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 Hodge-Lehmann location estimate is based on ranks. This makes it more resistant, as defined above, than the mean. This estimator also has high efficiency for symmetric disributions. It may be less successful with some skewed distributions.
Specifically, the Hodges-Lehmann estimate for location is defined as
Dataplot uses ACM algorithm 616 (HLQEST written by John Monohan) to compute the estimate. This is a fast, exact algoirthm. One modification is that for n <= 25 Dataplot computes the estimate directly from the definition. Syntax:
where <y1> is the response variable;
<par> is a parameter where the computed Hodges-Lehmann location estimate is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET A = HODGES LEHMANN Y1 SUBSET TAG > 2
Rand Wilcox (1997), "Introduction to Robust Estimation and Hypothesis Testing," Academic Press.
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 = HODGES LEHMANN Y1 LET A2 = HODGES LEHMANN Y2 LET A3 = HODGES LEHMANN Y3 LET A4 = HODGES LEHMANN Y4Program 2:
MULTIPLOT 2 2 MULTIPLOT CORNER COORDINATES 0 0 100 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 . BOOTSTRAP SAMPLES 500 BOOTSTRAP HODGES LEHMANN 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 HODGES LEHMANN BOOTSTRAP: CAUCHY MOVE 50 96 TEXT HODGES LEHMANN BOOTSTRAP: NORMAL