For n ≥ 2, let X₁, …, Xₙ be a random sample with density f(x|θ) = θx^{θ−1} on 0

Question

For n ≥ 2, let X₁, …, Xₙ be a random sample with density f(x|θ) = θx^{θ−1} on 0 < x < 1, θ > 0 unknown. Which of the following is the UMVUE for 1/θ?

Accepted Answer

A. −ln Xᵢ ~ Exp(θ) with mean 1/θ, and Σ ln Xᵢ is complete sufficient; the sample mean of −ln Xᵢ is unbiased for 1/θ.

Sufficiency, completeness, UMVUE, Cramér–Rao