Question

6.4.4. The Pareto distribution is frequently used a model in study of incomes and has the distribution function F(x;0,2)=1-(81/x)02 elsewhere, where 01 0 and 02 > 0 If X\,X2, ...,Xn is a random sample from this distribu- tion, find the maximum likelihood estimators of 01 and 02.

Answer 1

Answer:

The Pareto distribution's cumulative distribution function is given by F(x,0,6)=1_ Then, the density function of Pareto distribution is given by Since f (x)=(x) f(xe.)= d1- d 6(a) Given that a random sample of sizen drawn from a Pareto distribution with pdf ,0, sxs0, 01,6, > 0 f(x,0)=

Method of maximum likelihood Lilelihood function(L) =s(,a) xf(,a) xf(x,,a) X (e)() 1 (()IT In L (n+ n) In(6)- (+)n x

While finding MLE for e,, we assume that e, as constat а In L (n+ne) -fmx, Now the MLestimtor of , be the solution of the following likelihood equation d In L do (n+ n In x 0 i-1 = n+ n@ n x пе п+ Hence, the MLE of 6,, |e2 i-