Estimation of the Mean and Standard Deviation by Order Statistics. Part III

Abstract

In a previous work [7], the mean and standard deviation were estimated by arranging all the sample elements in ascending order and taking the best linear combination of them. We will use here the same principle to estimate the mean and standard deviation of certain populations from singly and doubly censored samples. Censored samples may be considered as truncated samples having a known number of unmeasured (missing) observations, i.e., those in which the total number of sample elements is known, but measurements on some of which are lacking. In life testing, fatigue testing, and in other tests of a destructive nature, we have $n$ items drawn at random from some population which when subjected to a test, fail in order of time. To save time and/or items, it is often required to stop the experiment (to censor the sample) after recording the first $r (<n)$ observations. This is a censored sample from the right. Again, censored samples are found frequently in biological data where some of the observations in a sample are either below or above a limit in the measure used. The values beyond this limit are believed to form a continuation of the scale of measurement but are unmeasurable in the experiment [6]. For example, in experimental biology, $n$ samples from each animal are tested for antibodies after a certain period of time. Only $r$ of these samples contain measurable amounts while $(n - r)$ of the animals develop the antigen at a level too low for measurement by the prevailing technique. This is a censored sample from the left. In fact, the estimation of the mean and standard deviation based on the linear combination of all sample elements is a special case, and the general one is considered here. Censored samples were considered recently in the work of Ipsen [6], Walsh [8], Hald [4], Gupta [3], Cohen [1], Halperin [5], and Epstein et al [2].