1, . . ., Xm) = 1 m! 5.2 U-Sta
Page 421 and 422:
where Pn is the ECDF. U(X1, . . .,
Page 423 and 424:
and T 2 r,S = C = Tr,S = C C TnT
Page 425 and 426:
Generalized U-Statistics 5.2 U-Stat
Page 427 and 428:
5.2 U-Statistics 409 Theorem 5.4 (H
Page 429 and 430:
5.3 Asymptotically Unbiased Estimat
Page 431 and 432:
5.3.2 Ratio Estimators 5.3 Asymptot
Page 433 and 434:
5.4 Asymptotic Efficiency 415 of a
Page 435 and 436:
⎧ ⎨ Tn = ⎩ n2 with probabilit
Page 437 and 438:
5.5 Applications 5.5 Applications 4
Page 439 and 440:
5.5 Applications 421 one aspect of
Page 441 and 442:
Proof. Because l ∈ span(X T ) = s
Page 443 and 444:
Optimal Properties of the Moore-Pen
Page 445 and 446:
5.5 Applications 427 implies implie
Page 447 and 448:
The “Sum of Squares” Quadratic
Page 449 and 450:
5.5 Applications 431 The question o
Page 451 and 452:
We now write the original model as
Page 453 and 454:
5.5 Applications 435 it from other
Page 455 and 456:
(it’s Bernoulli), and for i = j,
Page 457 and 458:
Fisher Efficient Estimators and Exp
Page 459 and 460:
6 Statistical Inference Based on Li
Page 461 and 462:
6.1 The Likelihood Function 443 is
Page 463 and 464: 6.2 Maximum Likelihood Parametric E
Page 471 and 472: 6.2.2 Finite Sample Properties of M
Page 491 and 492: where Now, consider This has two pa
Page 495 and 496: 6.3 Asymptotic Properties of MLEs,
Page 501 and 502: 6.4 Application: MLEs in Generalize
Page 511 and 512: Residuals 6.4 Application: MLEs in
Page 513: 6.5 Variations on the Likelihood 49
Page 517 and 518: Maximum Likelihood in Linear Models
Page 519 and 520: Exercises 501 b) Assume that σ 2 1
Page 521 and 522: 7 Statistical Hypotheses and Confid
Page 523 and 524: Tests of Hypotheses 7.1 Statistical
Page 525 and 526: 7.1 Statistical Hypotheses 507 Know
Page 527 and 528: Power of a Statistical Test 7.1 Sta
Page 529 and 530: An Optimal Test in a Simple Situati
Page 531 and 532: 7.2 Optimal Tests 513 All of these
Page 533 and 534: Use of Sufficient Statistics 7.2 Op
Page 535 and 536: 7.2 Optimal Tests 517 Syppose we as
Page 537 and 538: Nonexistence of UMP Tests 7.2 Optim
Page 539 and 540: H0 : θ = θ0 versus H1 : θ = θ0.
Page 541 and 542: 7.2.6 Equivariance, Unbiasedness, a
Page 543 and 544: 7.3 Likelihood Ratio Tests, Wald Te
Page 545 and 546: 7.3 Likelihood Ratio Tests, Wald Te
Page 547 and 548: Minimize 7.3 Likelihood Ratio Tests
Page 549 and 550: 7.4 Nonparametric Tests 531 I(0, ˆ
Page 551 and 552: Yij = µ + αi + ɛij, i = 1, . . .
Page 553 and 554: 7.7 The Likelihood Principle and Te
Page 555 and 556: 7.8 Confidence Sets 537 Monte Carlo
Page 557 and 558: 7.8 Confidence Sets 539 The concept
Page 559 and 560: 7.8 Confidence Sets 541 For any giv
Page 561 and 562: 7.9 Optimal Confidence Sets 543 Thi
Page 563 and 564: 7.9 Optimal Confidence Sets 545 Con
Page 565 and 566:
7.10 Asymptotic Confidence sets 547
Page 567 and 568:
7.11 Bootstrap Confidence Sets 549
Page 569 and 570:
Bias in Intervals Due to Bias in th
Page 571 and 572:
7.12 Simultaneous Confidence Sets 5
Page 573 and 574:
Sequential Tests Exercises 555 Wald
Page 575 and 576:
8 Nonparametric and Robust Inferenc
Page 577 and 578:
8.2 Inference Based on Order Statis
Page 579 and 580:
f ^ (x) Nonparametric Regression 0.
