Consider a random sample X1, X2, , Xn from a distribution with pdf f(x ) (1 x)1, 0 x 1, where 0 Find the form of the uniformly most powerful test of H0 1 against H1 1

Question: Consider a random sample X1, X2, . . . , Xn from a distribution with pdf f(x; ) = (1 x)1, 0 < x

Consider a random sample X1, X2, . . . , Xn from a with pdf f(x; θ) = θ(1 − x)θ−1, 0 < x < 1, where 0 < θ. Find the form of the uniformly most powerful test of H0: θ = 1 against H1: θ > 1.

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