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

(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, −∞

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 w1jβj ,...,xn− k j=1 wnjβj ) where k

(wij ) 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 + Σk j=1wijγj , −∞ < γ1,...,γk < ∞, and the most powerful invariant test is given by the rejection region

···
f1(x1 − w1jβj ,...,xn − wnjβj )dβ1, . . . , dβk
···
f0(x1 − w1jβj ,...,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,rxr



for suitably chosen constants air.]