Manuscript Oai280-B@1 · Oai280-B.tex · section · 6d92ab75

Exact binary inversion

Feasibility, binomial inversion, and channel structure › Exact binary inversion · lines 1580–1615

Oai280-B@1#eq:failureidentity

Suppose complete oriented scores are genuinely binary:

T_{A x} = Z_{A x}, T_{A y} = - Z_{A y}, T_{A z} = Z_{A z}, T_{A j} \in {\pm 1} .

Let $p_{A j} = Pr (T_{A j} = - 1 ∣ Z)$ . Then

1 - F_{obs, A} = \frac{1}{2} (p_{A x} + p_{A y} + p_{A z}) .

For an observed failure count $h$ out of $n$ , define the one-sided upper limit $u (n, h; a)$ by

u (n, n; a) = 1,

and, for $h < n$ , by the unique solution $u \in (0, 1)$ of

ℓ = 0 \sum^{h} (ℓ n) u^{ℓ} (1 - u)^{n - ℓ} = a .

Define the lower limit $ℓ (n, h; a)$ as zero for $h = 0$ , and otherwise by

ℓ = h \sum^{n} (ℓ n) ℓ (n, h; a)^{ℓ} (1 - ℓ (n, h; a))^{n - ℓ} = a .

These are the usual exact binomial inversions ClopperPearson.

Displayed formulas in this passage

Oai280-B@1#eq:failureidentity · line 1590
$1 - F_{obs, A} = \frac{1}{2} (p_{A x} + p_{A y} + p_{A z}) .$
unlabeled · line 1583
$T_{A x} = Z_{A x}, T_{A y} = - Z_{A y}, T_{A z} = Z_{A z}, T_{A j} \in {\pm 1} .$
unlabeled · line 1598
$u (n, n; a) = 1,$
unlabeled · line 1602
$ℓ = 0 \sum^{h} (ℓ n) u^{ℓ} (1 - u)^{n - ℓ} = a .$
unlabeled · line 1608
$ℓ = h \sum^{n} (ℓ n) ℓ (n, h; a)^{ℓ} (1 - ℓ (n, h; a))^{n - ℓ} = a .$