Problem 1.73 Suppose the data set Y = {y1,...,yn} is obtained from X = {x1,...,xn} by means of the linear transformation yi = axi +

b, where

a, b are constants and a > 0.

(a) Show that the order statistics for the data set Y are given by y(i) = ax(i) +

b.

(b) Let Qx(p) and Qy(p) denote the quantiles of order p for the X and Y data sets, respectively. Show that Qy(p) = aQx(p) +

b.

(c) Let Fˆn and Gˆn be the empirical distribution functions of the data sets {x1,...,xn} and {y1,...,yn}, respectively. Show that, Gˆn(y) = Fˆn y − b a  .