Praxisorientiertes Fachbuch der Interferenzstatistik mit den neuesten Entwicklungen aus diesem ständig wachsenden Wissensgebiet. Dieses übersichtlich und zugängliche Fachbuch richtet sich an Studenten höherer Semester, präsentiert die Interferenzstatistik ausführlich und praxisorientiert und stellt Ergebnisableitungen sowie MATLAB-Programme umfassend dar, ergänzt um Erläuterungen. Besonderes Augenmerk liegt auf einzelnen bedeutenden Aspekten, auf einer intuitiven Herangehensweise und auf Diskussionen. Der Blick auf die Interferenzstatistik ist dabei überaus modern. Inhalte neben den…mehr
Praxisorientiertes Fachbuch der Interferenzstatistik mit den neuesten Entwicklungen aus diesem ständig wachsenden Wissensgebiet. Dieses übersichtlich und zugängliche Fachbuch richtet sich an Studenten höherer Semester, präsentiert die Interferenzstatistik ausführlich und praxisorientiert und stellt Ergebnisableitungen sowie MATLAB-Programme umfassend dar, ergänzt um Erläuterungen. Besonderes Augenmerk liegt auf einzelnen bedeutenden Aspekten, auf einer intuitiven Herangehensweise und auf Diskussionen. Der Blick auf die Interferenzstatistik ist dabei überaus modern. Inhalte neben den klassischen Themen rund um die mathematische Statistik: intuitive Präsentation von Einfach-/Doppel-Bootstraps bei der Berechnung von Konfidenzintervallen, Schrumpfungsschätzung, Schätzung des maximalen Moments sowie eine Vielzahl vom Methoden der Punktschätzung, maximale Wahrscheinlichkeit, Anwendung von charakteristischen Funktionen und indirekte Interferenz. Zu allen Methoden gibt es praktische Beispiele. Ausführlich behandelt werden Schätzprobleme und deren Lösung in Verbindung mit der diskreten Mischung bei Normalverteilungen. Durchgängig liegt der Schwerpunkt auf nicht-Gaußschen Verteilungen, einschließlich der ausführlichen Behandlung der stabilen Pareto-Verteilung und der schnellen Berechnung von nicht-zentralen Student-t-Tests. Ein komplettes Kapitel widmet sich der Optimierung, darunter der Entwicklung von Hessian-Methoden, heuristische/genetische Algorithmen, die keine Kontinuität erfordern. Die entsprechenden MATLAB-Codes werden zur Verfügung gestellt. Der Fokus liegt auch auf Berechnungen, die das Thema greifbar und für die Studierenden zugänglich machen.
Marc S. Paolella, PhD, is a Professor at the Department of Banking and Finance, University of Zurich. He is also the Editor of Econometrics and an Associate Editor of the Royal Statistical Society Journal Series A.
Inhaltsangabe
Preface xi PART I ESSENTIAL CONCEPTS IN STATISTICS 1 Introducing Point and Interval Estimation 3 1.1 Point Estimation 4 1.1.1 Bernoulli Model 4 1.1.2 Geometric Model 6 1.1.3 Some Remarks on Bias and Consistency 11 1.2 Interval Estimation via Simulation 12 1.3 Interval Estimation via the Bootstrap 18 1.3.1 Computation and Comparison with Parametric Bootstrap 18 1.3.2 Application to Bernoulli Model and Modification 20 1.3.3 Double Bootstrap 24 1.3.4 Double Bootstrap with Analytic Inner Loop 26 1.4 Bootstrap Confidence Intervals in the Geometric Model 31 1.5 Problems 35 2 Goodness of Fit and Hypothesis Testing 37 2.1 Empirical Cumulative Distribution Function 38 2.1.1 The Glivenko-Cantelli Theorem 38 2.1.2 Proofs of the Glivenko-Cantelli Theorem 41 2.1.3 Example with Continuous Data and Approximate Confidence Intervals 45 2.1.4 Example with Discrete Data and Approximate Confidence Intervals 49 2.2 Comparing Parametric and Nonparametric Methods 52 2.3 Kolmogorov-Smirnov Distance and Hypothesis