Confidence bounds for a median. Let X1,..., Xn be a sample from a continuous cumulative distribution functions F. Let ξ be the unique median of F if it exists, or more generally let ξ = inf{ξ : F(ξ

) ≥ 1 2 }.

(i) If the ordered X’s are X(1) < ··· < X(n), a uniformly most accurate lower confidence bound for ξ is ξ = X(k) with probability ρ, ξ = X(k+1) with probability 1 − ρ, where k and ρ are determined by

ρ

n j=k n

j 1

2n + (1 − ρ)

n j=k+1 n

j 1

2n = 1 − α.

(ii) This bound has confidence coefficient 1 − α for any median of F.

(iii) Determine most accurate lower confidence bounds for the 100p-percentile ξ of F defined by ξ = inf{ξ : F(ξ

) = p}.

[For fixed ξ0, the problem of testing H : ξ = ξ0 to against K : ξ > ξ0 is equivalent to testing H : p = 1 2 against K : p < 1 2 .]