Question

Let X = ( X1, X2, X3, ,,,, Xn ) is iid,

f(x, a, b) = 1/ab * (x/a)^{(1-b)/b} 0 <= x <= a ,,,,, b < 1

then,

1. Show the density of the statistic T = X(n) is given by
   FX(n) (x) = n/ab * (x/a)^{n/(b-1}}   for 0 <= x <= a ; otherwise zero.
   # using the following
   P (X(n) < x ) = P (X1 < x, X2 < x, ,,,,,,,,, Xn < x ),
   
2. Then assume b is known. Show that X(n) is sufficient and complete for a.
   
3. Hence, or otherwise compute the UMVUE of a.
   # useful to compute E(X(n))

Answer 1

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