PERCENTAGE BEND MIDVARIANCEName:
Many statistics have one of these properties. However, it can be difficult to find statistics that are both resistant and have robustness of efficiency.
For scale estimaors, the variance is an optimal estimator for Gaussian data. However, it is not resistant and it does not have robustness of efficiency. The rationale for this estimate is given in these references.
The percentage bend midvariance estimator, discussed in Shoemaker and Hettmansperger and also by Wilcox, is both resistant and robust of efficiency.
The percentage bend midvariance of a a variable X is computed as follows:
The value of \( \beta \) is selected between 0 and 0.5. Higher values of \( \beta \) is selected result in a higher breakdown point at the expense of lower efficiency. Syntax:
where <y> is the response variable;
<par> is a parameter where the computed percentage bend midvariance is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET A = PERCENTAGE BEND MIDVARIANCE Y1 SUBSET TAG > 2
where <value> is greater than 0 and less than or equal to 0.5. The default value for \( \beta \) is 0.1.
Rand Wilcox (1997), "Introduction to Robust Estimation and Hypothesis Testing", Academic Press.
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 = PERCENTAGE BEND MIDVARIANCE Y1 LET A2 = PERCENTAGE BEND MIDVARIANCE Y2 LET A3 = PERCENTAGE BEND MIDVARIANCE Y3 LET A4 = PERCENTAGE BEND MIDVARIANCE 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 . BOOTSTRAP SAMPLES 500 BOOTSTRAP PERCENTAGE BEND MIDVARIANCE PLOT Y1 X1LABEL B025 = ^B025, B975=^B975 HISTOGRAM YPLOT X1LABEL . BOOTSTRAP PERCENTAGE BEND MIDVARIANCE PLOT Y1 X1LABEL B025 = ^B025, B975=^B975 HISTOGRAM YPLOT . END OF MULTIPLOT JUSTIFICATION CENTER MOVE 50 96 TEXT PERCENTAGE BEND MIDVARIANCE BOOTSTRAP: NORMAL MOVE 50 46 TEXT PERCENTAGE BEND MIDVARIANCE BOOTSTRAP: CAUCHY