Page 581 and 582:
8.3 Nonparametric Estimation of Fun
Page 583 and 584:
Page 585 and 586:
Page 587 and 588:
Page 589 and 590:
ISE θ = 8.3 Nonparametric Estim
Page 591 and 592:
8.4 Semiparametric Methods and Part
Page 593 and 594:
8.5 Nonparametric Estimation of PDF
Page 595 and 596:
Page 597 and 598:
Some Properties of the Histogram Es
Page 599 and 600:
Page 601 and 602:
AISB 1 pH = 12 8.5 Nonparametric
Page 603 and 604:
and the integrated variance is 8.5
Page 605 and 606:
and 8.5 Nonparametric Estimation o
Page 607 and 608:
h ∗ = 8.5 Nonparametric Estimatio
Page 609 and 610:
Computation of Kernel Density Estim
Page 611 and 612:
8.6 Perturbations of Probability Di
Page 613 and 614:
CDF 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Page 615 and 616:
8.6 Perturbations of Probability Di
Page 617 and 618:
8.7 Robust Inference 8.7 Robust Inf
Page 619 and 620:
p(y) (1-ε)p(y) The Influence Funct
Page 621 and 622:
8.7 Robust Inference 603 Notice tha
Page 623 and 624:
Nonparametric Statistics Exercises
Page 625 and 626:
0 Statistical Mathematics Statistic
Page 627 and 628:
0 Statistical Mathematics 609 x may
Page 629 and 630:
0.0 Some Basic Mathematical Concept
Page 631 and 632:
Because each is a subset of the oth
Page 633 and 634:
0.0.2 Sets and Spaces 0.0 Some Basi
Page 635 and 636:
0.0.2.4 Point Sequences in a Topolo
Page 637 and 638:
Page 639 and 640:
The sequence of sets {An} is said t
Page 641 and 642:
Page 643 and 644:
Page 645 and 646:
Page 647 and 648:
Page 649 and 650:
The proof of this is a classic: fir
Page 651 and 652:
Page 653 and 654:
• ∀x ∈ IR ∪ {∞}, x × ±
Page 655 and 656:
Page 657 and 658:
0.0.5.2 Sets of Reals; Open, Closed
Page 659 and 660:
Page 661 and 662:
Page 663 and 664:
We have 0.0 Some Basic Mathematical
Page 665 and 666:
Page 667 and 668:
Page 669 and 670:
Page 671 and 672:
Definition 0.0.14 (superharmonic fu
Page 673 and 674:
Page 675 and 676:
Ordering the Complex Numbers 0.0 So
Page 677 and 678:
Page 679 and 680:
Page 681 and 682:
Page 683 and 684:
Page 685 and 686:
Page 687 and 688:
Page 689 and 690:
Page 691 and 692:
0.0.9.5 Working with Real-Valued Fu
Page 693 and 694:
∞ 0.0 Some Basic Mathematical Con
Page 695 and 696:
Page 697 and 698:
Page 699 and 700:
Page 701 and 702:
Page 703 and 704:
a) Evaluate the integral (0.0.84):
Page 705 and 706:
0.1 Measure, Integration, and Funct
Page 707 and 708:
Page 709 and 710:
Page 711 and 712:
Page 713 and 714:
Page 715 and 716:
Page 717 and 718:
3. if A1, A2, . . . ∈ F are disjo
Page 719 and 720:
Page 721 and 722:
Page 723 and 724:
Page 725 and 726:
Page 727 and 728:
Page 729 and 730:
Page 731 and 732:
(why?), and so We also have λ 0.1
Page 733 and 734:
Page 735 and 736:
Page 737 and 738:
Page 739 and 740:
Page 741 and 742:
Proof. Exercise. 0.1 Measure, Integ
Page 743 and 744:
Hence, from inequality (0.1.47),
Page 745 and 746:
Page 747 and 748:
Page 749 and 750:
Page 751 and 752:
Page 753 and 754:
Page 755 and 756:
0.1.9.3 Norms of Functions 0.1 Meas
Page 757 and 758:
Page 759 and 760:
Page 761 and 762:
Page 763 and 764:
Page 765 and 766:
Page 767 and 768:
0.1.12 Transforms 0.1 Measure, Inte
Page 769 and 770:
Page 771 and 772:
Page 773 and 774:
Page 775 and 776:
0.2 Stochastic Processes and the St
