Non-Parametric Tolerance Limits

Abstract

In this note are presented graphs of minimum probable population coverage by sample blocks determined by the order statistics of a sample from a population with a continuous but unknown cumulative distribution function (c.d.f.). The graphs are constructed for the three tolerance levels .90, .95, and .99. The number, $m$, of blocks excluded from the tolerance region runs as follows: $m$ = 1(1)6(2)10(5)30(10)60(20)100, and the sample size, $n$, runs from $m$ to 500. Thus the curves show the solution, $\beta$, of the equation $1 - \alpha = I_\beta(n - m + 1, m)$ for $\alpha = .90, .95, .99$ over the range of $n$ and $m$ given above, where $I_x(p, q)$ is Pearson's notation for the incomplete beta function. Examples are cited below for the one- and two-variate cases. Finally, the exact and approximate formulae used in computations for these graphs are given.