54 Introduction to Robust Estimation and Hypothesis Testing 0.6 0.3 0.0 0 5 10 x 15 20 0.6 0.3 0.3 0.0 0 5 10 x 15 20 0 5 10 x 15 20 0 2 · · · · · · · · · · · · · 4 6 8 10 x · · 12 0.0 Figure 3.4: Some kernel density estimators can perform poorly when the variable under study is bounded, even with large sample sizes. The upper left panel is a plot based on the function skerd and n = 500 randomly sampled observations from the lognormal distribution in Figure 3.3. The upper right panel is Rosenblatt's shifted histogram (using on the function rdplot), the lower left panel shows the plot created by kdplot, and the lower right panel is based on the adaptive kernel estimator. were considered for the situation at hand, but similar results were obtained. So although situations are encountered where the R function skerd produces a smoother, more visually appealing plot versus rdplot, kdplot, and akerd, blind use of this function can be misleading. I 3.3 The Sample Trimmed Mean As already indicated, the standard error of the sample mean can be relatively large when sampling from a heavy-tailed distribution, and the sample mean estimates a nonrobust measure of location, µ. The sample trimmed mean addresses these problems. The sample trimmed mean, which estimates the population trimmed µ t (described in Section 2.2.3), is computed as follows. Let X 1 , . . . , X n be a random sample and let X (1) X (2) · · · X (n) be the observations written in ascending order. The value X (i) is called the ith order statistic. Suppose the desired amount of trimming has been chosen to be , 0 < 0.5. Let g = [ n], where [ n] is the value of n rounded down to the nearest integer. For example, [10.9] = 10. The sample trimmed mean is computed by removing the g largest and g smallest observations and averaging the values that remain. In symbols, the www.elsevierdirect.com