Let X, Y be independently normally distributed as N(, 1), and let V = Y X

Question:

Let X, Y be independently normally distributed as N(θ, 1), and let V = Y − X and W =

Y − X if X + Y > 0, X − Y if X + Y ≤ 0.

(i) Both V and W are ancillary, but neither is a function of the other.

(ii) (V,W) is not ancillary. [Basu (1959).]

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