The asymptotic distribution of weighted empirical distribution functions

Let Gn denote the empirical distribution based on n independent uniform (0, 1) random variables. The asymptotic distribution of the supremum of weighted discrepancies between Gn(u) and u of the forms ||wv(u)Dn(u)|| and ||wv(Gn(u))Dn(u)||, where Dn(u) = Gn(u)-u, wv(u) = (u(1-u))-1+v and 0 [less-than-or-equals, slant] v < is obtained. Goodness-of-fit tests based on these statistics are shown to be asymptotically sensitive only in the extreme tails of a distribution, which is exactly where such statistics that use a weight function wv with [less-than-or-equals, slant] v [less-than-or-equals, slant] 1 are insensitive. For this reason weighted discrepancies which use the weight function wv with 0 [less-than-or-equals, slant] v < are potentially applicable in the construction of confidence contours for the extreme tails of a distribution.