Suppose, in the notation previously established, that n = 3. Write the ancillary in the form {sign(X3

− X2

), sign(X2

− X1

)}, together with an additional component AX = (X(3) − X(2))/ (X(2) − X(1)), where X(j) are the ordered values of X. Show that ΑX is a function of the sufficient statistic, whereas the first two components are not. Let Yi(t) = 1/{Xi − t) and denote by A(t) the corresponding ancillary computed as a function of the transformed values.

Show that the function A(t) is continuous except at the three points t = Xi and hence deduce that the data values may be recovered from the set of ancillaries {A(t), −∞ < t <

∞}. [In fact, it is enough to know the values of A(t) at three distinct points interlacing the observed values. However, these cannot be specified in advance.]