Let X1 X2 Xn be a random sample from a distribution with

Let X1, X2, . . . , Xn be a random sample from a distribution with the following probability density function

where m is a known integer parameter and ? is an unknown positive parameter.

(a) Find the method of moments estimator, ? ?, of ?.

(b) Find the maximum likelihood estimator, ?ˆ, of ?.

(c) Are ? ? and ?ˆ unbiased estimators of ??

(d) Find the variance of ?ˆ.

Solution

by definition = =    dx

                                                       = dx

    the structure is gamma integral, hence arranging according to gamma integral we have.

                           =    (    d( )

                               =   ---(1)

the methods of moments consists replacing moments by sample moments

in (1) put r=1

therefore             =                     ^- = =

                                                                                                     ^-     =

2. to find maximum likelihood estimatior of ^

likelihood function of f(x) is

L( = l(x1,x2,..xn,L( =   i= 1,2,...n

              =         =

taking log on both the sides and partially with respect to and put  =0 (since log( and L() attains maximum together)

therefore ,     = = -       = 0

  =        ......(2)

3. from equations (1) and (2) are not unbiased estimators of since both estimators of lamba do not give true value of the parameter.

4. since ^ is not unbiased estimator of , cramer-Rao inquality cannot be applied to find variance of estimator (^). variance does not exists..