Question

2 Let X1, X2, ..., X., be independent continuous random variables from the following distribution: f(x)=ox-(-V where = 1 and a > 1 You may use the fact: E(X)= -1

Answer 1

Xe, x, ... Xn continuou random variables where x7,1 and a71 Given E(X) = x f(x)= xx (x-1) dh = 0 dlogl zo da ca 1) MLE u found by equating Likelihood , L = L (Ti) logt = nelogd - (x-1) laglo xi) =nlagd-(K-1) {log xi loglanlagd- & {log Xi + Elog Xi a leg l = 0 - - - - {log Xi = 0 = n = {log Xi ) 2 = {log Xi Also logh = -270 22² =) MLE = {log xi

2.) MOM estimator is found by equating x = f(x) Given E(X)= x-1 x = x » (2-1) X=4 = xĂ - x = x = x(x-1) = x 1x lxlik L MOM х 1

3) By füher Neymani factorization theorem to identify sufficient statistic A statutic Tú sufficient for oa) f(x,x ... xulo) = 4, CT,o) x ₂₂ (x, X. Xa) where L₂ ü either a constant or function depending on x, ,X, ... Xu Our Likelihood L = 2 (Xi) Logl- nelogat a log rixi - Lög (1 x) Clearly (Tixi) or (z log x.) ({ log xi ) is indipendent of O and hence the sufficient statistic only