Let X, Y have joint density p(x, y) = 2 f (x) f (y)F(x y), where f

Let X, Y have joint density p(x, y) = 2 f (x) f (y)F(θx y), where f is a known probability density symmetric about 0, and F its cumulative distribution function. Then

(i) p(x, y) is a probability density.
(ii) X and Y each have marginal density f and are therefore ancillary, but (X, Y )
is not.
(iii) X · Y is a sufficient statistic for θ. [Dawid (1977).]