In: Statistics and Probability
2. Let X1, . . . , Xn be a random sample from the distribution with pdf given by fX(x;β) = β 1(x ≥ 1).
xβ+1
(a) Show that T = ni=1 log Xi is a sufficient statistic for β.
Hint: Use
n1n1n=exp log=exp −logxi .i=1 xi i=1 xi i=1
(b) Find the pdf of Y = logX, where X ∼ fX(x;β).
(c) Find the distribution of T . Hint: Identify the distribution of
Y and use mgfs.
(d) Find E[1/T].
(e) Use your answer to the previous part to propose an unbiased
estimator of β based on T. (f) Argue that your answer to part (c)
is the MVUE of β.