Question: Let X, X,,..., X be a random sample from the distribution of a random variable X with a continuous cdf F(x), and let F,, (x)

Let X, X,,..., X be a random sample from the distribution of a random variable X with a continuous cdf F(x), and let F,, (x) be the empirical distribution function. Let C,,(F) =n [F,(x)-F(x)]'dF(x) be the Cramer-von Mises statistic. (a) Show that the distribution of the statistic C, (F) does not depend on F(x), that is, it is distribution-free. (b) Identify the limiting distribution of C, (F)

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