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

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.]