Let Y1 < Y2 < < Yn be the order statistics of a random sample from a uniform distribution on (0, ), where > 0.

(a)

Show that for testing H0 : =0 against H1 : =0 is =(Yn/0)n, Yn 0, and =0 if Yn >0.

(b)

When H0 is true, show that 2 log has an exact 2 (2) distribution, not 2(1). Note that the regularity conditions are not satisfied.