Manuscript Psy04-B@1 · Psy04-B-v1.tex · lemma · 62c2b6f0

Lemma: Bounded complete-record score stability

Complete records and robust score allowances › Complete-law robustness · lines 1603–1619 · proof included or linked

Psy04-B@1#lem:tv

For complete laws $P, P^{'}$ and a common score $f \in [0, 1]$ ,

∣ E_{P} f - E_{P^{'}} f ∣ \leq d_{TV} (P, P^{'}) .

Proof. Use

E_{P} f - E_{P^{'}} f = \int_{0}^{1} (P (f > t) - P^{'} (f > t)) d t

and the defining total-variation supremum.

Displayed formulas in this passage

unlabeled · line 1606
$∣ E_{P} f - E_{P^{'}} f ∣ \leq d_{TV} (P, P^{'}) .$
unlabeled · line 1613
$E_{P} f - E_{P^{'}} f = \int_{0}^{1} (P (f > t) - P^{'} (f > t)) d t$