Question

Let X = ( X1, X2, X3, ,,,, Xn ) is iid, f(x, a, b) =...

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))
Math   Statistics-And-Probability   
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