Some Estimates and Tests Based on the $r$ Smallest Values in a Sample

Abstract

Let us consider a situation where only the $r$ smallest values of a sample of size $n$ are available. This paper investigates the case where $n$ is large and $r$ is of the form $pn + O(\sqrt {n})$. Properties of some well known non-parametric point estimates, confidence intervals and significance tests for the 100p% point of the population are investigated. If the sample is from a normal population, these non-parametric estimates and tests have high efficiencies for small values of $p$ (at least 95% if $p \leqq 1/10)$. The other results of the paper are restricted to the special case of a normal population. Asymptotically "best" estimates and tests for the population percentage points are derived for the case in which the population standard deviation is known. For the case in which the population standard deviation is unknown, asymptotically most efficient estimates and tests can be obtained for the smaller population percentage points by suitable choice of $p$ and $O(\sqrt {n})$. The results derived have application in the field of life testing. There the variable associated with an item is the time to failure and the $r$ smallest sample values can be obtained without the necessity of obtaining the remaining values of the sample. By starting with a larger number of units but stopping the experiment when only a small percentage of the units have "died", it is often possible (using the results of this paper) to obtain the same amount of "information" with a substantial saving in cost and time over that which would be required if a smaller number of units were used and the experiment conducted until all the units have "died". Jacobson called attention to applications of this type in [1].