第七版 Indroduction_to_Mathematica_Statistics.Hogg,McKean,Craig seventh edition.pdf UIUC的 STAT401统计课程教材
Contents
Preface ix
1 Probability and Distributions 1
1.1 Introduction................................ 1
1.2 SetTheory ................................ 3
1.3 TheProbabilitySetFunction ...................... 10
1.4 ConditionalProbabilityandIndependence . . . . . . . . . . . . . . . 21
1.5 RandomVariables ............................ 32
1.6 DiscreteRandomVariables ....................... 40
1.6.1 Transformations ......................... 42
1.7 ContinuousRandomVariables...................... 44 1.7.1 Transformations ......................... 46
1.8 ExpectationofaRandomVariable ................... 52
1.9 SomeSpecialExpectations ....................... 57
1.10ImportantInequalities .......................... 68
2 Multivariate Distributions 73
2.1 DistributionsofTwoRandomVariables ................ 73 2.1.1 Expectation............................ 79
2.2 Transformations: Bivariate Random Variables . . . . . . . . . . . . . 84
2.3 Conditional Distributions and Expectations . . . . . . . . . . . . . . 94
2.4 TheCorrelationCoefficient ....................... 102
2.5 IndependentRandomVariables.....................110
2.6 ExtensiontoSeveralRandomVariables . . . . . . . . . . . . . . . . 117
2.6.1 ∗Multivariate Variance-Covariance Matrix . . . . . . . . . . . 123
2.7 Transformations for Several Random Variables . . . . . . . . . . . . 126
2.8 LinearCombinationsofRandomVariables. . . . . . . . . . . . . . . 134
3 Some Special Distributions 139
3.1 TheBinomialandRelatedDistributions . . . . . . . . . . . . . . . . 139
3.2 ThePoissonDistribution ........................ 150
3.3 TheΓ,χ2,andβDistributions ..................... 156
3.4 TheNormalDistribution......................... 168
3.4.1 ContaminatedNormals ..................... 174 v
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3.5 TheMultivariateNormalDistribution ................. 178 3.5.1 ∗Applications........................... 185
3.6 t-andF-Distributions .......................... 189 3.6.1 Thet-distribution ........................ 189 3.6.2 TheF-distribution........................ 191 3.6.3 Student’sTheorem........................ 193
3.7 MixtureDistributions .......................... 197
4 Some Elementary Statistical Inferences 203
4.1 SamplingandStatistics ......................... 203 4.1.1 HistogramEstimatesofpmfsandpdfs . . . . . . . . . . . . . 207
4.2 ConfidenceIntervals ........................... 214
4.2.1 Confidence Intervals for Difference in Means . . . . . . . . . . 217
4.2.2 Confidence Interval for Difference in Proportions . . . . . . . 219
4.3 Confidence Intervals for Parameters of Discrete Distributions . . . . 223
4.4 OrderStatistics.............................. 227 4.4.1 Quantiles ............................. 231 4.4.2 ConfidenceIntervalsforQuantiles . . . . . . . . . . . . . . . 234
4.5 IntroductiontoHypothesisTesting................... 240
4.6 Additional Comments About Statistical Tests . . . . . . . . . . . . . 248
4.7 Chi-SquareTests ............................. 254
4.8 TheMethodofMonteCarlo....................... 261
4.8.1 Accept–RejectGenerationAlgorithm. . . . . . . . . . . . . . 268 4.9 BootstrapProcedures .......................... 273 4.9.1 Percentile Bootstrap Confidence Intervals . . . . . . . . . . . 273 4.9.2 BootstrapTestingProcedures.................. 276 4.10∗ToleranceLimitsforDistributions................... 284
5 Consistency and Limiting Distributions 289
5.1 ConvergenceinProbability ....................... 289
5.2 ConvergenceinDistribution....................... 294
5.2.1 BoundedinProbability ..................... 300
5.2.2 ∆-Method............................. 301
5.2.3 MomentGeneratingFunctionTechnique............ 303
5.3 CentralLimitTheorem ......................... 307
5.4 ∗ExtensionstoMultivariateDistributions . . . . . . . . . . . . . . . 314
6 Maximum Likelihood Methods 321
6.1 MaximumLikelihoodEstimation .................... 321 6.2 Rao–Cram ́erLowerBoundandEfficiency . . . . . . . . . . . . . . . 327 6.3 MaximumLikelihoodTests ....................... 341 6.4 MultiparameterCase:Estimation.................... 350 6.5 MultiparameterCase:Testing...................... 359 6.6 TheEMAlgorithm............................ 367
