Statistics 3858 : Liklihood Ratio for Expotial Distributio I ths two xampl th rjctio rjctio rgio is of th form {x : 2 log (Λ(x)) > c} for a appropriat costat c. For a siz α tst, usig Thorm 9.5A w obtai this critical valu from a χ 2 () distributio. For α =.05 w obtai c = 3.84. O th surfac ths appar to b th sam, but th st of x i this rjctio rgio is diffrt for th o ad two sidd altrativs. O Sidd Altrativ X i, i =, 2,..., iid xpotial, λ. Cosidr H 0 : λ = vrsus th altrativ λ <. Fid th gralizd liklihood ratio tst ad show that it is quivalt to X > c, i th ss that th rjctio rgio is of th form X > c. Th ull hypothsis is H 0 : λ Θ 0 = { } ad th altrativ is H A : λ Θ A = {λ : λ < } = (0, ). Th liklihood fuctio is Th gralizd liklihood ratio is Th umrator is L(λ) = λ λ X max λ Θ 0 L(λ) max λ Θ0 Θ A L(λ) λ λ0 X 0 () For th domiator w d to fid argmax. Solvig log(l(λ)) λ = λ X = 0
Th solutio is λ = X. Thus for th domiator th argmax is giv by { λ = X if X < othrwis Th liklihood ratio is th giv by Aftr simplificatio this is Cosidr th fuctio g : (0, ) R giv by { ( λ0 X) X+ X X if X < ( ) X+λ0 X if X { ( λ0 X) X+ if X < if X g(y) = y λ0y+ Asid : This fuctio coms from oticig that for X < Λ / = g( X) By calculus th studt should vrify that g is mootoically icrasig for y < dcrasig for y >. Also show { g( X) if X < if X ad mootoically Th rjctio rgio for th GLR tst is R = {x = (x, x 2,..., x ) : Λ < c} whr c <. I th lctur w fill i th stp to show this is th sam st as {x : x > c } whr c solvs i trms of y > g(y) = c. Notic this is i fact ssibl sic it says that w rjct if X >, ad th altrativ is valus of λ such that λ >. 2
2 Two Sidd Altrativ This xampl is as abov but with H 0 : λ = vrsus H A : λ. Aftr appropriat calculatios th studt should show that th gralizd liklihood ratio is Λ(X) = ( X) X+ (2) Th studt should sktch th curv of g (sam fuctio as i sctio ). Th rjctio rgio corrspods to th st of x satisfyig x < yl, or x > yu whr th umbrs yl, y U ar solutios of g(y) = c Notic this is a diffrt st tha th rjctio rgio for th o sidd altrativ. I class w hav Thorm 9.5.A that stats, udr th ull hypothsis, 2 log(λ) covrgs i distributio to χ 2 (df) whr i this cas th dgrs of frdom is df = 0 =. I th rst of this sctio for this xampl w vrify th limit distributio dirctly by usig a Taylor s approximatio of ordr 2 ad a xtsio of th dlta mthod discussd arlir. I studyig this part w ca rcogiz that log(λ(x)) is a fuctio of X, ad so w will d to study th rol of this fuctio ad what rsults from it. As a asid otic that w d up with a fuctio of X sic it is a sufficit statistic for this statistical modl. Notic that log(λ(x)) = g( X) whr g(y) = log( y) ( y ) This is a diffrt fuctio g tha usd i th prvious part. Thorm 9.5A is byod what w ca study i our cours. Howvr w ca gt som udrstadig of this thorm by studyig th spcial cas i our problm. It ivolvs a Taylor s xpasio ad som proprtis of covrgc i distributio. ad Expad g about E 0 (X) =, th ma udr th assumptio that th ull hypothsis holds. For this fuctio w hav g (y) = y g (y) = y 2 3
so that g ( ) = 0 ad g ( ) = λ 2 0. Th first ordr Taylor approximatio is ) g (y) = g(/ ) + g (/ ) (y λ0 = 0 ad th scod ordr approximatio is g 2 (y) = g (y) + ( 2 g ( ) y ) 2 = ) 2 2 λ2 0 (y λ0 Th first ordr Taylor approximatio is ot vry usful so w us th xt ordr approximatio. Thus, otig that ˆλ = X 2 log(λ(x)) = 2g( X) 2g 2 ( X) = 2 ( 2 λ2 0 X ) 2 = ( X ) 2 Udr th assumptio that th ull hypothsis is tru, th trm i th last xprssio has th approximat distributio of Z 2, whr Z N(0, ), that is χ 2 (). 4
How would w us this i practic? H 0 : λ = 2 vrsus H A : λ 2. Th GLR is Λ(X) = ( X) X+ Udr H 0 this has a χ 2 () distributio. To prform this hypothsis tst at lvl α =.0 w would us th uppr.0 quatil as th critical valu, that is c = 6.63. Thus th rjctio rgio is R = {x : 2 log(λ(x)) > 6.63} 5