Let X1,...,Xn be a sample from (i) the normal distribution N(a, 2), with a fixed and 0

Let X1,...,Xn be a sample from (i) the normal distribution N(aσ, σ2), with a fixed and 0 <σ< ∞; (ii) the uniform distribution U(θ− 1 2 , θ+ 1 2 ), −∞ <θ< ∞; (iii) the uniform distribution U(θ1, θ2), ∞ < θ1 < θ2 < ∞.

For these three families of distributions the following statistics are sufficient: (i), T = (Xi,

X2 i ); (ii) and (iii), T = (min(X1,...,Xn), max(X1,...,Xn)). The family of distributions of T is complete for case (iii), but for (i) and (ii) it is not complete or even boundedly complete.

[(i): The distribution of Xi/

X2 i does not depend on σ.]