Question

Let X1,..., Xn be a random sample from the pdf f(x:0)-82-2, 0 < θ x < oo. (a) Find the method of moments estimator of θ. (b) Find the maxinum likelihood estimator of θ

Answer 1

a)

MoM estimator

$E(X)=\overline{X}$

$E(X)=\int _\theta ^{\infty \:}\frac{x\theta }{x^2}dx=\mathrm{diverges}$

hence there is no Method of moments estimator

b)

MLE estimation, we express the joint likelihood as proportional to We simply discard any factors of the joint density that are constant with respect to 0. Since this is nonzero if and only if 0 is positive but not exceeding the smallest observation in the sample, we seek to maximize θη subject to this constraint. Since n > 0, θη is a monotonically increasing function on θ > 0, hence L is greatest when θ-x(1): i.e., 0) is the MLE. It is trivially biased because the random variable X(1) is never smaller than θ and is almost surely strictly greater than θ; hence its expectation is almost surely greater than