ABSTRACT
Now we use condition (35.4.7) a i d (A.3) t o conch~le that P d ( O , IQ) has an extrcrnurn at uz = 0. Furtherrnorc condition (35.4.8), (A.5) and a
I"le00~ OF' r ~ ' ~ ~ ~ 0 1 1 f 5 1 \ . 1 35.4.2 We prove Lhe theorem by showing there exists a lest of the forrrl (35.2.4) sat,isf,yillg (35.4.4)-(35.4.8). Clearly airy test of lire fomi (35.2.4) s;rlisfies (35.4.3) ancl (35.4.5). Next we show tll i~t ;my such test satisfies (35.4.8). Note that for fixed X12 - Y Y 1 < <C:2 the test rejects if XI 1 > C:, . Howc:vcr thc contlitiona.1 clistrihtior~ of Xll IX12 when v - 0 is st,ochnsticdly tlccrcasiiig in X12. This irnplies that g(Xlz) increases a.s X i a v;trics from CY2 (C:l to srnirller values. F111.t~her.rrrore wlleri > C 5 C:, , the test rcjects whcri X I 1 I X12 > (72. But the corlditiorial distrihtiorl of X l 1 + X12 lXlz is stocll;tsli~i~lly incre;~sing as X 12 increases. This irnplies t,lrat g (XI2) increases a.s Xlz increases fi.orii ('2 - arid csta,blisllos cordition (35.4.8).