Testing 57 2.3.1 The Kolmogorov-Smirnov and Anderson-Darling Statistics 57 2.3.2 Significance and Hypothesis Testing 59 2.3.3 Small-Sample Correction 63 2.4 Testing Normality with KD and AD 65 2.5 Testing Normality with W² and U² 68 2.6 Testing the Stable Paretian Distributional Assumption: First Attempt 69 2.7 Two-Sample Kolmogorov Test 73 2.8 More on (Moron?) Hypothesis Testing 74 2.8.1 Explanation 75 2.8.2 Misuse of Hypothesis Testing 77 2.8.3 Use and Misuse of p-Values 79 2.9 Problems 82 3 Likelihood 85 3.1 Introduction 85 3.1.1 Scalar Parameter Case 87 3.1.2 Vector Parameter Case 92 3.1.3 Robustness and the MCD Estimator 100 3.1.4 Asymptotic Properties of the Maximum Likelihood Estimator 102 3.2 Cramér-Rao Lower Bound 107 3.2.1 Univariate Case 108 3.2.2 Multivariate Case 111 3.3 Model Selection 114 3.3.1 Model Misspecification 114 3.3.2 The Likelihood Ratio Statistic 117 3.3.3 Use of Information Criteria 119 3.4 Problems 120 4 Numerical Optimization 123 4.1 Root Finding 123 4.1.1 One Parameter 124 4.1.2 Several Parameters 131 4.2 Approximating the Distribution of the Maximum Likelihood Estimator 135 4.3 General Numerical Likelihood Maximization 136 4.3.1 Newton-Raphson and Quasi-Newton Methods 137 4.3.2 Imposing Parameter Restrictions 140 4.4 Evolutionary Algorithms 145 4.4.1 Differential Evolution 146 4.4.2 Covariance Matrix Adaption Evolutionary Strategy 149 4.5 Problems 155 5 Methods of Point Estimation 157 5.1 Univariate Mixed Normal Distribution 157 5.1.1 Introduction 157 5.1.2 Simulation of Univariate Mixtures 160 5.1.3 Direct Likelihood Maximization 161 5.1.4 Use of the EM Algorithm 169 5.1.5 Shrinkage-Type Estimation 174 5.1.6 Quasi-Bayesian Estimation 176 5.1.7 Confidence Intervals 178 5.2 Alternative Point Estimation Methodologies 184 5.2.1 Method of Moments Estimator 185 5.2.2 Use of Goodness-of-Fit Measures 190 5.2.3 Quantile Least Squares 191 5.2.4 Pearson Minimum Chi-Square 193 5.2.5 Empirical Moment Generating Function Estimator 195 5.2.6 Empirical Characteristic Function Estimator 198 5.3 Comparison of Methods 199 5.4 A Primer on Shrinkage Estimation 200 5.5 Problems 202 PART II FURTHER FUNDAMENTAL CONCEPTS IN STATISTICS 6 Q-Q Plots and Distribution Testing 209 6.1 P-P Plots and Q-Q Plots 209 6.2 Null Bands 211 6.2.1 Definition and Motivation 211 6.2.2 Pointwise Null Bands via Simulation 212 6.2.3 Asymptotic Approximation of Pointwise Null Bands 213 6.2.4 Mapping Pointwise and Simultaneous Significance Levels 215 6.3 Q-Q Test 217 6.4 Further P-P and Q-Q Type Plots 219 6.4.1 (Horizontal) Stabilized P-P Plots 219 6.4.2 Modified S-P Plots 220 6.4.3 MSP Test for Normality 224 6.4.4 Modified Percentile (Fowlkes-MP) Plots 228 6.5 Further Tests for Composite Normality 231 6.5.1 Motivation 232 6.5.2 Jarque-Bera Test 234 6.5.3 Three Powerful (and More Recent) Normality Tests 237 6.5.4 Testing Goodness of Fit via Binning: Pearson's X P² Test 240 6.6 Combining Tests and Power Envelopes 247 6.6.1 Combining Tests 248 6.6.2 Power Comparisons for Testing Composite Normality 252 6.6.3 Most Powerful Tests and Power Envelopes 252 6.7 Details of a Failed Attempt 255 6.8 Problems 260 7 Unbiased Point Estimation and Bias Reduction 269 7.1 Sufficiency 269 7.1.1 Introduction 269 7.1.2 Factorization 272 7.1.3 Minimal Sufficiency 276 7.1.4 The Rao-Blackwell Theorem 283 7.2 Completeness and the Uniformly Minimum Variance Unbiased Estimator 286 7.3 An Example with i.i.d. Geometric Data 289 7.4 Methods of Bias Reduction 293 7.4.1 The