Let X, Y have joint density p(x, y)=2f(x)f(y)F(xy), where f is a known probability density symmetric about

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