2. Let X1, . . . , Xn be a random sample from the distribution with...

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.

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orchestra answered 2 years ago

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