Bias-Function Approach 293 7.4.2 Median-Unbiased Estimation 296 7.4.3 Mode-Adjusted Estimator 297 7.4.4 The Jackknife 302 7.5 Problems 305 8 Analytic Interval Estimation 313 8.1 Definitions 313 8.2 Pivotal Method 315 8.2.1 Exact Pivots 315 8.2.2 Asymptotic Pivots 318 8.3 Intervals Associated with Normal Samples 319 8.3.1 Single Sample 319 8.3.2 Paired Sample 320 8.3.3 Two Independent Samples 322 8.3.4 Welch's Method for my1 - my2 when sigma1² sigma2² 323 8.3.5 Satterthwaite's Approximation 324 8.4 Cumulative Distribution Function Inversion 326 8.4.1 Continuous Case 326 8.4.2 Discrete Case 330 8.5 Application of the Nonparametric Bootstrap 334 8.6 Problems 337 PART III ADDITIONAL TOPICS 9 Inference in a Heavy-Tailed Context 341 9.1 Estimating the Maximally Existing Moment 342 9.2 A Primer on Tail Estimation 346 9.2.1 Introduction 346 9.2.2 The Hill Estimator 346 9.2.3 Use with Stable Paretian Data 349 9.3 Noncentral Student's t Estimation 351 9.3.1 Introduction 351 9.3.2 Direct Density Approximation 352 9.3.3 Quantile-Based Table Lookup Estimation 353 9.3.4 Comparison of NCT Estimators 354 9.4 Asymmetric Stable Paretian Estimation 358 9.4.1 Introduction 358 9.4.2 The Hint Estimator 359 9.4.3 Maximum Likelihood Estimation 360 9.4.4 The McCulloch Estimator 361 9.4.5 The Empirical Characteristic Function Estimator 364 9.4.6 Testing for Symmetry in the Stable Model 366 9.5 Testing the Stable Paretian Distribution 368 9.5.1 Test Based on the Empirical Characteristic Function 368 9.5.2 Summability Test and Modification 371 9.5.3 ALHADI: The alpha-Hat Discrepancy Test 375 9.5.4 Joint Test Procedure 383 9.5.5 Likelihood Ratio Tests 384 9.5.6 Size and Power of the Symmetric Stable Tests 385 9.5.7 Extension to Testing the Asymmetric Stable Paretian Case 395 10 The Method of Indirect Inference 401 10.1 Introduction 401 10.2 Application to the Laplace Distribution 403 10.3 Application to Randomized Response 403 10.3.1 Introduction 403 10.3.2 Estimation via Indirect Inference 406 10.4 Application to the Stable Paretian Distribution 409 10.5 Problems 416 A Review of Fundamental Concepts in Probability Theory 419 A.1 Combinatorics and Special Functions 420 A.2 Basic Probability and Conditioning 423 A.3 Univariate Random Variables 424 A.4 Multivariate Random Variables 427 A.5 Continuous Univariate Random Variables 430 A.6 Conditional Random Variables 432 A.7 Generating Functions and Inversion Formulas 434 A.8 Value at Risk and Expected Shortfall 437 A.9 Jacobian Transformations 451 A.10 Sums and Other Functions 453 A.11 Saddlepoint Approximations 456 A.12 Order Statistics 460 A.13 The Multivariate Normal Distribution 462 A.14 Noncentral Distributions 465 A.15 Inequalities and Convergence 467 A.15.1 Inequalities for Random Variables 467 A.15.2 Convergence of Sequences of Sets 469 A.15.3 Convergence of Sequences of Random Variables 473 A.16 The Stable Paretian Distribution 483 A.17 Problems 492 A.18 Solutions 509 References 537 Index 561
Preface xi PART I ESSENTIAL CONCEPTS IN STATISTICS 1 Introducing Point and Interval Estimation 3 1.1 Point Estimation 4 1.1.1 Bernoulli Model 4 1.1.2 Geometric Model 6 1.1.3 Some Remarks on Bias and Consistency 11 1.2 Interval Estimation via Simulation 12 1.3 Interval Estimation via the Bootstrap 18 1.3.1 Computation and Comparison with Parametric Bootstrap 18 1.3.2 Application to Bernoulli Model and Modification 20 1.3.3 Double Bootstrap 24 1.3.4 Double Bootstrap with Analytic Inner Loop 26 1.4 Bootstrap Confidence