Page 777 and 778:
Page 779 and 780:
Variation of Functionals 0.2 Stocha
Page 781 and 782:
Page 783 and 784:
0.2.1.3 Ito Processes 0.2 Stochasti
Page 785 and 786:
Page 787 and 788:
0.2.2.2 Ito’s Lemma 0.2 Stochasti
Page 789 and 790:
Page 791 and 792:
0.3 Some Basics of Linear Algebra 0
Page 793 and 794:
Page 795 and 796:
0.3.2.2 The Trace and Some of Its P
Page 797 and 798:
Page 799 and 800:
The Fourier Expansion 0.3 Some Basi
Page 801 and 802:
Page 803 and 804:
Page 805 and 806:
0.3.2.8 Spectral Decomposition 0.3
Page 807 and 808:
0.3.2.11 Inverses of Matrices 0.3 S
Page 809 and 810:
Page 811 and 812:
Page 813 and 814:
Page 815 and 816:
we have 0.3 Some Basics of Linear A
Page 817 and 818:
Page 819 and 820:
Page 821 and 822:
Page 823 and 824:
f(x) ≈ f(x∗) + (x − x∗) T
Page 825 and 826:
Page 827 and 828:
Page 829 and 830:
and so E T π (I + A) n−1 Eπ =
Page 831 and 832:
Page 833 and 834:
0.4.1.2 Methods 0.4 Optimization 81
Page 835 and 836:
0.4.1.7 The Steps in Iterative Algo
Page 837 and 838:
0.4.1.11 Modifications of Newton’
Page 839 and 840:
We generally require g x (k) ; x (
Page 841 and 842:
0.4 Optimization 823 3. Generate st
Page 843 and 844:
Theory of Statistics c○2000-2013
Page 845 and 846:
A Important Probability Distributio
Page 847 and 848:
Appendix A. Important Probability D
Page 849 and 850:
Page 851 and 852:
Page 853 and 854:
Page 855 and 856:
B Useful Inequalities in Probabilit
Page 857 and 858:
B.3 Pr(X ∈ A) and E(f(X)) 839 Pr(
Page 859 and 860:
B.4 E(f(X)) and f(E(X)) 841 Theorem
Page 861 and 862:
B.4 E(f(X)) and f(E(X)) 843 A relat
Page 863 and 864:
B.5 E(f(X,Y )) and E(g(X)) and E(h(
Page 865 and 866:
Now assume p > 1. Now, E(|X + Y | p
Page 867 and 868:
C Notation and Definitions All nota
Page 869 and 870:
C.1 General Notation 851 constant;
Page 871 and 872:
C.2 General Mathematical Functions
Page 873 and 874:
Functions of Convenience C.2 Genera
Page 875 and 876:
C.2 General Mathematical Functions
Page 877 and 878:
A1 ∩ A2 A1 − A2 A1∆A2 A1 × A
Page 879 and 880:
C.4 Linear Spaces and Matrices 861
Page 881 and 882:
a(ij) C.4 Linear Spaces and Matrice
Page 883 and 884:
References The number(s) following
Page 885 and 886:
References 867 Lyle D. Broemeling.
Page 887 and 888:
References 869 William Feller. An I
Page 889 and 890:
References 871 H. O. Hartley. In Dr
Page 891 and 892:
References 873 Feldman, and Murad S
Page 893 and 894:
References 875 Roger B. Nelsen. An
Page 895 and 896:
References 877 Glenn Shafer. A Math
Page 897 and 898:
References 879 Abraham Wald. Sequen
Page 899 and 900:
Index a.e. (almost everywhere), 704
Page 901 and 902:
cartesian product measurable space,
Page 903 and 904:
derivative of a functional, 752-753
Page 905 and 906:
Euclidean norm, 774 Euler’s formu
Page 907 and 908:
unbiased test, 292, 519 uniform con
Page 909 and 910:
sequence of sets, 620-623 limit poi
Page 911 and 912:
natural parameter space, 173 neglig
Page 913 and 914:
proportional hazards, 573 pseudoinv
Page 915 and 916:
square root matrix, 783 squared-err
Page 917:
weak convergence in mean square, 56
show all

Theory of Statistics - George Mason University

Create successful ePaper yourself

Delete template?

Save as template?