Quantile processes with statistical applications by Miklos Csorgo

By Miklos Csorgo

Presents a entire concept of the approximations of quantile strategies in gentle of modern advances, in addition to a few of their statistical purposes.

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9) to hold true, it is sufficient that we have g(y) (cf. 10)) tend to oo as y -> 0. 19) and the Hajek-Renyi inequality (cf. 2). 10) (cf. 3). We have for each n = l , 2 , . . , where {B(y);OSy = l} is a Brownian bridge. As in Shorack (1979), let G(t): = inf (g(y); 0 < y =gf}. 10), G(t)t°° as fj,0. 5) and because G(a n )t°° as n^oo. 14). 1. 1 is of interest on its own in that it calls attention to the fact that in the middle [o^, 1 - a^\ of (0,1) it is not also necessary to divide by g(y) for the sake of weak convergence.

1 on F, for every continuous h: D(0,1) —» R1 and c fixed. Proof. 5) holds true with cn — c. 24). 2 hold true also with cn = l/(n + l). 1. 1. Then as n —»°o where pn(y) and Un(y) are respectively defined in terms of Xk:n and F(Xk,n) = Uk:n (fc = 1, 2 , . . , n). 13), we have also Proof. Let 0

4. 4 (from conversation with Pal Revesz at ETH Zurich, June 1981). }. , h(x) = x +log (1/x)-1, x § 1). 6) for every fixed c 6 (0,1) and large enough n, it is not of much use when letting c = cn 10. 30). 30) is not the right way of approach for this problem, and that a better form of the said inequality may exist. 4 we refer to Jan Bierlant and Lajos Horvath (1982 (preprint), Thm. B). 5. } was also studied by Shorack (1972a), (1972b). Under conditions somewhat different from ours, he proved a number of results.

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