Manuscript En10-B@1 · En10-B.tex · lemma · a1105bad

Lemma: Calibration conversion

Calibration and complete-record finite-shot inference › An explicit calibration allowance · lines 803–842 · proof included or linked

En10-B@1#lem:calibration, En10-B@1#eq:calibration-conversion

Back to manuscript

Uniformly for $η \in K, θ \in [0, 1]$ ,

∣ q_{η} (θ) - (a_{0} + b_{0} θ) ∣ \leq Δ_{cal} \leq ϵ_{γ} + ϵ_{ϕ} + κ + z_{*}^{2} .

Moreover,

b_{η} \geq b_{*} := (1 - γ_{+}) (1 - 2 t_{+}) > 0.

Proof. The difference of the two affine mean functions is affine in $θ$ . Its largest absolute value on $[0, 1]$ occurs at an endpoint. At $θ = 0$ and $1$ those differences are precisely the two expressions in (eq:calibration-allowance).

The function $sin^{2} x$ is globally $1$ -Lipschitz. Hence

∣ t_{k} - u_{0} ∣ \leq ∣ ϕ - ϕ_{0} + ξ_{k} ∣ + n_{z}^{2} \leq ϵ_{ϕ} + κ + z_{*}^{2} .

Averaging gives the same bound on $∣ u - u_{0} ∣$ . Writing

q_{η} (θ) = γ + (1 - γ) [u + (1 - 2 u) θ],

its derivative with respect to $u$ has absolute value at most one, and its derivative with respect to $γ$ lies in $[0, 1]$ . Comparing first $u$ , then $γ$ , proves the displayed upper bound. The slope bound follows from (eq:moment-bounds).

Displayed formulas in this passage

En10-B@1#eq:calibration-conversion · line 806
$∣ q_{η} (θ) - (a_{0} + b_{0} θ) ∣ \leq Δ_{cal} \leq ϵ_{γ} + ϵ_{ϕ} + κ + z_{*}^{2} .$
unlabeled · line 813
$b_{η} \geq b_{*} := (1 - γ_{+}) (1 - 2 t_{+}) > 0.$
unlabeled · line 826
$∣ t_{k} - u_{0} ∣ \leq ∣ ϕ - ϕ_{0} + ξ_{k} ∣ + n_{z}^{2} \leq ϵ_{ϕ} + κ + z_{*}^{2} .$
unlabeled · line 833
$q_{η} (θ) = γ + (1 - γ) [u + (1 - 2 u) θ],$

Nearby in this section

section: An explicit calibration allowance