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. Forθ1> θ2, we must showF(x|θ1)≤F(x|θ2). Nowddx[F(x|θ1)-F(x|θ2)] =f(x|θ1)-f(x|θ2) =f(x|θ2)f(x|θ1)f(x|θ2)-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 at∞or-∞, 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|θ) = (π[1+(x-θ)2])-1is stochasticallyincreasing. The family does not have MLR.8.27 Forθ2> θ1,g(t|θ2)g(t|θ1)=c(θ2)c(θ1)e[w(θ2)-w(θ1)]twhich is increasing intbecausew(θ2)-w(θ1)>0. Examples include n(θ,1), beta(θ,1), andBernoulli(θ)....
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- Spring '12
- Statistics, Convex function, discrete MLR families, location param2 eter, location Cauchy family