Question: 1. Let X be a continuous random variable whose probability distribution depends on a single unknown parameter 6 ( O. Let F(a|0) be the cumulative

1. Let X be a continuous random variable whose probability distribution depends on a single unknown parameter 6 ( O. Let F(a|0) be the cumulative distribution function (cdf) of X and Qp(0) be the pth quantile of X for a specified p E (0, 1). Assume that the family of cdfs {F(r|0), 0 ( O} is stochastically nondecreasing in 0, i.e., for each r, F(x 0) is a nonincreasing function of 0. Thus, 81 F(x|62) for all r. (a) Show that Qp(0) is a nondecreasing function of 0. (b) Suppose U is a 100(1 - a)% upper confidence bound for 0. Show that Qp(U) is a 100(1 - a)% upper confidence bound for Qp(8). (c) What does (b) suggest regarding constructing confidence bounds for a quan- tile? (d) As an example, give a distribution for X whose cdf is stochastically nonde- creasing in d

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