33. Confidence bounds for a median. Let XI" ' " Xn be a sample from a continuous cumulative distribution function F. Let be the unique median of F if it exists, or more generally let = inf{~' : F( 0 = t}. (i) If the ordered X's are X(I) < . . . < X(n)' a uniformly most accurate lower confidence bound for is §= X(k) with probability p, §= X(k+ I) with probability 1 - p, where k and p are determined by PLn (n) 1 n (n) 1 . ---;;+(1-p) L . ---;;=1-a

(ii) This bound has confidence coefficient 1 - a for any median of F. (iii) Determine most accurate lower confidence bounds for the 100p-percentile of F defined by = inf{ f : F(n = p} . [For fixed ~o the problem of testing H : = ~o against K : > ~o is equivalent to testing H' : P = t against K' : P < t·)