Probability bounds analysis: Difference between revisions

Revision as of 17:39, 14 May 2012

Revision as of 17:13, 13 October 2012

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and ''Y'' is actually easier than the problem assuming independence.

Makarov<ref name=Makarov/><ref name=Franketal87/><ref name=WilliamsonDowns/> showed that

:''Z'' ~ <big>[ sup</big>_x+y=z max(''F''(''x'') + ''G''(''y'') &minus; 1, 0), <big>inf</big>_x+y=z ''F''(''x'') + ''G''(''y'') &minus; max(''F''(''x'') + ''G''(''y'') &minus; 1, 0) <big>]</big>.

These bounds are implied by the [[copula#Fr.C3.A9chet.E2.80.93Hoeffding_copula_bounds|Fréchet–Hoeffding]] [[copula]] bounds. The problem can also be solved using the methods of [[mathematical programming]]<ref name=BerleantGoodmanStrauss />.

The convolution under the intermediate assumption that ''X'' and ''Y'' have [[positive quadrant dependence|positive dependence]] is likewise easy to compute, as is the convolution under the extreme assumptions of [[Comonotonicity|perfect positive]] or [[countermonotonicity|perfect negative]] dependency between ''X'' and ''Y''.<ref name=Fersonetal04 />