Manuscript VII-H@1 · VII-H.tex · proposition · d20d2a9f

Proposition: Uniform sensitivity bound

Finite banks, calibration, and error control › Imperfect annihilation and numerical errors · lines 1413–1441 · proof included or linked

VII-H@1#prop:error, VII-H@1#eq:annihilation-error

Suppose an actual evaluated prescription difference is $M θ$ with $∥ θ ∥ \leq C_{0}$ . Then

∣ Δ I_{w} ∣ \leq C_{0} M^{T} w .

If a numerical matrix $M$ satisfies $M - M_{op} \leq ϵ_{M}$ and $M^{T} w = 0$ , then

∣ Δ I_{w} ∣ \leq C_{0} ϵ_{M} ∥ w ∥ .

If each estimated response pairing has deterministic error at most $ϵ_{i}$ , the contrast error is at most $\sum_{i} ∣ w_{i} ∣ ϵ_{i}$ .

Proof. Use

Δ I_{w} = w^{T} M θ = (M^{T} w)^{T} θ

and Cauchy–Schwarz. The matrix and pairing estimates follow from the operator norm and triangle inequality.

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