Dr. Hackney STA Solutions pg 148

Dr. Hackney STA Solutions pg 148 - 9-6Solutions Manual for...

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Unformatted text preview: 9-6Solutions Manual for Statistical Inference9.15 Fiellers Theorem says that a 1-confidence set for=Y/Xis:x2-t2n-1,/2n-1s2X!2-2xy-t2n-1,/2n-1sY X!+y2-t2n-1,/2n-1s2Y!.a. Definea= x2-ts2X,b= xy-tsY X,c= y2-ts2Y, wheret=t2n-1,/2n-1. Then the parabolaopens upward ifa >0. Furthermore, ifa >0, then there always exists at least one real root.This follows from the fact that at= y/x, the value of the function is negative. For= y/xwe have(x2-ts2X)yx2-2(xy-tsXY)yx+(y2-as2Y)=-ty2x2s2X-2yxsXY+s2Y=-t"nXi=1y2x2(xi-x)2-2yx(xi-x)(yi-y) + (yi-y)2#=-t"nXi=1yx(xi-x)-(yi-y)2#which is negative.b. The parabola opens downward ifa <0, that is, if x2< ts2X. This will happen if the test ofH:X= 0 acceptsHat level.c. The parabola has no real roots ifb2< ac. This can only occur ifa <0.9.16 a. The LRT (see Example 8.2.1) has rejection region{x:|x-|> z/2/n}, acceptanceregionA() ={x:-z/2/...
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.

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