25--- Asap

Let > 0 be an unknown parameter and T > 0 be a statistic. Suppose that T / is a pivotal quantity having Lebesgue density f and that x2f(x) is unimodal at x0 in the sense that f(x) is nondecreasing for x x0 and f(x) is nonincreasing for x x0. Consider the following class of confidence intervals for : C = [b1T,a1T] : a > 0,b> 0, b a f(x)dx = 1 " . Show that if [b1 T,a1 T] C, a2 f(a) = b2 f(b) > 0, and a x0 b, then the interval [b1 T,a1 T] has the shortest length within C.