Let X1,... , Xn be iid observations from a location-scale family. Let T1 (X1,... , Xn) and T2(X1, ... , Xn) be two statistics that both satisfy
Ti(ax1 + b,..., axn + b) = aTi(xi,... ,xn)
for all values of x1,..., xn and b and for any a > 0.
a. Show that T1/T2 is an ancillary statistic.
b. Let R be the sample range and S be the sample standard deviation. Verify that R and S satisfy the above condition so that R/S is an ancillary statistic.