What is the maximum likelihood estimator for a given density function?

[sara_87]

Homework Statement

pdf: f(x)=ax^(a-1) ; 0<x<1, a>0
estimate a by maximum likelihood

Homework Equations

let L be maximum likelihood
L=(a(x[1])^(a-1))(a(x[2])^(a-1))...(a(x[n])^(a-1))

The Attempt at a Solution

Im trying to make this into a nicer expression:
L=a^n... (now I am stuck)

Any help would be v much appreciated.
Thank you.

 

[statdad]

Remember that if your density is $$f(x)$$, then the likelihood function for an i.i.d. sample is

$$L(x_1, \dots, x_n) = \prod_{i=1}^n f(x_i)$$

For the density you give this is

$$L(x_1, \dots, x_n) = \prod_{i=1}^n a x_i^{a-1} = a^n \prod_{i=1}^n x_i^{a-1}$$

What do you know about simplifying a product of different bases when each is raised to the same power?