
BIWEIGHT SCALEName:
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 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 scale 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 scale estimator can be considered for situations where high performance is needed. The biweight scale 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 location is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET A = BIWEIGHT SCALE Y1 SUBSET TAG > 2
LET Y1 = NORMAL RANDOM NUMBERS FOR I = 1 1 10000 LET Y2 = LOGISTIC RANDOM NUMBERS FOR I = 1 1 10000 LET Y3 = CAUCHY RANDOM NUMBERS FOR I = 1 1 10000 LET Y4 = DOUBLE EXPONENTIAL RANDOM NUMBERS FOR I = 1 1 10000 LET A1 = BIWEIGHT SCALE Y1 LET A2 = BIWEIGHT SCALE Y2 LET A3 = BIWEIGHT SCALE Y3 LET A4 = BIWEIGHT SCALE Y4 LET B1 = STANDARD DEVIATION Y1 LET B2 = STANDARD DEVIATION Y2 LET B3 = STANDARD DEVIATION Y3 LET B4 = STANDARD DEVIATION Y4 LET C1 = MAD Y1 LET C2 = MAD Y2 LET C3 = MAD Y3 LET C4 = MAD Y4 PRINT "BIWEIGHT SCALE ESTIMATE FOR NORMAL RANDOM NUMBERS = ^A1" PRINT "STANDARD DEVIATION ESTIMATE FOR NORMAL RANDOM NUMBERS = ^B1" PRINT "MAD ESTIMATE FOR NORMAL RANDOM NUMBERS = ^C1" PRINT " " PRINT "BIWEIGHT SCALE ESTIMATE FOR LOGISTIC RANDOM NUMBERS = ^A2" PRINT "STANDARD DEVIATION ESTIMATE FOR LOGISTIC RANDOM NUMBERS = ^B2" PRINT "MAD ESTIMATE FOR LOGISTIC RANDOM NUMBERS = ^C2" PRINT " " PRINT "BIWEIGHT SCALE ESTIMATE FOR CAUCHY RANDOM NUMBERS = ^A3" PRINT "STANDARD DEVIATION ESTIMATE FOR CAUCHY RANDOM NUMBERS = ^B3" PRINT "MAD ESTIMATE FOR CAUCHY RANDOM NUMBERS = ^C3" PRINT " " PRINT "BIWEIGHT SCALE ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = ^A4" PRINT "STANDARD DEVIATION ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = ^B4" PRINT "MAD ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = ^C4"Dataplot generates the following output: BIWEIGHT SCALE ESTIMATE FOR NORMAL RANDOM NUMBERS = 1.016386 STANDARD DEVIATION ESTIMATE FOR NORMAL RANDOM NUMBERS = 0.9975 MAD ESTIMATE FOR NORMAL RANDOM NUMBERS = 0.681249 BIWEIGHT SCALE ESTIMATE FOR LOGISTIC RANDOM NUMBERS = 3.066369 STANDARD DEVIATION ESTIMATE FOR LOGISTIC RANDOM NUMBERS = 1.817945 MAD ESTIMATE FOR LOGISTIC RANDOM NUMBERS = 1.116496 BIWEIGHT SCALE ESTIMATE FOR CAUCHY RANDOM NUMBERS = 3.480419 STANDARD DEVIATION ESTIMATE FOR CAUCHY RANDOM NUMBERS = 998.389 MAD ESTIMATE FOR CAUCHY RANDOM NUMBERS = 1.015878 BIWEIGHT SCALE ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = 1.529625 STANDARD DEVIATION ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = 1.424258 MAD ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = 0.684497Program 2: SKIP 25 READ GEAR.DAT DIAMETER BATCH TITLE AUTOMATIC XLIMITS 1 10 MAJOR XTIC MARK NUMBER 10 MINOR XTIC MARK NUMBER 0 XTIC OFFSET 1 1 X1LABEL BATCH Y1LABEL BIWEIGHT SCALE OF DIAMETER BIWEIGHT SCALE PLOT DIAMETER BATCH Program 3: MULTIPLOT 2 1 MULTIPLOT CORNER COORDINATES 0 0 100 100 LET Y = CAUCHY RANDOM NUMBERS FOR I = 1 1 1000 TITLE AUTOMATIC BOOTSTRAP BIWEIGHT SCALE PLOT Y X1LABEL B025 = ^B025, B975 = ^B975 TITLE BOOTSTRAP OF BIWEIGHT SCALE: CAUCHY RANDOM NUMBERS HISTOGRAM YPLOT END OF MULTIPLOT
 
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Date created: 11/20/2001 