Weak convergence of empirical Wasserstein type distances
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We estimate contrasts ∫_0 ^1 ρ(F^-1(u)-G^-1(u))du between two continuous distributions F and G on R such that the set {F=G} is a finite union of intervals, possibly empty or R. The non-negative convex cost function ρ is not necessarily symmetric and the sample may come from any joint distribution H on R^2 with marginals F and G having light enough tails with respect to ρ. The rates of weak convergence and the limiting distributions are derived in a wide class of situations including the classical Wasserstein distances W_1 and W_2. The new phenomenon we describe in the case F=G involves the behavior of ρ near 0, which we assume to be regularly varying with index ranging from 1 to 2 and to satisfy a key relation with the behavior of ρ near ∞ through the common tails. Rates are then also regularly varying with powers ranging from 1/2 to 1 also affecting the limiting distribution, in addition to H.
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