Consider a random sample X1 ,X2 ,• • • , X,, from the Weibull distribution with p.d.f. f( x; 0, -r) = 0-rxi - 1e-0xr, 0 x oo, where O 0, 0 -r, zero elsewhere.

(a) If -r is known, find the m.l.e. of 0.

(b) Ifthe parameter O has a prior gamma p.d.f. g(8) with parameters IX and p• = 1/P ,show that the compound distribution is a Burr type with p.d.f. h( x) = 1X -r/J' xi - 1 /( xr + P)a.+ 1 , 0 x oo , zero elsewhere.

(c) If,in the Burr distribution, -r and P are known, find the m.l.e. of IX based on a random sample of size n.