( 1 ) Show that the power function IT of the test $ ( * ) is nonincreasing . Recall IT is defined by TT ( O ) = Ed ( $ (* ) ) , a lt; lt; b . ( 2 )...

Let X have density fθ with a < θ < b. Suppose we are interested in testing H0 : θ ≥ θ0 versus H1 : θ < θ0, where θ0 belongs to the interval (a, b).

Suppose the test φ(X) is NP(fθ2 , fθ1 ) for every a < θ1 < θ2 < b.

( 1 ) Show that the power function IT of the test $ ( * ) is nonincreasing . Recall IT is defined byTT ( O ) = Ed ( $ (* ) ) , a &lt; &lt; b .( 2 ) Find the size of the test $ ( * ).( 3 ) Supposed ( * ) is another test with satisfies ED, ( 8 ( * ) ) &lt; Foo ( $ ( * ) ) . Show that {` ( 8 ( * ) ) &lt;IT ( D ) holds for every $ &lt; Do .