Mathematical Statistics with R I: Foundations

Summary statistics, visualisation, and the art of understanding data before modelling it.

1. 1.1 Types of Data and Central Tendency
2. 1.2 Spread and Shape
3. 1.3 Visualisation and Relationships

Sample spaces, axioms, conditional probability, independence, and Bayes theorem.

1. 2.1 Sets, Axioms, and Counting
2. 2.2 Conditional Probability and Independence
3. 2.3 Bayes' Theorem

Discrete and continuous random variables, expectation, variance, moments, and moment generating functions.

1. 3.1 Discrete and Continuous Random Variables
2. 3.2 Expectation and Variance
3. 3.3 Higher Moments and Moment Generating Functions

Bernoulli, binomial, geometric, hypergeometric, and Poisson — each derived from first principles.

1. 4.1 Bernoulli and Binomial
2. 4.2 Geometric, Negative Binomial, and Hypergeometric
3. 4.3 The Poisson Distribution and Relationships

Uniform, exponential, gamma, normal, chi-square, t, F, and beta distributions — derived and connected.

1. 5.1 Uniform, Exponential, and Gamma
2. 5.2 Normal and Derived Distributions
3. 5.3 Beta, Transformations, and Relationships

Joint, marginal, and conditional distributions. Covariance, correlation, and the multivariate normal.

1. 6.1 Joint, Marginal, and Conditional Distributions
2. 6.2 Covariance and Correlation
3. 6.3 The Multivariate Normal

Modes of convergence, the law of large numbers, the central limit theorem, and the delta method.

1. 7.1 Inequalities and Convergence Types
2. 7.2 The Law of Large Numbers
3. 7.3 The Central Limit Theorem and the Delta Method

Method of moments, maximum likelihood, bias, consistency, efficiency, Fisher information, and the Cramer-Rao bound.

1. 8.1 Estimator Properties and Method of Moments
2. 8.2 Maximum Likelihood Estimation
3. 8.3 Fisher Information, Cramér-Rao, and Sufficiency

Confidence intervals derived from first principles. Pivotal quantities and sample size determination.

1. 9.1 Confidence Intervals for Means
2. 9.2 Proportions and Variance
3. 9.3 Pivotal Quantities and Sample Size

The logic of testing, p-values, Type I and II errors, power, the Neyman-Pearson lemma, and likelihood ratio tests.

1. 10.1 Logic and P-values
2. 10.2 Errors and Power
3. 10.3 Neyman-Pearson and the Likelihood Ratio Test

t-tests, chi-square tests, F-tests — each derived from the theory, not presented as recipes.

1. 11.1 t-Tests
2. 11.2 Proportions and Chi-Square
3. 11.3 F-Test and Checking Assumptions

Family-wise error rate, false discovery rate, and the replication crisis.

1. 12.1 FWER and Bonferroni
2. 12.2 FDR and Replication

Permutation tests, rank-based tests, the bootstrap, and cross-validation.

1. 13.1 Permutation and Rank Tests
2. 13.2 Bootstrap and Cross-Validation

Vectors, matrices, projection, eigenvalues — taught in service of linear models.

1. 14.1 Vectors, Matrices, and Systems
2. 14.2 Projection and Eigenvalues

Least squares derived by calculus, properties of estimators, inference, and residual analysis.

1. 15.1 Model and Estimation
2. 15.2 Inference and Prediction
3. 15.3 R-squared and Residuals

The matrix formulation, Gauss-Markov theorem, inference, multicollinearity, and model selection.

1. 16.1 Matrix Formulation
2. 16.2 Gauss-Markov and Inference
3. 16.3 Model Selection

Assumption checking, influential observations, heteroscedasticity, and remedial measures.

1. 17.1 Assumptions and Residuals
2. 17.2 Influence and Remedies

ANOVA as a special case of linear models. One-way, two-way, interactions, and ANCOVA.

1. 18.1 One-Way ANOVA and Comparisons
2. 18.2 Factorial Designs and ANCOVA

The exponential family, link functions, iteratively reweighted least squares, and deviance.

1. 19.1 The Exponential Family
2. 19.2 Link Functions and IRLS

The model derived from exponential family theory. Inference, model assessment, and extensions.

1. 20.1 Model and Inference
2. 20.2 Assessment and Extensions

Poisson regression, overdispersion, negative binomial, and zero-inflated models.

1. 21.1 Poisson Regression
2. 21.2 Overdispersion and Extensions

Priors, posteriors, conjugate analysis, credible intervals, and Bayes factors.

1. 22.1 Framework and Conjugate Priors
2. 22.2 Credible Intervals and Testing

Monte Carlo integration, Metropolis-Hastings, Gibbs sampling, and diagnostics.

1. 23.1 MCMC Algorithms
2. 23.2 Diagnostics and Practice

Randomisation, blocking, sample size determination, and common pitfalls.

1. 24.1 Principles and Practice

The statistical workflow, case studies, and the road ahead.

1. 25.1 Workflow and Case Studies