Mathematical Statistics

676 Pages·2018·8.1 MB·English

by  Dieter Rasch

Mathematical Statistics
Mathematical Statistics
Dieter Rasch
University of Natural Resources and Life Sciences,
Institute of Applied Statistics and Computing
Vienna, Austria
Dieter Schott
Faculty of Engineering, Hochschule Wismar, University of Applied
Sciences: Technology, Business and Design
Wismar, Germany
Thiseditionfirstpublished2018
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LibraryofCongressCataloging-in-PublicationData
Names:Rasch,Dieter,author.|Schott,Dieter,author.
Title:Mathematicalstatistics/byDieterRasch,DieterSchott.
Description:Hoboken,NJ:JohnWiley&Sons,2018.|Includes
bibliographicalreferencesandindex.|
Identifiers:LCCN2017039506(print)|LCCN2017046479(ebook)|ISBN
9781119385264(pdf)|ISBN9781119385233(epub)|ISBN9781119385288(cloth)
Subjects: LCSH:Mathematicalstatistics.
Classification:LCCQA276(ebook)|LCCQA276.R3622018(print)|DDC
519.5–dc23
LCrecordavailableathttps://lccn.loc.gov/2017039506
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v
Contents
Preface xiii
1 BasicIdeasofMathematicalStatistics 1
1.1 Statistical Population and Samples 2
1.1.1 Concrete Samples and Statistical Populations 2
1.1.2 SamplingProcedures 4
1.2 Mathematical Models forPopulation and Sample 8
1.3 Sufficiency and Completeness 9
1.4 The Notion of Information inStatistics 20
1.5 Statistical Decision Theory 28
1.6 Exercises 32
References 37
2 PointEstimation 39
2.1 Optimal UnbiasedEstimators 41
2.2 Variance-Invariant Estimation 53
2.3 Methodsfor Construction and Improvement of Estimators 57
2.3.1 Maximum Likelihood Method 57
2.3.2 Least Squares Method 60
2.3.3 MinimumChi-Squared Method 61
2.3.4 Methodof Moments 62
2.3.5 Jackknife Estimators 63
2.3.6 Estimators Based on Order Statistics 64
2.3.6.1 Order and Rank Statistics 64
2.3.6.2 L-Estimators 66
2.3.6.3 M-Estimators 67
2.3.6.4 R-Estimators 68
2.4 Properties of Estimators 68
2.4.1 Small Samples 69
vi Contents
2.4.2 Asymptotic Properties 71
2.5 Exercises 75
References 78
3 StatisticalTestsandConfidenceEstimations 79
3.1 Basic Ideas of Test Theory 79
3.2 The Neyman–Pearson Lemma 87
3.3 Tests forComposite Alternative Hypotheses and One-Parametric
Distribution Families 96
3.3.1 Distributions with Monotone Likelihood Ratio and Uniformly Most
Powerful Tests for One-Sided Hypotheses 96
3.3.2 UMPU-Tests for Two-Sided Alternative Hypotheses 105
3.4 Tests forMulti-Parametric Distribution Families 110
3.4.1 General Theory 111
3.4.2 The Two-SampleProblem: Properties of Various Tests and
Robustness 124
3.4.2.1 Comparison of Two Expectations 125
3.4.3 Comparison of Two Variances 137
3.4.4 Table for Sample Sizes 138
3.5 Confidence Estimation 139
3.5.1 One-Sided Confidence Intervals inOne-Parametric Distribution
Families 140
3.5.2 Two-Sided Confidence Intervals inOne-Parametric and Confidence
Intervals in Multi-Parametric Distribution Families 143
3.5.3 Table for Sample Sizes 146
3.6 Sequential Tests 147
3.6.1 Introduction 147
3.6.2 Wald’s Sequential Likelihood Ratio Test for One-Parametric
Exponential Families 149
3.6.3 Test about Mean Values for Unknown Variances 153
3.6.4 Approximate Tests for the Two-Sample Problem 158
3.6.5 Sequential Triangular Tests 160
3.6.6 A Sequential Triangular Test for the Correlation Coefficient 162
