AlgorithmAlgorithm: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Split-sample univariate winsorized trimmed mean

Definition

Let $N \geq 1$ be an integer, let $η \geq 0$ , $0 < δ < 1$ , and set

ε = 8 η + \frac{12 \log (4 / δ)}{N} .

The procedure is admissible when $0 < ε \leq 1 / 4$ . Its input is two observed real samples ${\tilde{X}}_{1}, \dots, {\tilde{X}}_{N}$ and ${\tilde{Y}}_{1}, \dots, {\tilde{Y}}_{N}$ . Sort the second sample nondecreasingly as ${\tilde{Y}}_{(1)}^{*} \leq \dots \leq {\tilde{Y}}_{(N)}^{*}$ and define $r (u) = ⌈ u N ⌉$ for $0 < u \leq 1$ . Set

α = {\tilde{Y}}_{(r (ε))}^{*}, β = {\tilde{Y}}_{(r (1 - ε))}^{*},

and, for $α \leq β$ , let $ϕ_{α, β} (x) = \min {β, \max {α, x}}$ . Return

\hat{μ} = \frac{1}{N} \sum_{i = 1}^{N} ϕ_{α, β} ({\tilde{X}}_{i}) .

The operation is defined for every admissible input, including tied observations.

Remarks

The probability guarantee is stated separately under the absolutely continuous clean-sample and sample-replacement hypotheses of Univariate trimmed-mean bound under sample replacement.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Gábor Lugosi and Shahar Mendelson, Robust Multivariate Mean Estimation: The Optimality of Trimmed Mean, §2 (standard reference, not scraped)
Gábor Lugosi, Mean Estimation in High Dimension, §2 (standard reference, not scraped)