(i) Let X = (X1,..., Xn) have probability density (1/n) f [(x1 )/,...,(xn )/], where...

(i) Let X = (X1,..., Xn) have probability density (1/θn) f [(x1 −

ξ)/θ,...,(xn − ξ)/θ], where −∞ < ξ < ∞, 0 < θ are unknown, and where f is even. The problem of testing f = f0 against f = f1 remains invariant under the transformations x i = axi + b (i = 1,..., n), a = 0, −∞ < b < ∞ and the most powerful invariant test is given by the rejection region

 ∞

−∞

 ∞

0 vn−2 f1(vx1 + u,...,vxn + u) dv du

> C

 ∞

−∞

 ∞

0 vn−2 f0(vx1 + u,...,vxn + u) dv du.

(ii) Let X = (X1,..., Xn) have probability density f (x1 − k j=1 w1 jβj,..., xn − k j=1 wnjβj) where k < n, thew’s are given constants, the matrix (wi j)is of rank k, the β’s are unknown, and we wish to test f = f0 against f = f1. The problem remains invariant under the transformations x i = xi + k j=1wi jγj, −∞ <
γ1,..., γk < ∞, and the most powerful invariant test is given by the rejection region  ···  f1(x1 − w1 jβj,..., xn − wnjβj)dβ1,..., dβk  ···  f0(x1 − w1 jβj,..., xn − wnjβj)dβ1,..., dβk > C.
[A maximal invariant is given by y = "
x1 − n r=n−k+1 a1r xr, x2 − n r=n−k+1 a2r xr,..., xn−k − n r=n−k+1 an−k,r xr #
for suitably chosen constants air.]