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Distributionshard
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Let X1,X2,…,XnX_1, X_2, \dots, X_nX1​,X2​,…,Xn​ be a random sample from a distribution with PDF f(x;θ)=θxθ−1f(x; \theta) = \theta x^{\theta-1}f(x;θ)=θxθ−1 for x∈(0,1)x \in (0, 1)x∈(0,1) and θ>0\theta > 0θ>0. What is the distribution of the sufficient statistic T=−∑i=1nln⁡(Xi)T = -\sum_{i=1}^n \ln(X_i)T=−∑i=1n​ln(Xi​)?