3.7 Remarks about Interpretation 169
3.8 Exercises 170
References 176
4 LinearModels–GeneralTheory 179
4.1 Linear Models with Fixed Effects 179
4.1.1 Least Squares Method 180
4.1.2 Maximum Likelihood Method 184
Contents vii
4.1.3 Tests of Hypotheses 185
4.1.4 Construction of Confidence Regions 190
4.1.5 Special Linear Models 191
4.1.6 The Generalised Least Squares Method(GLSM) 198
4.2 Linear Models with Random Effects: Mixed Models 199
4.2.1 Best Linear UnbiasedPrediction (BLUP) 200
4.2.2 Estimation of Variance Components 202
4.3 Exercises 203
References 204
5 AnalysisofVariance(ANOVA)–FixedEffectsModels(ModelIof
AnalysisofVariance) 207
5.1 Introduction 207
5.2 Analysis of Variance with One Factor (Simple- or One-Way Analysis
of Variance) 215
5.2.1 The Model and the Analysis 215
5.2.2 Planning the Size of anExperiment 228
5.2.2.1 General Description for All Sections of This Chapter 228
5.2.2.2 The Experimental Size for the One-Way Classification 231
5.3 Two-Way Analysis of Variance 232
5.3.1 Cross-Classification (A×B) 233
5.3.1.1 Parameter Estimation 236
5.3.1.2 TestingHypotheses 244
5.3.2 Nested Classification (A B) 260
5.4 Three-Way Classification 272
5.4.1 Complete Cross-Classification (A×B×C) 272
5.4.2 Nested Classification (C≺B≺A) 279
5.4.3 MixedClassification 282
5.4.3.1 Cross-Classification betweenTwo Factors Where One of Them Is
Subordinatedto a Third Factor B≺A ×C 282
5.4.3.2 Cross-Classification of Two Factors inWhich a Third Factor Is
Nested C≺ A×B 288
5.5 Exercises 291
References 291
6 AnalysisofVariance:EstimationofVarianceComponents
(ModelIIoftheAnalysisofVariance) 293
6.1 Introduction: Linear Models with Random Effects 293
6.2 One-Way Classification 297
6.2.1 Estimation of Variance Components 300
6.2.1.1 Analysis of Variance Method 300
viii Contents
6.2.1.2 Estimators in Case of Normally Distributed Y 302
6.2.1.3 REML Estimation 304
6.2.1.4 Matrix Norm Minimising Quadratic Estimation 305
6.2.1.5 Comparison of Several Estimators 306
6.2.2 Tests of Hypotheses and Confidence Intervals 308
6.2.3 Variances and Properties of the Estimators of the Variance
Components 310
6.3 Estimators of Variance Components in the Two-Way and
Three-Way Classification 315
6.3.1 General Description for Equal and Unequal Subclass Numbers 315
6.3.2 Two-Way Cross-Classification 319
6.3.3 Two-Way Nested Classification 324
6.3.4 Three-Way Cross-Classification with Equal
Subclass Numbers 326
6.3.5 Three-Way Nested Classification 333
6.3.6 Three-Way Mixed Classification 335
6.4 Planning Experiments 336
6.5 Exercises 338
References 339
7 AnalysisofVariance–ModelswithFiniteLevelPopulations
andMixedModels 341
7.1 Introduction: Models with Finite Level Populations 341
7.2 Rules forthe Derivation of SS,df,MS and E(MS) in Balanced
ANOVA Models 343
7.3 Variance Component Estimators inMixedModels 348
7.3.1 AnExample fortheBalanced Case 349
7.3.2 The Unbalanced Case 351
7.4 Tests forFixed Effects and Variance Components 353
7.5 Variance Component Estimation and Tests of Hypotheses in
Special Mixed Models 354
7.5.1 Two-Way Cross-Classification 355
7.5.2 Two-Way Nested Classification B≺A 358
7.5.2.1 Levels of A Random 360
7.5.2.2 Levels of B Random 361
7.5.3 Three-Way Cross-Classification 362
7.5.4 Three-Way Nested Classification 365
7.5.5 Three-Way Mixed Classification 368
