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Unformatted text preview: 8-12Solutions Manual for Statistical Inference8.26 a. We will prove the result for continuous distributions. But it is also true for discrete MLRfamilies. For1> 2, we must showF(x|1)F(x|2). Nowddx[F(x|1)-F(x|2)] =f(x|1)-f(x|2) =f(x|2)f(x|1)f(x|2)-1.Becausefhas MLR, the ratio on the right-hand side is increasing, so the derivative can onlychange sign from negative to positive showing that any interior extremum is a minimum.Thus the function in square brackets is maximized by its value ator-, which is zero.b. From Exercise 3.42, location families are stochastically increasing in their location param-eter, so the location Cauchy family with pdff(x|) = ([1+(x-)2])-1is stochasticallyincreasing. The family does not have MLR.8.27 For2> 1,g(t|2)g(t|1)=c(2)c(1)e[w(2)-w(1)]twhich is increasing intbecausew(2)-w(1)>0. Examples include n(,1), beta(,1), andBernoulli()....
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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.
- Spring '12