Let Xi ∼ U(0, 1), i = 1, . . . , 20, iid. IQR = $F^{−1}(.75)−F^{−1}(.25)$ = $X_{(15)}−X_{(5)}$ in this example as n = 20.
a. Find the distribution of the random variable W = IQR.
b. Devise a function of this random variable such that it has expectation of the length of the support of distribution. Do the same thing for the range
c. Consider both of these quantities from (b) as ways to guess (estimate) the length of the support. Compute the relative efficiency (the ratio of mean square errors, it doesn’t matter which is in the numerator) of these two quantities.
I could figure out solution to (a) by using formulas for order statistic, but I'm stuck on (b) and (c). How do I devise such functions of IQR and range? And how do I make use of results from (b) to solve (c)?
Any hint is greatly appreciated!