On the Risk of Some Strategies for Outlying Observations

Abstract

Many procedures have been proposed for handling outlying observations. Most authors use various statistics $T$ to reject one or more observations, if $T$ is too large (or too small), and compute percentage points of the distribution function of $T$. Some authors, as in [11], [13], [15], [16], find statistics with the optimum property of minimizing for certain alternative hypotheses the error of the second kind given the error of the first kind. The observations that are not rejected are used to estimate unknown parameters e.g. the mean. In this paper, we shall consider one of these procedures which gives rise to a one-parameter family of estimators for the mean and compare their risks with those of the Bayes solutions with respect to a one-parameter family of prior distributions. Two simplified models of outliers as will be specified in (2.1) and (2.2) and a quadratic loss function will be investigated. For sample sizes of 3, 6, and 10, the risks are tabulated and plotted. For Model (2.1) the risks of the first procedure are only a little larger than those of the Bayes solutions; in other words, to each admissible strategy there exists an estimator arising from the first procedure with approximately the same risk. Winsorizing the observations [18] should presumably result in an even better approximation; however, numerical results are not available. For Model (2.2), the risks differ considerably. Although this model is rather unrealistic it is included here because of its prominence in the literature. References for tables of percentage points and the distribution functions of various statistics are collected in the table guide of J. A. Greenwood and H. O. Hartley [7]. Bayes solutions are considered by S. Karlin and D. Truax [10], B. de Finetti [3], and T. E. Ferguson [4]. Some other important papers are [1], [12], [14], [18]. In this paper, "stragglers" are random variables that obey a probability law with a larger variance or with a larger or smaller expectation than the "typical" random variables. An "outlier" is an observation that because of its magnitude may be suspected of being a straggler. Thus the term "outlier" is not precisely defined.