Simplified Estimation from Censored Normal Samples

Abstract

0. Summary. Estimators of mean and standard deviation for censored normal samples which are based on linear systematic statistics and which use simple coefficients are almost as efficient as estimators using the best possible coefficients. Estimators are given for samples of size N < 20 for censoring at one extreme and for several types of censoring at both extremes. 1. Introduction. A censored sample is a sample lacking one or more observations at either or both extremes with the number and positions of the missing observations known. Censoring may take place naturally i.e., an observation has a magnitude known only to be more extreme than the other observations in the sample. Censoring may also be imposed by the experimenter who from past experience knows that extreme observations are so unreliable that their magnitudes should not be used as observed. The experimenter may impose censoring to reduce the duration of an experiment and obtain estimates before the extreme cases are determined. Estimation of the mean and standard deviation of a normal distribution from a sample which is censored has been considered by Sarhan and Greenberg [1], who obtained coefficients for best linear systematic statistics. They also record efficiencies of these estimators compared to the case of no censoring. Winsor [4} and perhaps others have suggested using for the magnitude of an extreme, poorly known, or unknown observation the magnitude of the next largest (or smallest) observation. We shall show that when symmetry is maintained (or proper adjustment is made) this practice results in estimators of the mean whose efficiencies are scarcely distinguishable from those of best linear estimators. For non-symmetrical censoring, it is demonstrated that optimum simple estimators of the mean result from these Winsorized estimators. Also presented are estimators of the standard deviations using one or two ranlges (not necessarily symmetrical) which have efficiency .94 or greater when compared with the best linear systematic statistics. The variances of the proposed estimators were computed from an original 21 decimal tabulation of the means variances and covariances of the order statistics made available by Dan Teichroew. These tables are described in reference [5]. The efficiencies are the ratios of variances of corresponding estimators givenl by Sarhan and Greenberg [1]. 2. Symmetrical censoring. Estimation of mean. If natural or imposed censoring of the sample results in the same number of observations censored from each extreme of the sample the practice of using for each missing observation the magnitude of its nearest neighbor whose magnitude is known has a minimum