Mathematical Statistics and Data Analysis

685 Pages·2010·4.8 MB·English

by  John A. Rice

THIRD EDITION
Mathematical
Statistics and
Data Analysis
John A. Rice
University of California, Berkeley
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MathematicalStatisticsandDataAnalysis,ThirdEdition
JohnA.Rice
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We must be careful not to confuse data with the
abstractions we use to analyze them.
WILLIAM JAMES (1842–1910)
Contents
Preface xi
1 Probability 1
1.1 Introduction 1
1.2 SampleSpaces 2
1.3 ProbabilityMeasures 4
1.4 ComputingProbabilities:CountingMethods 6
1.4.1 TheMultiplicationPrinciple 7
1.4.2 PermutationsandCombinations 9
1.5 ConditionalProbability 16
1.6 Independence 23
1.7 ConcludingRemarks 26
1.8 Problems 26
2 RandomVariables 35
2.1 DiscreteRandomVariables 35
2.1.1 BernoulliRandomVariables 37
2.1.2 TheBinomialDistribution 38
2.1.3 TheGeometricandNegativeBinomialDistributions 40
2.1.4 TheHypergeometricDistribution 42
2.1.5 ThePoissonDistribution 42
2.2 ContinuousRandomVariables 47
2.2.1 TheExponentialDensity 50
2.2.2 TheGammaDensity 53
iv
Contents v
2.2.3 TheNormalDistribution 54
2.2.4 TheBetaDensity 58
2.3 FunctionsofaRandomVariable 58
2.4 ConcludingRemarks 64
2.5 Problems 64
3 JointDistributions 71
3.1 Introduction 71
3.2 DiscreteRandomVariables 72
3.3 ContinuousRandomVariables 75
3.4 IndependentRandomVariables 84
3.5 ConditionalDistributions 87
3.5.1 TheDiscreteCase 87
3.5.2 TheContinuousCase 88
3.6 FunctionsofJointlyDistributedRandomVariables 96
3.6.1 SumsandQuotients 96
3.6.2 TheGeneralCase 99
3.7 ExtremaandOrderStatistics 104
3.8 Problems 107
4 ExpectedValues 116
4.1 TheExpectedValueofaRandomVariable 116
4.1.1 ExpectationsofFunctionsofRandomVariables 121
4.1.2 ExpectationsofLinearCombinationsofRandomVariables 124
4.2 VarianceandStandardDeviation 130
4.2.1 AModelforMeasurementError 135
4.3 CovarianceandCorrelation 138
4.4 ConditionalExpectationandPrediction 147
4.4.1 DefinitionsandExamples 147
4.4.2 Prediction 152
4.5 TheMoment-GeneratingFunction 155
4.6 ApproximateMethods 161
4.7 Problems 166
vi Contents
5 LimitTheorems 177
5.1 Introduction 177
5.2 TheLawofLargeNumbers 177
5.3 ConvergenceinDistributionandtheCentralLimitTheorem 181
5.4 Problems 188
6 DistributionsDerivedfromtheNormalDistribution 192
6.1 Introduction 192
6.2 χ2,t,and F Distributions 192
6.3 TheSampleMeanandtheSampleVariance 195
6.4 Problems 198
7 SurveySampling 199
7.1 Introduction 199
7.2 PopulationParameters 200
7.3 SimpleRandomSampling 202
7.3.1 TheExpectationandVarianceoftheSampleMean 203
7.3.2 EstimationofthePopulationVariance 210
7.3.3 TheNormalApproximationtotheSamplingDistributionofX 214
7.4 EstimationofaRatio 220
7.5 StratifiedRandomSampling 227
7.5.1 IntroductionandNotation 227
7.5.2 PropertiesofStratifiedEstimates 228
7.5.3 MethodsofAllocation 232
7.6 ConcludingRemarks 238
7.7 Problems 239
8 EstimationofParametersandFittingofProbabilityDistributions 255
8.1 Introduction 255
8.2 FittingthePoissonDistributiontoEmissionsofAlphaParticles 255
8.3 ParameterEstimation 257
8.4 TheMethodofMoments 260
8.5 TheMethodofMaximumLikelihood 267
Contents vii
8.5.1 MaximumLikelihoodEstimatesofMultinomialCellProbabilities 272
8.5.2 LargeSampleTheoryforMaximumLikelihoodEstimates 274
