6.13 In some applications the quantity jp FX(kp), where kp is the pth quantile of FY,...

6.13 In some applications the quantity jp ¼ FX(kp), where kp is the pth quantile of FY, is of interest. Let limn!1 (m=n) ¼ l, where l is a fixed quantity, and let {rn} be a sequence of positive integers such that limn!1 (rn=n) ¼ p. Finally let Vm, n be the number of X observations that do not exceed Y(rn).

(a) Show that m1Vm,n is a consistent estimator of jp.

(b) Show that the random variable m1=2[m1Vm,n  jp] is asymptotically normally distributed with mean zero and variance jp(1  jp) þ lp(1  p)
f2X (kp)
f 2 Y(kp)
where fX and fY are the density functions corresponding to FX and FY, respectively (Gastwirth, 1968; Chakraborti and Mukerjee, 1990).