1. What is the significance of the empirical cdf F n in establishing non-parametric statistical inference for the integral 1 p F -1 (u) - F -1 n (u) du?
The empirical cdf F n plays a crucial role in establishing non-parametric statistical inference for the integral 1 p F -1 (u) - F -1 n (u) du. It is defined as F n (x) = 1 n n i=1 1{X i <= x}, where X 1 , . . ., X n are independent copies of X. The empirical u th quantile F -1 (u) is then defined as inf{x R : F n (x) >= u}. The challenge lies in establishing the limiting distribution for the difference F -1 (u) - F -1 n (u) as n approaches infinity. While it is tempting to conclude that the integral 1 p F -1 (u) - F -1 n (u) du is the average of certain transformations of X 1 , . . ., X n asymptotically when n - , the same holds true under much weaker assumptions. This highlights the significance of the empirical cdf F n in non-parametric statistical inference.
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2. What is the asymptotic variance of the Expected Shortfall at a given probability level p*?
The asymptotic variance of the Expected Shortfall at a given probability level p* is given by s2 F = 1 (1 - p*)2 p*1 p* (st - st)dF -1 (s)dF -1 (t), provided that the cdf F has a density f. This expression does not rely on the existence of f and serves as an alternative way of writing the asymptotic variance s2 F in the form of a Lebesgue-Stieltjes integral. It indicates that absolute continuity of the cdf F is not needed, and this fact has been established in the present paper.
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