Question

Let ?1,?2, … , ?? be a random sample of size ? from a population that can be modeled by the following probability model: ??(?) = ?? ?−1 ? ? , 0 < ? < ?, ? > 0 a) Find the probability density function of ?(?) = max(?1,?2, … , ??). b) Is ?(?) an unbiased estimator for ?? If not, suggest a function of ?(?) that is an unbiased estimator for ?.

Answer 1

Let X1,X2,X3,..........,Xn be a random sample of size n from a population where,

f(x)= ??^(?−1) ?^(?) , 0 < ? < ?, ? > 0

The cummulative distribution function of Xn is

F(x)= (??)^?

The cummulative distribution function of X(n) is

F1(x)= P[X(n) x] =P[X1x, X2X, ................ , Xnx]

     = {P[Xnx]}^n

    = {(??)^?}^n

    = (??)^n?

The probability distribution function of X(n) is the derivative of cummulative distribution function of X(n) which is

f1(x)=n??^(n?−1) ?^(n?) , 0 < ? < ?, ? > 0

YES, X(n) is an unbiased estimator of ?.

Since from X1 to Xn all of them will have range less than ?.

Hence, the maximum among them will also have value less than ?.