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In-depth, derivation-first explanations of statistical methods -- free to read, and citable.
Likelihood: How One Idea Reshaped Statistics
From tally tables to modern estimation -- and the much older twin the idea turns out to share with Bayesian inference
Traces statistics from pure data organization (tally tables, means, variances) through curve fitting -- correcting the record to show Gauss and Laplace attached a Normal-error probabilistic model to least squares within seven years of Legendre's model-free 1805 publication, not a century later -- through Pearson's 1894 method of moments, to Fisher's 1921-22 formalization of likelihood. Corrects a common historical inversion: inverse probability (Bayes 1763, Laplace, De Morgan's 1837 naming) predates Fisher's likelihood by a century and a half, and Fisher built likelihood in explicit opposition to it rather than statisticians later discovering Bayesian methods share the same machinery. Closes with likelihood's 20th-century descendants -- quasi-likelihood, partial likelihood, REML, and empirical likelihood -- and where each is used today.
Time Series and Stationarity
A single history is both all the data you will ever get and never enough, on its own, to define a distribution
Works out what makes time series data structurally different from cross-sectional data: a stochastic process has an ensemble dimension (fixed time, varying realization, where distributions live) and a temporal dimension (fixed realization, varying time, where the data live), and a single history offers no replication at any one time point. Covers autocorrelation as the structure that substitutes for that missing replication, why stationarity is required before prediction is possible at all, strict vs. weak (covariance) stationarity and why practice targets the latter, ergodicity as a separate assumption from stationarity (with a worked example of a stationary but non-ergodic process, verified by simulation), and a taxonomy of how real series actually become non-stationary -- trends, level shifts, regime shifts, variance changes, and unit roots -- each with a different diagnostic signature. Closes by explaining why no single test can certify stationarity, previewing a dedicated future article on the tests themselves.
An Introduction to Regularization
Why ordinary regression can break in two genuinely different ways -- and how a single penalty term fixes both
Motivates regularization from a precise failure of ordinary least squares: correlated predictors inflate coefficient variance for purely statistical reasons that hold even under exact arithmetic, while extreme ill-conditioning separately threatens the floating-point computation itself, at a much higher threshold, verified here by direct simulation. Also covers the sharper p >= n case (true non-identifiability, not just instability), introduces the general penalized-estimation framework and the bias-variance tradeoff behind it, and previews the three specific penalties -- Ridge, Lasso, Elastic Net -- each covered in its own dedicated article next.
The Kaplan-Meier Estimator
Building a survival curve one risk set at a time, and why the bookkeeping is the whole idea
An introduction to survival analysis -- what makes time-to-event data different from ordinary regression or time series, and a taxonomy of nonparametric, semiparametric, and parametric survival models -- followed by a from-scratch derivation of the Kaplan-Meier estimator. Builds the product-limit formula from risk-set bookkeeping, works a real 21-patient leukemia remission dataset, derives Greenwood's variance formula and the log-log confidence interval, and shows numerically why two common shortcuts for handling censored data produce badly biased survival curves.
The Compendium of Statistical Tests
A comprehensive reference covering 255+ statistical tests across 25 chapters, with conceptual primers, assumptions, hypotheses, formulas, and implementation guidance.