TRIMMED MEAN STANDARD ERRORName:
Mosteller and Tukey (see Reference section below) define two types of robustness:
For location estimaors, the mean is the optimal estimator for Gaussian data. However, it is not resistant and it does not have robustness of efficiency. The trimmed mean estimator is both resistant and robust of efficiency.
The standard error of the trimmed mean can be used to estimate the uncertainty of the trimmed mean estimate (and to create confidence intervals). The trimmed mean standard error is defined as:
where sw is the Winsorized standard deviation (enter HELP WINSORIZED STANDARD DEVIATION for details), \( \gamma_1 \) is the lower trimming fraction, \( \gamma_2 \) is the upper trimming fraction, and n is the sample size.
Tukey and Mclaughlin suggest the following confidence interval for the trimmed mean:
where alpha is the desired significance level, t is the student t-distribution, and \( g = [\gamma n] \) (the integer portion of the trimming fraction times the sample size). Note that we are assuming equal trimming on both tails (\( \gamma \) = .10 means we trim 10% on both tails).
An alternative method for confidence intervals is to use the BOOTSTRAP TRIMMED MEAN PLOT command and use appropriate percentiles of the generated bootstrap trimmed mean values. Wilcox suggests a refinement of the standard bootstrap, which he calls he percentile t bootstrap, which has better performance than the standard bootstrap. Dataplot does not currently support this refinement. Syntax:
where <y1> is the response variable;
<par> is a parameter where the computed trimmed mean standard error is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET A = TRIMMED MEAN STANDARD ERROR Y1 SUBSET TAG > 2
LET P2 = 10
LET A = TRIMMED MEAN STANDARD ERROR Y
Tukey and McLaughlin, "Less Vunerable Confidence and Significance Procedures for Location Based on a Single Sample: Trimming/Winsorization", Sankhya A, 25, pp. 331-352.
Mosteller and Tukey (1977), "Data Analysis and Regression: A Second Course in Statistics", Addison-Wesley, pp. 203-209.
LET Y1 = NORMAL RANDOM NUMBERS FOR I = 1 1 100 LET Y2 = LOGISTIC RANDOM NUMBERS FOR I = 1 1 100 LET Y3 = CAUCHY RANDOM NUMBERS FOR I = 1 1 100 LET Y4 = DOUBLE EXPONENTIAL RANDOM NUMBERS FOR I = 1 1 100 LET A1 = TRIMMED MEAN STANDARD ERROR Y1 LET A2 = TRIMMED MEAN STANDARD ERROR Y2 LET A3 = TRIMMED MEAN STANDARD ERROR Y3 LET A4 = TRIMMED MEAN STANDARD ERROR Y4Program 2:
MULTIPLOT 2 2 MULTIPLOT CORNER COORDINATES 0 0 100 100 MULTIPLOT SCALE FACTOR 2 X1LABEL DISPLACEMENT 12 . LET Y1 = NORMAL RANDOM NUMBERS FOR I = 1 1 200 LET Y2 = CAUCHY RANDOM NUMBERS FOR I = 1 1 200 LET P1 = 10 LET P2 = 10 . BOOTSTRAP SAMPLES 500 BOOTSTRAP TRIMMED MEAN STANDARD ERROR PLOT Y1 X1LABEL B025 = ^B025, B975=^B975 HISTOGRAM YPLOT X1LABEL . BOOTSTRAP BIWEIGHT MIDVARIANCE PLOT Y1 X1LABEL B025 = ^B025, B975=^B975 HISTOGRAM YPLOT . END OF MULTIPLOT . JUSTIFICATION CENTER MOVE 50 46 TEXT TRIMMED MEAN SE BOOTSTRAP: CAUCHY MOVE 50 96 TEXT TRIMMED MEAN SE BOOTSTRAP: NORMAL