62 Introduction to Robust Estimation and Hypothesis Testing 3.3.3 Estimating the Standard Error of the Sample Winsorized Mean ¯ An estimate of the standard error of the sample Winsorized mean, X w , can be derived from the influence function of the population Winsorized mean given in Section 2.2.2. Dixon and Tukey (1968) suggest a simpler estimate: n - 1 s w × , n - 2g - 1 n where g = [ n] is the number of observations Winsorized in each tail, so n - 2g is the number of observations that are not Winsorized. 3.3.4 R Functions winmean, winvar, trimse, and winse Included in the R functions written for this book is a function called winmean that computes the Winsorized mean. If the data are stored in the R variable x, it has the form winmean(x,tr=0.2). The optional argument tr is the amount of Winsorizing, which defaults to 0.2 if unspecified. (The R function win also computes the Winsorized mean.) For example, the command winmean(dat) computes the 20% Winsorized mean for the data in the R vector dat. The command winmean(x,0.1) computes the 10% Winsorized mean. If there are any missing values (stored as NA in R), the function automatically removes them. 2 The function winvar computes the Winsorized sample variance, s w . It has the form winvar(x,tr=0.2). Again, tr is the amount of Winsorization which defaults to 0.2 if unspecified. The function trimse(x,tr=0.2) estimates the standard error of the trimmed mean and winse(x,tr=.2) estimates the standard error of the Winsorized mean. For example, the R command trimse(x,0.1) estimates the standard error of the 10% trimmed mean for the data stored in the vector x, and winvar(x,0.1) computes the Winsorized sample variance using 10% 2 Winsorization. The R command winvar(x) computes s w using 20% Winsorization. 3.3.5 Estimating the Standard Error of the Sample Median, M Trimmed means contain the usual sample median, M, as a special case where the maximum amount of trimming is used. When using M and the goal is to estimate its standard error, www.elsevierdirect.com