Outer and Inner Confidence Intervals for Finite Population Quantile Intervals

Abstract

Abstract Let π N be a finite population of size N whose elements have distinct X values X (1) < … < X (N). Let x (1) < … < x (n) denote the order statistics of a simple random sample of size n taken from π N without replacement. An outer confidence interval is formed for the quantile interval [X (t), X (u)] (1 ≤ t < u ≤ N) of the form [x (r),x(s)], where 1 ≤ r < s ≤ n and r ≤ t, and an exact expression is derived for the associated confidence coefficient as Pr[x (r) ≤ X (t) < X (u) ≤ x (s)]. A brief table (see Table 1) of confidence coefficients is included, along with several extensions. For example, consider a population of size N = 399 distinct elements. Suppose that we want a 95% or greater confidence interval for the interval in which the middle half of the population lies: [X (100), X (300) Table 1 shows that, based on a simple random sample of size n = 20, the second and nineteenth order statistics of the sample yield a confidence interval [x (2), x (19)] with confidence coefficient 95.7%. In a similar way, the exact expression is given for the confidence coefficient of the inner confidence interval, as follows: Pr[X (t) ≤ x (r) < x (s) ≤ X (u)]. A brief table (see Table 2) is also given of confidence coefficients for the inner confidence interval. Key Words: Nonparametric confidence intervalsOrder statistics