Question

Suppose that X1, X2, ..., Xn are i.i.d. from Unif[α, 0].
(a) Find ˆαMM, which is the estimator using method of moments.
(b) Compute E(ˆαMM) and V ar(ˆαMM)
(c) Find ˆαML, which is the estimate using maximum likelihood method.
(d) Determine the density of X(1), the smallest of X1, · · · , Xn, by solving the following:
i. Find P(X(1) ≥ x) as a function of x, where x ≥ 0.
(Hint: X(1) is defined to be the smallest. If the smallest is at least x, then X1 ≥ x and
X2 ≥ x and · · · Xn ≥ x.)
ii. Find the cdf of X(1), i.e. P(X(1) ≤ x) as a function of x. Consider three cases x < α,
α < x < 0 and x > 0.
iii. Obtain the density of X(1) by differentiating your answer in (ii).
(e) Compute E(ˆαML). Is ˆαML unbiased? If not, make it unbiased, and denote the unbiased
estimate as ˆα.
(f) Compute V ar(ˆα). Which one is more efficient, ˆαMM or ˆα?
(g) Compute the MSEs (Mean Squared Error) of ˆαMM and ˆαML. Which one has the smaller MSE?
(Then, that is the better one according to MSE criterion.)

Answer 1