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7 Sufficiency 375
7.1 MeasuresofQualityofEstimators ................... 375 7.2 ASufficientStatisticforaParameter.................. 381 7.3 PropertiesofaSufficientStatistic.................... 388 7.4 CompletenessandUniqueness...................... 392 7.5 TheExponentialClassofDistributions. . . . . . . . . . . . . . . . . 397 7.6 FunctionsofaParameter ........................ 402 7.7 TheCaseofSeveralParameters..................... 407 7.8 MinimalSufficiencyandAncillaryStatistics . . . . . . . . . . . . . . 415 7.9 Sufficiency, Completeness, and Independence . . . . . . . . . . . . . 421
8 Optimal Tests of Hypotheses 429
8.1 MostPowerfulTests ........................... 429
8.2 UniformlyMostPowerfulTests ..................... 439
8.3 LikelihoodRatioTests.......................... 447
8.4 TheSequentialProbabilityRatioTest ................. 459
8.5 MinimaxandClassificationProcedures. . . . . . . . . . . . . . . . . 466
8.5.1 MinimaxProcedures....................... 466 8.5.2 Classification ........................... 469
9 Inferences About Normal Models 473
9.1 QuadraticForms ............................. 473 9.2 One-WayANOVA ............................ 478 9.3 Noncentralχ2andF-Distributions...................484 9.4 MultipleComparisons .......................... 486 9.5 TheAnalysisofVariance ........................ 490 9.6 ARegressionProblem .......................... 497 9.7 ATestofIndependence ......................... 506 9.8 TheDistributionsofCertainQuadraticForms. . . . . . . . . . . . . 509 9.9 TheIndependenceofCertainQuadraticForms . . . . . . . . . . . . 516
10 Nonparametric and Robust Statistics 525
10.1LocationModels ............................. 525 10.2SampleMedianandtheSignTest.................... 528 10.2.1 AsymptoticRelativeEfficiency ................. 533 10.2.2 Estimating Equations Based on the Sign Test . . . . . . . . . 538 10.2.3 ConfidenceIntervalfortheMedian . . . . . . . . . . . . . . . 539 10.3Signed-RankWilcoxon.......................... 541 10.3.1 AsymptoticRelativeEfficiency ................. 546 10.3.2 Estimating Equations Based on Signed-Rank Wilcoxon . . . 549 10.3.3 ConfidenceIntervalfortheMedian . . . . . . . . . . . . . . . 549 10.4Mann–Whitney–WilcoxonProcedure.................. 551 10.4.1 AsymptoticRelativeEfficiency ................. 555 10.4.2 Estimating Equations Based on the Mann–Whitney–Wilcoxon 556 10.4.3 Confidence Interval for the Shift Parameter ∆ . . . . . . . . . 557 10.5GeneralRankScores........................... 559
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10.5.1 Efficacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 10.5.2 Estimating Equations Based on General Scores . . . . . . . . 563 10.5.3 Optimization:BestEstimates.................. 564
10.6AdaptiveProcedures........................... 571 10.7SimpleLinearModel........................... 576 10.8MeasuresofAssociation ......................... 581
10.8.1 Kendall’sτ ............................ 582
10.8.2 Spearman’sRho ......................... 584 10.9RobustConcepts ............................. 588 10.9.1 LocationModel.......................... 589 10.9.2 LinearModel ........................... 595
11 Bayesian Statistics 605
11.1SubjectiveProbability .......................... 605 11.2BayesianProcedures ........................... 608 11.2.1 PriorandPosteriorDistributions . . . . . . . . . . . . . . . . 609 11.2.2 BayesianPointEstimation.................... 612 11.2.3 BayesianIntervalEstimation .................. 615 11.2.4 BayesianTestingProcedures .................. 616 11.2.5 BayesianSequentialProcedures................. 617 11.3MoreBayesianTerminologyandIdeas ................. 619 11.4GibbsSampler .............................. 626 11.5ModernBayesianMethods........................ 632 11.5.1 EmpiricalBayes ......................... 636
A Mathematical Comments 641
A.1 RegularityConditions .......................... 641 A.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
B R Functions 645
C Tables of Distributions 655
D Lists of Common Distributions 665
E References 669
F Answers to Selected Exercises 673
Index 683
