X1, ..., Xn are a random sample from a uniform distribution between 0 and θ. X(1) is defined as the minimum value of X1, ..., Xn and X(n) is defined as the maximum value. The ratio of the minimum value to the maximum value, U=X(1)/X(n), is an ancillary statistic.

1. Let X 1 , . . . , X n be a random sample from an uniform distribution U(0, ), > 0. Let X (1) = min(X
1 ,...,X n ) and X (n) = max(X 1 ,...,X n ).
Show that U = X (1) /X (n) is an ancillary statistic.