\( \newcommand{\E}{\mathbb{E}} \newcommand{\Var}{\operatorname{Var}} \newcommand{\Cov}{\operatorname{Cov}} \newcommand{\Corr}{\operatorname{Corr}} \newcommand{\Prob}{\mathbb{P}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\iid}{\overset{\text{i.i.d.}}{\sim}} \newcommand{\dto}{\xrightarrow{d}} \newcommand{\pto}{\xrightarrow{p}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} \)

Statistical tests

Derivations of test statistics, null distributions and power.

Derivations of test statistics, null distributions and power.

Title Description Date
Rank tests (Mann–Whitney, Wilcoxon, Kruskal–Wallis) Comparing location without distributional assumptions. The null distribution of ranks, the relation between U and the rank sum, means, variances and recursions for the exact distributions, tie corrections, power compared with the t-test, and the pitfall of unequal variances. 2026-09-30
Shapiro–Wilk test A test of normality. We derive the W statistic from generalized least squares on the order statistics, and cover its location–scale invariance, its independence of (X̄, S²), its exact distribution for n = 3, Royston’s approximation, and power. 2026-09-30
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