Consider n tosses with a biased die, for which the probabilities of 1,..., 6 points are given by 123456 1−θ

12 2−θ

12 3−θ

12 1+θ

12 2+θ

12 3+θ

12 and let Xi be the number of tosses showing i points.

(i) Show that the triple Z1 = X1 + X5, Z2 = X2 + X4, Z3 = X3 + X6 is a maximal ancillary; determine its distribution and the distribution of X1,..., X6 given Z1 = z1, Z2 = z2, Z3 = z3.

(ii) Exhibit five other maximal ancillaries. [Basu (1964).]