
BIWEIGHT MIDVARIANCEName:
For scale estimaors, the standard deviation (or variance) is the optimal estimator for Gaussian data. However, it is not resistant and it does not have robustness of efficiency. The median absolute deviation (MAD) is a resistant estimate, but it has only modest robustness of efficiency. The biweight midvariance estimator is both resistant and robust of efficiency. Mosteller and Tukey recommend using the MAD or interquartile range for exploratory work where moderate efficiency in a variety of situations is adequate. The biweight midvariance estimator can be considered for situations where high performance is needed. The biweight midvariance estimate is defined as:
where the summation is restricted to \( u_{i}^2 \le 1 \) and
and
where MAD is the median absolute deviation.
<SUBSET/EXCEPT/FOR qualification> where <y> is the response variable; <par> is a parameter where the computed biweight midvariance is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET A = BIWEIGHT MIDVARIANCE Y1 SUBSET TAG > 2
Mosteller and Tukey (1977), "Data Analysis and Regression: A Second Course in Statistics," AddisonWesley, pp. 203209.
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 = BIWEIGHT MIDVARIANCE Y1 LET A2 = BIWEIGHT MIDVARIANCE Y2 LET A3 = BIWEIGHT MIDVARIANCE Y3 LET A4 = BIWEIGHT MIDVARIANCE Y4Program 2: MULTIPLOT 2 2 MULTIPLOT CORNER COORDINATES 0 0 100 MULTIPLOT SCALE FACTOR 2 . LET Y1 = NORMAL RANDOM NUNBERS FOR I = 1 1 200 LET Y2 = CAUCHY RANDOM NUNBERS FOR I = 1 1 200 . BOOTSTRAP SAMPLES 500 TITLE BIWEIGHT MIDVARIANCE BOOTSTRAP: CAUCHY BOOTSTRAP BIWEIGHT MIDVARIANCE PLOT Y1 X1LABEL B025 = ^B025, B975=^B975 TITLE HISTOGRAM YPLOT . TITLE BIWEIGHT MIDVARIANCE BOOTSTRAP: NORMAL BOOTSTRAP BIWEIGHT MIDVARIANCE PLOT Y1 X1LABEL B025 = ^B025, B975=^B975 TITLE HISTOGRAM YPLOT . END OF MULTIPLOT  
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Date created: 07/26/2002 