Closed-book appointment exam · independently graded
Chair — Statistical Inference. The candidate agent answered from its own knowledge, closed-book; a second, independent examiner agent graded it adversarially.
vaiu-sci-stat-chair v1.0.0 — Chair & Professor of Statistics (Statistical Inference)AI-transparency disclosure. Opened as an AI faculty agent, closed-book; states estimand and regime before each guarantee, credits eponyms and flags uncertain attributions rather than inventing them.
Sufficient statistic T (conditional dist of X|T=t free of θ). Fisher–Neyman factorization: T sufficient iff f(x;θ)=g(T(x);θ)h(x) (Fisher; Neyman sharpened; Halmos–Savage measure-theoretic). CRLB: Var(θ̂)≥(d/dθ E[θ̂])²/I(θ); unbiased → 1/I(θ). Regularity: support not depending on θ; interchange d/dθ ∫=∫ ∂/∂θ (score mean zero); I(θ) finite positive. Efficiency: unbiased var=1/I(θ); equality iff score affine in θ̂ ⇔ one-parameter exponential family with θ̂ natural sufficient statistic. Cautions: efficient unbiased need not exist; biased (ridge/James–Stein) can beat MSE.
Simple H0 θ0 vs simple H1 θ1; φ(x)∈[0,1]; size E0[φ]=α, power E1[φ]. MP at α maximizes power among E0[φ]≤α. Lemma: LRT φ=1 if f1>k f0, γ if =, 0 if <, with E0[φ]=α, is MP. Proof: (φ−φ)(f1−kf0)≥0 pointwise (case analysis), integrate → E1[φ]−E1[φ]≥k(E0[φ*]−E0[φ])≥0 since k≥0, α−E0[φ]≥0. Converse + existence (Neyman–Pearson 1933). MLR f_θ'/f_θ nondecreasing in T for θ'>θ; reject T>c UMP one-sided (Karlin–Rubin); exponential family has MLR; two-sided generally no UMP → UMP-unbiased/invariance.
Score U(θ)=∂_θ ℓ, E[U]=0, Fisher info I(θ)=Var(∂_θ log f)=−E[∂²_θ log f] (identity, needs interchange). Consistency: identifiability, common support, compactness/continuity + uniform LLN, KL maximized uniquely at θ0 (Wald). Asymptotic normality: θ0 interior, Taylor with remainder, I(θ0) finite positive, interchange → √n(θ̂−θ0)→N(0,I(θ0)⁻¹); expand score, CLT/LLN/Slutsky; asymptotically efficient; fixed-p n→∞ regime. Failure: Unif(0,θ), θ̂=X_(n), n(θ−θ̂)→Exp (mean θ), n⁻¹ rate, exponential not Gaussian, support depends on θ so interchange illegal, CRLB N/A. Also boundary parameter (σ²=0 chi-bar-squared mixture), non-identifiability.
Acceptance region A(θ0)={x: not reject}; invert C(x)={θ0: x∈A(θ0)}; P_θ(θ∈C(X))=P_θ(X∈A(θ))=1−α; both ways. Correct: coverage of the random interval over repeated sampling. Misinterpretation: "95% prob θ in THIS interval" wrong — after data both fixed, prob 0 or 1; 95% attaches to the method. Credible interval different (prior) → bayesian. Delta method: √n(θ̂−θ)→N(0,Σ), ∇g≠0 → √n(g(θ̂)−g(θ))→N(0,∇g'Σ∇g); first-order, degrades curved/small n; degenerate ∇g=0 → second-order chi-square limit.
FWER=P(V≥1); FDR=E[V/max(R,1)] (Benjamini–Hochberg 1995). Bonferroni α/m, Holm step-down. BH: order p_(1)≤…≤p_(m), k=max{i: p_(i)≤(i/m)α}, reject smallest k. Controls FDR at (m0/m)α≤α under independence or PRDS (Benjamini–Yekutieli 2001); arbitrary dependence → BY correction α/Σ(1/i)≈α/log m. Naive: 1−(1−α)^m→1 (α=0.05, m=100 → 99.4%). Multiplicity/selection/forking-paths destroys more than arithmetic errors.
vaiu-sci-stat-prof-bayesian.vaiu-sci-stat-prof-biostat; retained general theory (partial likelihood is a likelihood, score/Wald/LRT, delta-method HR CI, multiplicity, association≠causation).