Fundamentals of Mathematical Statistics

Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
List of Figures
List of Tables
1. Statistical Models
	1.1. Models and parametrizations
		1.1.1. Examples of statistical models
		1.1.2. Reparametrization
		1.1.3. Parameter functions
		1.1.4. Nuisance parameters and parameters of interest
	1.2. Likelihood, score, and information
		1.2.1. The likelihood function
			1.2.1.1. Formal definition
			1.2.1.2. Equivariance of the likelihood function
			1.2.1.3. Likelihood as a random variable
		1.2.2. Score and information
		1.2.3. Reparametrization and repetition
	1.3. Exercises
2. Linear Normal Models
	2.1. The multivariate normal distribution
	2.2. The normal distribution on a vector space
		2.2.1. Random vectors in V
		2.2.2. Projections with respect to the concentration
		2.2.3. Derived distributions
	2.3. The linear normal model
		2.3.1. Basic structure
		2.3.2. Likelihood, score, and information
	2.4. Exercises
3. Exponential Families
	3.1. Regular exponential families
	3.2. Examples of exponential families
	3.3. Properties of exponential families
	3.4. Constructing exponential families
		3.4.1. Product families
		3.4.2. Repeated observations
		3.4.3. Transformations
		3.4.4. Affine subfamilies
	3.5. Moments, score, and information
	3.6. Curved exponential families
	3.7. Exercises
4. Estimation
	4.1. General concepts and exact properties
	4.2. Various estimation methods
		4.2.1. The method of least absolute deviations
		4.2.2. The method of least squares
		4.2.3. M-estimators
		4.2.4. The method of moments
	4.3. The method of maximum likelihood
		4.3.1. General considerations
		4.3.2. Maximum likelihood in regular exponential families
	4.4. Exercises
5. Asymptotic Theory
	5.1. Asymptotic consistency and normality
	5.2. Asymptotics of moment estimators
	5.3. Asymptotics in regular exponential families
		5.3.1. Asymptotic consistency of maximum likelihood
		5.3.2. Asymptotic normality of maximum likelihood
		5.3.3. Likelihood ratios and quadratic forms
	5.4. Asymptotics in curved exponential families
		5.4.1. Consistency of the maximum-likelihood estimator
		5.4.2. Asymptotic normality in curved exponential families
		5.4.3. Geometric interpretation of the score equation
		5.4.4. Likelihood ratios and quadratic forms
	5.5. More about asymptotics
	5.6. Exercises
6. Set Estimation
	6.1. Basic issues and definition
	6.2. Exact confidence regions by pivots
	6.3. Likelihood-based regions
	6.4. Confidence regions by asymptotic pivots
		6.4.1. Asymptotic likelihood-based regions
		6.4.2. Quadratic score regions
		6.4.3. Wald regions
		6.4.4. Confidence regions for parameter functions
	6.5. Properties of set estimators
		6.5.1. Reparametrization
		6.5.2. Coverage and length
	6.6. Credibility regions
	6.7. Exercises
7. Significance Testing
	7.1. The problem
	7.2. Hypotheses and test statistics
		7.2.1. Formal concepts
		7.2.2. Classifying hypotheses by purpose
		7.2.3. Mathematical classification of hypotheses
	7.3. Significance and p-values
	7.4. Critical regions, power, and error types
	7.5. Set estimation and testing
	7.6. Test in linear normal models
		7.6.1. The general case
			7.6.1.1. Linear hypothesis, variance known
			7.6.1.2. Linear subhypothesis, variance unknown
		7.6.2. Some standard tests
			7.6.2.1. Z-test for a given mean, variance known
			7.6.2.2. T-test for a given mean, variance unknown
			7.6.2.3. T-test for comparing means
			7.6.2.4. T-test for paired comparisons
	7.7. Determining p-values
		7.7.1. Monte Carlo p-values
			7.7.1.1. Simple hypotheses
			7.7.1.2. Composite hypotheses
		7.7.2. Asymptotic p-values
			7.7.2.1. Simple hypotheses
			7.7.2.2. Composite hypotheses
			7.7.2.3. Smooth hypotheses
	7.8. Exercises
8. Models for Tables of Counts
	8.1. Multinomial exponential families
		8.1.1. The unrestricted multinomial family
		8.1.2. Curved multinomial families
			8.1.2.1. Score and information
			8.1.2.2. Likelihood ratio
			8.1.2.3. Wald statistics
		8.1.3. Residuals
		8.1.4. Weldon’s dice
	8.2. Genetic equilibrium models
		8.2.1. Hardy–Weinberg equilibrium
		8.2.2. The ABO blood type system
	8.3. Contingency tables
		8.3.1. Comparing multinomial distributions
			8.3.1.1. Comparing two multinomial distributions
			8.3.1.2. Comparing two proportions
			8.3.1.3. Comparing several multinomial distributions
		8.3.2. Independence of classification criteria
		8.3.3. Poisson models for contingency tables
			8.3.3.1. The simple multiplicative Poisson model
			8.3.3.2. The shifted multiplicative Poisson model
		8.3.4. Sampling models for tables of counts
			8.3.4.1. From multiplicative Poisson to independence
			8.3.4.2. From independence to homogeneity
			8.3.4.3. Exact conditional tests
		8.3.5. Fisher’s exact test
		8.3.6. Rank statistics for ordinal data
	8.4. Exercises
A. Auxiliary Results
	A.1. Euclidean vector spaces
	A.2. Convergence of random variables
		A.2.1. Convergence in probability
		A.2.2. Convergence in distribution
		A.2.3. The delta method
	A.3. Results from real analysis
		A.3.1. Inverse and implicit functions
		A.3.2. Taylor approximation
	A.4. The information inequality
	A.5. Trace of a matrix
B. Technical Proofs
	B.1. Analytic properties of exponential families
		B.1.1. Integrability of derivatives
		B.1.2. Quadratic regularity of exponential families
	B.2. Asymptotic existence of the MLE
	B.3. Iterative proportional scaling
Bibliography
Index