Recall that for a continuous random variable with cdf F, the median
. More generally, let
denote the population quantile of order . That is
A confidence interval for a population quantile can be obtained using order statistics as follows. Suppose we take a sample of size from the given population and write the ordered sample as . Then for integers , , ( ) the probability can be determined. It is equal to
Alternatively, for a given value of , we'd like to be able to determine and . Substituting observed values and for and , the % CI for is (). Note that we can't usually choose , to give the conventional , , , values for . Note also that the above has assumed no particular form for , and is, in that sense, a distribution-free property.
Let denote the order statistics of a random sample of size from a distribution of the continuous type. Compute