Introduction

Let be the order statistics of a sample of real-valued random variables. A linear combination of (transformed) order statistics, or L-statistic, is a statistic of the form

The coefficients Cni are a triangular array of constants and a is some fixed function. This “score function” can without much loss of generality be taken equal to the identity function, for an L-statistic with monotone function a can be viewed as a linear combination of the order statistics of the variables and an L-statistic with a function a of bounded variation can be dealt with similarly, by splitting the L-statistic into two parts.

22.1 Example (Trimmed and Winsorized means). The simplest example of an L-statistic is the sample mean. More interesting are the a-trimmed meanst and the a-Winsorized means

The a-trimmed mean is the average of the middle -th fraction of the observations, the a-Winsorized mean replaces the ath fractions of smallest and largest data by and, respectively, and next takes the average. Both estimators were already used in the early days of statistics as location estimators in situations in which the data were suspected to contain outliers. Their properties were studied systematicclIly in the context of robust estimation in the 1960s and 1970s. The estimators were shown to have good properties in situations in which the data follows a heavier tailed distribution than the normal one. Figure 22.1 shows the asymptotic variances of the trimmed means as a function of a for four distributions. (A formula for the asymptotic variance is given in Example 22.11.) The four graphs suggest that 10% to 15% trimming may give an improvement over the sample mean in some cases and does not cost much even for the normal distribution.