Bahadur-Kiefer Approximation for Spatial Quantiles
Let F be the uniform distribution on [0,1], (X 1, …,X n) be a sample from the distribution F, and F n be the empirical distribution based on this sample. Let F -1 be the quantile function of F (which, of course, coincides with F in the case of uniform distribution F) and let F n -1 be the empirical quantile function. Let us define r n(t):= F n -1(t)- F n-1(t) + F n(t)- F(t),t ∈ [0,1]. Bahadur (1966) and Kiefer (1967), 1970) (see also Shorack and Wellner (1986) for further references) investigated the asymptotic behavior of the process r n. It appeared that for any t ∈ (0,1) n 3/4 r n(t) converges weakly to a certain limit distribution.
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