On the basis of a sample X = (X1,...,Xn) of fixed size from N(, 2) there do

On the basis of a sample X = (X1,...,Xn) of fixed size from N(ξ, σ2) there do not exist confidence intervals for ξ with positive confidence coefficient and of bounded length.13

[Consider any family of confidence intervals δ(X) ± L/2 of constant length L.

Let ξ1,...ξ2n be such that |ξi − ξj | > L whenever i = j. Then the sets Si{x :

|δ(x) − ξi| ≤ L/2} (i = 1,..., 2N) are mutually exclusive. Also, there exists

σ0 > 0 such that

|Pξi,σ{X ∈ Si} − Pξ1,σ{X ∈ Si}| ≤ 1 2N for σ>σ0,

as is seen by transforming to new variables Yj = (Xj − ξ1)/σ and applying Lemmas 5.5.1 and 5.11.1 of the Appendix. Since mini Pξ1,σ{X ∈ Si} ≤ 1/(2N), it follows for σ>σ0 that mini Pξ1,σ{X ∈ Si} ≤ 1/N, and hence that inf ξ,σ
Pξ,σ 
|δ(X) − ξ| ≤ L 2 
≤ 1 N The confidence coefficient associated with the intervals δ(X) ± L/2 is therefore zero, and the same must be true a fortiori of any set of confidence intervals of length ≤ L.]