7.5.5.1 The Type (B≺A)×C 368
7.5.5.2 The Type C≺AB 371
7.6 Exercises 374
References 374
Contents ix
8 RegressionAnalysis–LinearModelswithNon-randomRegressors
(ModelIofRegressionAnalysis)andwithRandomRegressors
(ModelIIofRegressionAnalysis) 377
8.1 Introduction 377
8.2 Parameter Estimation 380
8.2.1 Least Squares Method 380
8.2.2 Optimal Experimental Design 394
8.3 TestingHypotheses 397
8.4 Confidence Regions 406
8.5 Models with Random Regressors 410
8.5.1 Analysis 410
8.5.2 Experimental Designs 415
8.6 MixedModels 416
8.7 Concluding Remarks about Models of Regression Analysis 417
8.8 Exercises 419
References 419
9 RegressionAnalysis–IntrinsicallyNon-linearModelI 421
9.1 Estimating by the Least Squares Method 424
9.1.1 Gauss–Newton Method 425
9.1.2 InternalRegression 431
9.1.3 Determining Initial Values for Iteration Methods 433
9.2 GeometricalProperties 434
9.2.1 Expectation Surface and Tangent Plane 434
9.2.2 Curvature Measures 440
9.3 Asymptotic Properties and the Bias of LS Estimators 443
9.4 Confidence Estimations and Tests 447
9.4.1 Introduction 447
9.4.2 Tests and Confidence Estimations Based on the Asymptotic
Covariance Matrix 451
9.4.3 Simulation Experiments to Check Asymptotic Tests and
Confidence Estimations 452
9.5 Optimal Experimental Design 454
9.6 Special Regression Functions 458
9.6.1 Exponential Regression 458
9.6.1.1 Point Estimator 458
9.6.1.2 Confidence Estimations and Tests 460
9.6.1.3 Results of Simulation Experiments 463
9.6.1.4 Experimental Designs 466
9.6.2 The Bertalanffy Function 468
9.6.3 The Logistic (Three-Parametric Hyperbolic Tangent) Function 473
9.6.4 The Gompertz Function 476
x Contents
9.6.5 The Hyperbolic Tangent Function with Four Parameters 480
9.6.6 The Arc Tangent Function with Four Parameters 484
9.6.7 The Richards Function 487
9.6.8 Summarisingthe Results of Sections 9.6.1–9.6.7 487
9.6.9 Problems of Model Choice 488
9.7 Exercises 489
References 490
10 AnalysisofCovariance(ANCOVA) 495
10.1 Introduction 495
10.2 General Model I–I of the Analysis of Covariance 496
10.3 Special Models of the Analysis of Covariance forthe Simple
Classification 503
10.3.1 One Covariable with Constant γ 504
10.3.2 ACovariablewithRegressionCoefficientsγ DependingontheLevels
i
of the Classification Factor 506
10.3.3 A Numerical Example 507
10.4 Exercises 510
References 511
11 MultipleDecisionProblems 513
11.1 Selection Procedures 514
11.1.1 Basic Ideas 514
11.1.2 Indifference Zone Formulation forExpectations 516
11.1.2.1 Selection of Populations with Normal Distribution 517
11.1.2.2 ApproximateSolutionsforNon-normalDistributionsandt=1 529
11.1.3 Selection of a Subset Containing the Best Population with Given
Probability 531
11.1.3.1 Selection of the Normal Distribution with the Largest
Expectation 534
11.1.3.2 Selection of the Normal Distribution with Smallest Variance 534
11.2 Multiple Comparisons 539
11.2.1 Confidence Intervals for All Contrasts: Scheffé’sMethod 542
11.2.2 Confidence Intervals for Given Contrasts: Bonferroni’s and Dunn’s
Method 548
11.2.3 Confidence Intervals for All Contrasts for n =n:Tukey’s
i
Method 550
11.2.4 Confidence Intervals for All Contrasts: Generalised Tukey’s
Method 553
11.2.5 ConfidenceIntervalsfortheDifferencesofTreatmentswithaControl:
Dunnett’s Method 554
11.2.6 Multiple Comparisons and Confidence Intervals 556