An Introduction to Mathematical Statistics

Fetsje Bijma worked as an assistant professor of mathematics at the Vrije Universiteit Amsterdam for 10 years. She studied mathematics at the University of Groningen and received a Ph.D. at the Faculty of Medicine of the Vrije Universiteit Amsterdam. She now works in industry. Marianne Jonker is a biostatistician at the Radboud University Medical Center in Nijmegen. She studied mathematics and received her Ph.D. at the Vrije Universiteit Amsterdam, where she worked as an assistant professor in mathematics for some time. She has done work as a biostatistician for the Leiden University Medical Center in Leiden and the VU University Medical Center in Amsterdam. Aad van der Vaart is a professor of stochastics at Leiden University and member of the Royal Netherlands Academy of Arts and Sciences. He studied mathematics and psychology at Leiden University and received a Ph.D. in mathematics at the same university. He held positions at universities in the United States and in Paris (France) and at the Vrije Universiteit Amsterdam. In 2015 he won the highest scientific award in the Netherlands, the NWO Spinoza Prize, for his groundbreaking research in statistics.

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1[-]1.1. WhatIsStatistics? . . . . . . . . . . . . . . . . . . . . . 1[-]1.2. StatisticalModels . . . . . . . . . . . . . . . . . . . . . 2[-]Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 12[-]Application: Cox Regression . . . . . . . . . . . . . . . . . 15[-]2. DescriptiveStatistics . . . . . . . . . . . . . . . . . . . . . . 21[-]2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 21[-]2.2. UnivariateSamples . . . . . . . . . . . . . . . . . . . . . 21[-]2.3. Correlation . . . . . . . . . . . . . . . . . . . . . . . . 32[-]2.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . 38[-]Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 39[-]Application: Benford's Law . . . . . . . . . . . . . . . . . 41[-]3. Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . 45[-]3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 45[-]3.2. MeanSquareError . . . . . . . . . . . . . . . . . . . . . 46[-]3.3. Maximum Likelihood Estimators . . . . . . . . . . . . . . . 54[-]3.4. MethodofMomentsEstimators . . . . . . . . . . . . . . . . 72[-]3.5. BayesEstimators . . . . . . . . . . . . . . . . . . . . . . 75[-]3.6. M-Estimators . . . . . . . . . . . . . . . . . . . . . . . 88[-]3.7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . 93[-]Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 94[-]Application: Twin Studies . . . . . . . . . . . . . . . . . 100[-]4. Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . 105[-]4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 105[-]4.2. Null Hypothesis and Alternative Hypothesis . . . . . . . . . . 105[-]4.3. SampleSizeandCriticalRegion . . . . . . . . . . . . . . 107[-]4.4. Testing with p-Values . . . . . . . . . . . . . . . . . . . 121[-]4.5. StatisticalSignificance . . . . . . . . . . . . . . . . . . 126[-]4.6. SomeStandardTests . . . . . . . . . . . . . . . . . . . 127[-]4.7. Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . 143[-]4.8. ScoreandWaldTests . . . . . . . . . . . . . . . . . . . 150[-]4.9. Multiple Testing . . . . . . . . . . . . . . . . . . . . . 153[-]4.10. Summary . . . . . . . . . . . . . . . . . . . . . . . . 159[-]Exercises . . . . . . . . . . . . . . . . . . . . . . . . 160[-]Application: Shares According to Black-Scholes . . . . . . . . 169[-]5. ConfidenceRegions . . . . . . . . . . . . . . . . . . . . . 174[-]5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 174[-]5.2. Interpretation of a Confidence Region . . . . . . . . . . . . 174[-]5.3. PivotsandNear-Pivots . . . . . . . . . . . . . . . . . . 177[-]5.4. Maximum Likelihood Estimators as Near-Pivots . . . . . . . . 181[-]5.5. ConfidenceRegionsandTests . . . . . . . . . . . . . . . 195[-]5.6. Likelihood Ratio Regions . . . . . . . . . . . . . . . . . 198[-]5.7. BayesianConfidenceRegions . . . . . . . . . . . . . . . . 201[-]5.8. Summary . . . . . . . . . . . . . . . . . . . . . . . . 205[-]Exercises . . . . . . . . . . . . . . . . . . . . . . . . 206[-]Application: The Salk Vaccine . . . . . . . . . . . . . . . 209[-]6. Optimality Theory . . . . . . . . . . . . . . . . . . . . . . 212[-]6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 212[-]6.2. SufficientStatistics . . . . . . . . . . . . . . . . . . . . 212[-]6.3. EstimationTheory . . . . . . . . . . . . . . . . . . . . 219[-]6.4. TestingTheory . . . . . . . . . . . . . . . . . . . . . 231[-]6.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . 245[-]Exercises . . . . . . . . . . . . . . . . . . . . . . . . 246[-]Application: High Water in Limburg . . . . . . . . . . . . . 250[-]7. RegressionModels . . . . . . . . . . . . . . . . . . . . . . 259[-]7.1. Introduction