In: Statistics and Probability
For a fixed θ>0, let X1,X2,…,Xn be i.i.d., each with the beta (1,θ) density.
i) Find θ^ that is the maximum likelihood estimate of θ.
ii) Let X have the beta (1,θ) density. Find the density of −log(1−X). Recognize this as one of the famous ones and provide its name and parameters.
iii) Find f that is the density of the MLE θ^ in part (i).