Bijma F., Jonker M., Aad van der Vaart. An Introduction to Mathematical Statistics

Amsterdam University, 2018. — 384 p. — ISBN: 9462985103.The field of statistics focuses on drawing conclusions from data by modeling and analyzing the data using probabilistic models. The authors of this introductory text describe three key concepts from statistics—estimators, tests, and confidence regions—which they demonstrate and apply in an extensive variety of examples and case studies. An entire chapter covers regression models, including linear regression and analysis of variance. This book, designed for students, assumes a basic knowledge of probability theory, calculus, and linear algebra
About the Author
Fetsje Bijma worked as an assistant professor of mathematics at the Vrije Universiteit Amsterdam for ten years.
Marianne Jonker is a biostatistician at the Radboud University Medical Center in Nijmegen.
Aad van der Vaart is professor of stochastics at Leiden University.What Is Statistics?
StatisticalModels
Exercises
Application: Cox RegressionDescriptive StatisticsUnivariate Samples
CorrelationExercises
Application: Benford’s LawEstimatorsMeanSquareError
Maximum Likelihood Estimators
Methodof Moments Estimators
Bayes Estimators
M-EstimatorsExercises
Application: Twin StudiesHypothesis TestingNull Hypothesis and Alternative Hypothesis
Sample Sizeand Critical Region
Testing with p-Values
Statistical Significance
Some Standard Tests
Likelihood Ratio Tests
Scoreand Wald Tests
Multiple TestingExercises
Application: Shares According to Black–ScholesConfidence RegionsInterpretation of a Confidence Region
Pivotsand Near-Pivots
Maximum Likelihood Estimators as Near-Pivots
Confidence Regionsand Tests
Likelihood Ratio Regions
Bayesian Confidence RegionsExercises
Application: The Salk VaccineOptimality TheorySufficientStatistics
EstimationTheory
TestingTheoryExercises
Application: High Water in LimburgRegression ModelsLinear Regression
Analysisof Variance
Nonlinear and Nonparametric Regression
Classification
Cox Regression Model
Mixed ModelsExercises
Application: Regression Models and CausalityModel SelectionGoalof Model Selection
Test Methods
Penalty Methods
Bayesian Model Selection
Cross-Validation
Post-Model Selection AnalysisApplication: Air PollutionProbability TheoryDistributions
Expectationand Variance
Standard Distributions
Multivariateand Marginal Distributions
Independence and Conditioning
Limit Theorems and the Normal Approximation
ExercisesMultivariate Normal DistributionCovariance Matrices
Definition and Basic Properties
Conditional Distributions
Multivariate Central Limit Theorem
Derived Distributions
Tables
Normal Distribution
t-Distribution
Chi-Square Distribution
Binomial Distribution (n = 10)
AnswerstoExercises