Statistical Inference Course

Lesson 1 • Probability Review for Inference

Reviews probability rules essential for deriving sampling distributions. Connects axiomatic probability to the uncertainty quantification central to inference.

Lesson 2 • Random Variables and Expectation

Formalizes random variables, expected value, and variance as tools for characterizing estimator behavior. Provides the algebra needed for deriving sampling distributions.

Lesson 3 • Sampling Distributions

Derives the distribution of sample statistics under repeated sampling. Directly motivates the standard error concept used in hypothesis tests and confidence intervals.

Lesson 4 • Key Reference Distributions

Introduces the t, chi-squared, and F distributions derived from normal samples. These distributions underpin the test statistics covered in later chapters.

Lesson 1 • Maximum Likelihood Estimation

Constructs the likelihood function and maximizes it to obtain MLEs. MLE is the dominant estimation method referenced throughout applied chapters.

Lesson 2 • Cramér-Rao Lower Bound and UMVUE

Establishes the theoretical minimum variance for unbiased estimators. Identifies uniformly minimum variance unbiased estimators as the gold standard.

Lesson 3 • Estimator Properties and Criteria

Defines unbiasedness, consistency, and mean squared error as formal evaluation criteria. Establishes the framework for comparing competing estimators objectively.

Lesson 4 • Method of Moments Estimation

Derives estimators by equating population and sample moments. Provides an intuitive entry point before the more powerful maximum likelihood approach.

Lesson 1 • Confidence Interval Concepts

Defines coverage probability and the frequentist interpretation of confidence. Corrects common misconceptions before students apply interval methods.

Lesson 2 • Intervals for Means and Proportions

Derives z and t intervals for single and two-sample means and proportions. These are the most frequently applied intervals in professional practice.

Lesson 3 • Likelihood-Based Confidence Regions

Builds confidence sets from profile likelihood and likelihood ratio inversion. Provides asymptotically valid intervals without requiring pivotal quantities.

Lesson 4 • Sample Size Determination

Derives required sample sizes to achieve target margin of error and coverage. Connects interval width to study design decisions made before data collection.

Lesson 1 • Tests for Variances

Applies chi-squared and F tests to variance hypotheses. Prepares students for ANOVA and regression diagnostics introduced in later chapters.

Lesson 2 • p-Values and Their Interpretation

Defines the p-value as the probability of data at least as extreme under the null. Addresses widespread misinterpretations that lead to flawed conclusions.

Lesson 3 • Type I and Type II Errors

Quantifies the probability of false rejection and false acceptance. Introduces the power function as the tool for evaluating test quality.

Lesson 4 • Logic and Structure of Hypothesis Tests

Frames testing as a decision problem with null and alternative hypotheses. Establishes the rejection region paradigm that all subsequent tests follow.

Lesson 1 • Score and Wald Tests

Presents score and Wald tests as asymptotically equivalent alternatives to the LRT. Highlights computational advantages in complex models.

Lesson 2 • Neyman-Pearson Lemma

Proves that the likelihood ratio test is most powerful for simple hypotheses. Provides the theoretical foundation for all optimal test derivations.

Lesson 3 • Uniformly Most Powerful Tests

Extends most powerful tests to composite alternatives via monotone likelihood ratios. Identifies when UMP tests exist and when they do not.

Lesson 4 • Unbiased and Invariant Tests

Introduces unbiasedness and invariance as additional optimality criteria for tests. Resolves cases where UMP tests do not exist by imposing structural constraints.

Lesson 1 • Empirical Bayes and Hierarchical Models

Estimates hyperparameters from data and structures priors hierarchically. Bridges full Bayes and frequentist shrinkage estimators used in applied settings.

Lesson 2 • Bayesian Framework and Prior Distributions

Formalizes Bayes' theorem as the engine of inference and introduces prior specification. Contrasts subjective, conjugate, and non-informative prior choices.

Lesson 3 • Posterior Distributions and Summaries

Derives posterior distributions and extracts point and interval summaries. Connects posterior mean and MAP estimates to frequentist counterparts.

Lesson 4 • Bayesian Hypothesis Testing

Introduces Bayes factors and posterior odds as alternatives to p-values. Demonstrates how Bayesian testing avoids some frequentist decision-making pitfalls.

Lesson 1 • Foundations of Nonparametric Inference

Motivates distribution-free methods when normality or homoscedasticity cannot be assumed. Introduces order statistics and ranks as the building blocks of nonparametric tests.

Lesson 2 • Permutation Tests

Generates the exact null distribution by permuting observed data labels. Provides exact p-values for any test statistic without asymptotic approximations.

Lesson 3 • Bootstrap Methods

Uses resampling from the empirical distribution to estimate sampling variability. Constructs bootstrap confidence intervals without closed-form standard error formulas.

Lesson 4 • Nonparametric Tests for Multiple Groups

Extends rank-based inference to k-sample and block designs via Kruskal-Wallis and Friedman tests. Provides nonparametric alternatives to one-way and two-way ANOVA.

Lesson 1 • Statistical Decision Theory

Frames estimation and testing as decisions minimizing expected loss. Unifies frequentist and Bayesian approaches under a common decision-theoretic umbrella.

Lesson 2 • Multiple Comparisons Problem

Quantifies how familywise error rate inflates with the number of simultaneous tests. Motivates the need for multiplicity correction in any multi-hypothesis setting.

Lesson 3 • Replication, Effect Size, and Power

Connects statistical significance to practical importance via effect size measures. Addresses the replication crisis and best practices for reproducible inference.

Lesson 4 • False Discovery Rate Control

Introduces FDR as a less conservative alternative to FWER for large-scale testing. Derives the Benjamini-Hochberg procedure and its assumptions.