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

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.14

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

ξ1,... ξ2n be such that |ξi − ξ j| > 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 = (X j − 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.]