Suppose X1, ,Xn are i i d with c d f F on the real line The problem is to test the null hypothesis H0 that the Xi are uniform on (0, for some Let n max(X1, ,Xn), and let Fn be the empirical distribution function Let dK(F, G) be the Kolmogorov Smirnov distance between F and G Consider the test stati...

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Suppose X1,...,Xn are i.i.d. with c.d.f F on the real line. The problem is to test the

Question:

Suppose X1,...,Xn are i.i.d. with c.d.f F on the real line. The problem is to test the null hypothesis H0 that the Xi are uniform on (0, θ] for some θ. Let ˆθn = max(X1,...,Xn), and let Fˆn be the empirical distribution function. Let dK(F, G) be the Kolmogorov-Smirnov distance between F and G.

Consider the test statistic Tn = n1/2 dK(Fˆn, Fθ

ˆn ) , where Fθ is the uniform (0, θ) c.d.f. Under H0, what is the limiting distribution of Tn?

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