Question

Suppose X1,··· ,Xn are i.i.d. with pdf $f(x)=\theta x^{\theta-1}$ if 0 < x < 1 and 0 otherwise.

(a) Construct the MP test for the hypothesis $H_0:\theta=1$ v.s. $H_1:\theta=2$ with α=0.05.

(b) Derive the power function of the test in (a).

Answer 1

a) MP test for Ho ; 8=1 vs H.;0=2, . d=0.05 8(x) = f(x01) - 10 | (xi) 0 1 0 of f (x 100) 100 å en xi z ki (as Oi = 270o = 1) E-lnxi cc . . . 12 . XM 1 N Now, XN B(0,1). ->y-t n pareto =) en yn Exponential a x 2 - 2 luan x ² . { (-2dnxi) N.X² MP critical region of its size is W=&a & -lnxice} mp text at Å=0.05 is 2 Pho [XtW] =P0=1 { & -lnxice}

b) fx (8,0) = 50x8-1 ocx cl 10 . Otherwise let y=-0 lux ) fyly,0)=dey ,0<4.00 Co otherwise Hence, y = -0 en sin exp (1) independent : Y 2 = -0 en X2 n exp ll .. yit YqN Gamma (2) power function = B (@) = Po [Xixzz?] = Pol-olux,-0 dnxzs-Ben(2)] 1.2 Po[Yi+y29.colu()] => e-Hyeny dy BOOM(3) =[-(4+1) 8-87-8 3 () = {1-1-oen() +) (%)'). which is pino Тенте 25м