The following problems are from FA&JR:
1. Let Y1;:::;Yn U(0;#).
(a) Derive the MP test at level for testing H0 : # = #0 vs. H1 : # =
#1, where #1 >#0.
(b) Calculate the power of the MP test.
2. Let Y1;:::;Yn be a random sample from a distribution with the density
function
f#(y) = y#e y
2
2#; y 0; #> 0
(a) Find the MP test at level for testing two simple hypotheses
H0 : # = #0 vs. H1 : # = #1, where #1 >#0.
(b) Is there a UMP test at level for testing the one-sided composite
hypotheses H0 : # #0 vs. H1 : # > #0? What is its power
function?
(Hint: show that Y2i exp(1=(2#)) and, therefore, Pni=1Y2i # 22n.)
3. Let Y1;:::;Yn N( ; 2).
(a) Assume rst that is known.
i. Find an MP test at level for testing H0 : 2 = 20 vs.
H1 : 2 = 21, where 1 > 0.
ii. Show that the MP test from the previous paragraph is a UMP
test for testing H0 : 2 20 vs. H1 : 2 > 20.
(Hint: recall that Pni=1(Yi )2 2 2n.)
(b) Consider now an unknown . See also Exercise ??.
i. Is there an MP test for testing H0 : 2 = 20 vs. H1 : 2 = 21,
where 1 > 0, in this case? If not, nd the corresponding
GLRT test.
ii. Is the above test also a GLRT for testing the one-sided hy-
potheses H0 : 2 20 vs. H1 : 2 > 20?
1
iii. Find the GLRT test at level to test H0 : 2 20 vs. H1 :
2 < 20?
(Hint: recall that Pni=1(Yi Y)2 2 2n 1.)