Question: Let y1,.., yn be a random sample from a population with density function f0(v) = eΘ-v for v≥ Θ and f(v) = 0 for v<Θ, where Θ is an unknown parameter.
a. Determine the method of moments estimator of θ, based on the first moment. Determine the mean and variance of this estimator.
b. Prove that this estimator is consistent.
c. Prove that the maximum likelihood estimator of θ is given by the minimum value of y1,.., yn. Give explicit proofs that this estimator is biased but consistent.
d. Discuss which of the two estimators in a and c you would prefer. Consider in particular the two extreme cases of a single observation (n = 1) and the asymptotic case (for n →∞).