Manuscript Psy02-A@1 · Psy02-A.tex · proposition · d92ce402

Proposition: Bounded statistic from a mean contrast

Causal support and mechanism identification › Unbounded records require clipping and tail control · lines 2215–2253 · proof included or linked

Psy02-A@1#prop:clipping

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The statistic lies in $[0, 1]$ , and

E_{h} f = \frac{1}{2} + \frac{1}{2 R C _{w}} i \sum w_{i} E_{h} c_{R} (Z_{i}) .

If the ideal oriented mean-contrast gap is $G > 0$ , and $E_{h} Z_{i}^{2} \leq M_{h, i}$ , put

B = i \sum \abs w_{i} (M_{0, i} + M_{1, i}) .

Then its clipped expectation gap is at least

\frac{1}{2 R C _{w}} (G - \frac{B}{R}) .

For independent setting-independent erasure with observation probability $γ$ , score erased runs by $1/2$ . The gap is multiplied by $γ$ . Actual total-variation implementation errors then subtract $e_{0} + e_{1}$ .

Proof. The definition of $C_{w}$ proves boundedness. Average over the randomized setting. Pointwise,

\abs z - c_{R} (z) \leq \frac{z ^{2}}{R},

so each mean changes by at most $M_{h, i} / R$ . Apply the weighted triangle inequality. Independent erasure multiplies the nonconstant expectation by $γ$ . Finally use total-variation control.

Displayed formulas in this passage

unlabeled · line 2218
$E_{h} f = \frac{1}{2} + \frac{1}{2 R C _{w}} i \sum w_{i} E_{h} c_{R} (Z_{i}) .$
unlabeled · line 2226
$B = i \sum \abs w_{i} (M_{0, i} + M_{1, i}) .$
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$\frac{1}{2 R C _{w}} (G - \frac{B}{R}) .$
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$\abs z - c_{R} (z) \leq \frac{z ^{2}}{R},$

Proposition: Bounded statistic from a mean contrast

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