8.5.3 ConfidenceIntervalsfromMaximumLikelihoodEstimates 279
8.6 TheBayesianApproachtoParameterEstimation 285
8.6.1 FurtherRemarksonPriors 294
8.6.2 LargeSampleNormalApproximationtothePosterior 296
8.6.3 ComputationalAspects 297
8.7 EfficiencyandtheCrame´r-RaoLowerBound 298
8.7.1 AnExample:TheNegativeBinomialDistribution 302
8.8 Sufficiency 305
8.8.1 AFactorizationTheorem 306
8.8.2 TheRao-BlackwellTheorem 310
8.9 ConcludingRemarks 311
8.10 Problems 312
9 TestingHypothesesandAssessingGoodnessofFit 329
9.1 Introduction 329
9.2 TheNeyman-PearsonParadigm 331
9.2.1 SpecificationoftheSignificanceLevelandtheConceptofa p-value 334
9.2.2 TheNullHypothesis 335
9.2.3 UniformlyMostPowerfulTests 336
9.3 TheDualityofConfidenceIntervalsandHypothesisTests 337
9.4 GeneralizedLikelihoodRatioTests 339
9.5 LikelihoodRatioTestsfortheMultinomialDistribution 341
9.6 ThePoissonDispersionTest 347
9.7 HangingRootograms 349
9.8 ProbabilityPlots 352
9.9 TestsforNormality 358
9.10 ConcludingRemarks 361
9.11 Problems 362
10 SummarizingData 377
10.1 Introduction 377
10.2 MethodsBasedontheCumulativeDistributionFunction 378
viii Contents
10.2.1 TheEmpiricalCumulativeDistributionFunction 378
10.2.2 TheSurvivalFunction 380
10.2.3 Quantile-QuantilePlots 385
10.3 Histograms,DensityCurves,andStem-and-LeafPlots 389
10.4 MeasuresofLocation 392
10.4.1 TheArithmeticMean 393
10.4.2 TheMedian 395
10.4.3 TheTrimmedMean 397
10.4.4 MEstimates 397
10.4.5 ComparisonofLocationEstimates 398
10.4.6 EstimatingVariabilityofLocationEstimatesbytheBootstrap 399
10.5 MeasuresofDispersion 401
10.6 Boxplots 402
10.7 ExploringRelationshipswithScatterplots 404
10.8 ConcludingRemarks 407
10.9 Problems 408
11 ComparingTwoSamples 420
11.1 Introduction 420
11.2 ComparingTwoIndependentSamples 421
11.2.1 MethodsBasedontheNormalDistribution 421
11.2.2 Power 433
11.2.3 ANonparametricMethod—TheMann-WhitneyTest 435
11.2.4 BayesianApproach 443
11.3 ComparingPairedSamples 444
11.3.1 MethodsBasedontheNormalDistribution 446
11.3.2 ANonparametricMethod—TheSignedRankTest 448
11.3.3 AnExample—MeasuringMercuryLevelsinFish 450
11.4 ExperimentalDesign 452
11.4.1 MammaryArteryLigation 452
11.4.2 ThePlaceboEffect 453
11.4.3 TheLanarkshireMilkExperiment 453
11.4.4 ThePortacavalShunt 454
11.4.5 FD&CRedNo.40 455
11.4.6 FurtherRemarksonRandomization 456
Contents ix
11.4.7 ObservationalStudies,Confounding,andBiasinGraduateAdmissions 457
11.4.8 FishingExpeditions 458
11.5 ConcludingRemarks 459
11.6 Problems 459
12 TheAnalysisofVariance 477
12.1 Introduction 477
12.2 TheOne-WayLayout 477
12.2.1 NormalTheory;theFTest 478
12.2.2 TheProblemofMultipleComparisons 485
12.2.3 ANonparametricMethod—TheKruskal-WallisTest 488
12.3 TheTwo-WayLayout 489
12.3.1 AdditiveParametrization 489
12.3.2 NormalTheoryfortheTwo-WayLayout 492
12.3.3 RandomizedBlockDesigns 500
12.3.4 ANonparametricMethod—Friedman’sTest 503
12.4 ConcludingRemarks 504
12.5 Problems 505
13 TheAnalysisofCategoricalData 514
13.1 Introduction 514
13.2 Fisher’sExactTest 514
13.3 TheChi-SquareTestofHomogeneity 516
13.4 TheChi-SquareTestofIndependence 520
13.5 Matched-PairsDesigns 523
13.6 OddsRatios 526
13.7 ConcludingRemarks 530
13.8 Problems 530
14 LinearLeastSquares 542
14.1 Introduction 542
14.2 SimpleLinearRegression 547
14.2.1 StatisticalPropertiesoftheEstimatedSlopeandIntercept 547