Question

Problem 1.2 Let Xi, X2, ..., Xn be a random sample from the pdf a) Find the maximum likelihood estimator of. θΜΕ- b) Find the method of moments estimator of 0. NDM c) If a random sample of n - 4 yields the following data: method of moments estimate of θ would be θΜΟΜ- MOM 7.50, 3.73, 4.52, 3.35 then the maximumn likelihood estimate of θ would be éMLE-- and the

Answer 1

a)

Hence, MLEX) (A)

b)

MoM

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

E(X) =

$\int _\theta ^{\infty \:}2\cdot \frac{x\theta ^2}{x^3}dx=2\theta$

hence

$\widehat{\theta } = \frac{\overline{X}}{2}$

c)
MLE =3.35
MoM = (7.50 + 3.73 + 4.52 + 3.35)/8
= 2.3875