Intervals in the Geometric Model 31 1.5 Problems 35 2 Goodness of Fit and Hypothesis Testing 37 2.1 Empirical Cumulative Distribution Function 38 2.1.1 The Glivenko-Cantelli Theorem 38 2.1.2 Proofs of the Glivenko-Cantelli Theorem 41 2.1.3 Example with Continuous Data and Approximate Confidence Intervals 45 2.1.4 Example with Discrete Data and Approximate Confidence Intervals 49 2.2 Comparing Parametric and Nonparametric Methods 52 2.3 Kolmogorov-Smirnov Distance and Hypothesis Testing 57 2.3.1 The Kolmogorov-Smirnov and Anderson-Darling Statistics 57 2.3.2 Significance and Hypothesis Testing 59 2.3.3 Small-Sample Correction 63 2.4 Testing Normality with KD and AD 65 2.5 Testing Normality with W² and U² 68 2.6 Testing the Stable Paretian Distributional Assumption: First Attempt 69 2.7 Two-Sample Kolmogorov Test 73 2.8 More on (Moron?) Hypothesis Testing 74 2.8.1 Explanation 75 2.8.2 Misuse of Hypothesis Testing 77 2.8.3 Use and Misuse of p-Values 79 2.9 Problems 82 3 Likelihood 85 3.1 Introduction 85 3.1.1 Scalar Parameter Case 87 3.1.2 Vector Parameter Case 92 3.1.3 Robustness and the MCD Estimator 100 3.1.4 Asymptotic Properties of the Maximum Likelihood Estimator 102 3.2 Cramér-Rao Lower Bound 107 3.2.1 Univariate Case 108 3.2.2 Multivariate Case 111 3.3 Model Selection 114 3.3.1 Model Misspecification 114 3.3.2 The Likelihood Ratio Statistic 117 3.3.3 Use of Information Criteria 119 3.4 Problems 120 4 Numerical Optimization 123 4.1 Root Finding 123 4.1.1 One Parameter 124 4.1.2 Several Parameters 131 4.2 Approximating the Distribution of the Maximum Likelihood Estimator 135 4.3 General Numerical Likelihood Maximization 136 4.3.1 Newton-Raphson and Quasi-Newton Methods 137 4.3.2 Imposing Parameter Restrictions 140 4.4 Evolutionary Algorithms 145 4.4.1 Differential Evolution 146 4.4.2 Covariance Matrix Adaption Evolutionary Strategy 149 4.5 Problems 155 5 Methods of Point Estimation 157 5.1 Univariate Mixed Normal Distribution 157 5.1.1 Introduction 157 5.1.2 Simulation of Univariate Mixtures 160 5.1.3 Direct Likelihood Maximization 161 5.1.4 Use of the EM Algorithm 169 5.1.5 Shrinkage-Type Estimation 174 5.1.6 Quasi-Bayesian Estimation 176 5.1.7 Confidence Intervals 178 5.2 Alternative Point Estimation Methodologies 184 5.2.1 Method of Moments Estimator 185 5.2.2 Use of Goodness-of-Fit Measures 190 5.2.3 Quantile Least Squares 191 5.2.4 Pearson Minimum Chi-Square 193 5.2.5 Empirical Moment Generating Function Estimator 195 5.2.6 Empirical Characteristic Function Estimator 198 5.3 Comparison of Methods 199 5.4 A Primer on Shrinkage Estimation 200 5.5 Problems 202 PART II FURTHER FUNDAMENTAL CONCEPTS IN STATISTICS 6 Q-Q Plots and Distribution Testing 209 6.1 P-P Plots and Q-Q Plots 209 6.2 Null Bands 211 6.2.1 Definition and Motivation 211 6.2.2 Pointwise Null Bands via Simulation 212 6.2.3 Asymptotic Approximation of Pointwise Null Bands 213 6.2.4 Mapping Pointwise and Simultaneous Significance Levels 215 6.3 Q-Q Test 217 6.4 Further P-P and Q-Q Type Plots 219 6.4.1 (Horizontal) Stabilized P-P Plots 219 6.4.2 Modified S-P Plots 220 6.4.3 MSP Test for Normality 224 6.4.4 Modified Percentile (Fowlkes-MP) Plots 228 6.5 Further Tests for Composite Normality 231 6.5.1 Motivation 232 6.5.2 Jarque-Bera Test 234 6.5.3 Three Powerful (and More Recent) Normality Tests 237 6.5.4 Testing Goodness of Fit via Binning: Pearson's X P² Test 240 6.6 Combining Tests and Power Envelopes 247 6.6.1 Combining Tests 248 6.6.2 Power Comparisons for Testing