題組內容

4、Let f(x; θ) = (1/θ)x⁽¹⁻⁰⁾/θ, 0 < x < 1, 0 < θ < ∞. Let X₁, X₂, ..., Xₙ denote a random sample of size n from this distribution.



(1) \( (5\%) \) Show that the maximum likelihood estimator of \( \theta \) is \( \hat{\theta} = -\left(\frac{1}{n}\right)\sum{i=1}^{n}\ln(Xi) \).