Composite Normality 252 6.6.3 Most Powerful Tests and Power Envelopes 252 6.7 Details of a Failed Attempt 255 6.8 Problems 260 7 Unbiased Point Estimation and Bias Reduction 269 7.1 Sufficiency 269 7.1.1 Introduction 269 7.1.2 Factorization 272 7.1.3 Minimal Sufficiency 276 7.1.4 The Rao-Blackwell Theorem 283 7.2 Completeness and the Uniformly Minimum Variance Unbiased Estimator 286 7.3 An Example with i.i.d. Geometric Data 289 7.4 Methods of Bias Reduction 293 7.4.1 The Bias-Function Approach 293 7.4.2 Median-Unbiased Estimation 296 7.4.3 Mode-Adjusted Estimator 297 7.4.4 The Jackknife 302 7.5 Problems 305 8 Analytic Interval Estimation 313 8.1 Definitions 313 8.2 Pivotal Method 315 8.2.1 Exact Pivots 315 8.2.2 Asymptotic Pivots 318 8.3 Intervals Associated with Normal Samples 319 8.3.1 Single Sample 319 8.3.2 Paired Sample 320 8.3.3 Two Independent Samples 322 8.3.4 Welch's Method for my1 - my2 when sigma1² sigma2² 323 8.3.5 Satterthwaite's Approximation 324 8.4 Cumulative Distribution Function Inversion 326 8.4.1 Continuous Case 326 8.4.2 Discrete Case 330 8.5 Application of the Nonparametric Bootstrap 334 8.6 Problems 337 PART III ADDITIONAL TOPICS 9 Inference in a Heavy-Tailed Context 341 9.1 Estimating the Maximally Existing Moment 342 9.2 A Primer on Tail Estimation 346 9.2.1 Introduction 346 9.2.2 The Hill Estimator 346 9.2.3 Use with Stable Paretian Data 349 9.3 Noncentral Student's t Estimation 351 9.3.1 Introduction 351 9.3.2 Direct Density Approximation 352 9.3.3 Quantile-Based Table Lookup Estimation 353 9.3.4 Comparison of NCT Estimators 354 9.4 Asymmetric Stable Paretian Estimation 358 9.4.1 Introduction 358 9.4.2 The Hint Estimator 359 9.4.3 Maximum Likelihood Estimation 360 9.4.4 The McCulloch Estimator 361 9.4.5 The Empirical Characteristic Function Estimator 364 9.4.6 Testing for Symmetry in the Stable Model 366 9.5 Testing the Stable Paretian Distribution 368 9.5.1 Test Based on the Empirical Characteristic Function 368 9.5.2 Summability Test and Modification 371 9.5.3 ALHADI: The alpha-Hat Discrepancy Test 375 9.5.4 Joint Test Procedure 383 9.5.5 Likelihood Ratio Tests 384 9.5.6 Size and Power of the Symmetric Stable Tests 385 9.5.7 Extension to Testing the Asymmetric Stable Paretian Case 395 10 The Method of Indirect Inference 401 10.1 Introduction 401 10.2 Application to the Laplace Distribution 403 10.3 Application to Randomized Response 403 10.3.1 Introduction 403 10.3.2 Estimation via Indirect Inference 406 10.4 Application to the Stable Paretian Distribution 409 10.5 Problems 416 A Review of Fundamental Concepts in Probability Theory 419 A.1 Combinatorics and Special Functions 420 A.2 Basic Probability and Conditioning 423 A.3 Univariate Random Variables 424 A.4 Multivariate Random Variables 427 A.5 Continuous Univariate Random Variables 430 A.6 Conditional Random Variables 432 A.7 Generating Functions and Inversion Formulas 434 A.8 Value at Risk and Expected Shortfall 437 A.9 Jacobian Transformations 451 A.10 Sums and Other Functions 453 A.11 Saddlepoint Approximations 456 A.12 Order Statistics 460 A.13 The Multivariate Normal Distribution 462 A.14 Noncentral Distributions 465 A.15 Inequalities and Convergence 467 A.15.1 Inequalities for Random Variables 467 A.15.2 Convergence of Sequences of Sets 469 A.15.3 Convergence of Sequences of Random Variables 473 A.16 The Stable Paretian Distribution 483 A.17 Problems 492 A.18 Solutions 509 References 537 Index 561
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