
BIWEIGHT LOCATIONName:
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 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 median is a resistant estimate, but it has only moderate robustness of efficiency. The biweight location estimator is both resistant and robust of efficiency. Mosteller and Tukey recommend using the median for exploratory work where moderate efficiency in a variety of situations is adequate and the biweight in situations when high performance is needed. The biweight location estimate is defined as:
where
\( w_{i} = 0 \hspace{0.5in} \mbox{otherwise} \) and \( S = \mbox{median}\{y_{i}  y*\} \) c = 6 (using 6 means that residuals up to approximately \( 4 \sigma \) are included) Note that this is an iterative estimate since y* depends on w_{i} and w_{i} depends on y*. Dataplot will compute up to 10 iterations (computation is terminated if the biweight location estimate does not change in value by more than 0.000001).
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 LOCATION 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 LOCATION Y1 LET A2 = BIWEIGHT LOCATION Y2 LET A3 = BIWEIGHT LOCATION Y3 LET A4 = BIWEIGHT LOCATION Y4 LET B1 = MEAN Y1 LET B2 = MEAN Y2 LET B3 = MEAN Y3 LET B4 = MEAN Y4 LET C1 = MEDIAN Y1 LET C2 = MEDIAN Y2 LET C3 = MEDIAN Y3 LET C4 = MEDIAN Y4 PRINT "BIWEIGHT LOCATION ESTIMATE FOR NORMAL RANDOM NUMBERS = ^A1" PRINT "MEAN ESTIMATE FOR NORMAL RANDOM NUMBERS = ^B1" PRINT "MEDIAN ESTIMATE FOR NORMAL RANDOM NUMBERS = ^C1" PRINT " " PRINT "BIWEIGHT LOCATION ESTIMATE FOR LOGISTIC RANDOM NUMBERS = ^A2" PRINT "MEAN ESTIMATE FOR LOGISTIC RANDOM NUMBERS = ^B2" PRINT "MEDIAN ESTIMATE FOR LOGISTIC RANDOM NUMBERS = ^C2" PRINT " " PRINT "BIWEIGHT LOCATION ESTIMATE FOR CAUCHY RANDOM NUMBERS = ^A3" PRINT "MEAN ESTIMATE FOR CAUCHY RANDOM NUMBERS = ^B3" PRINT "MEDIAN ESTIMATE FOR CAUCHY RANDOM NUMBERS = ^C3" PRINT " " PRINT "BIWEIGHT LOCATION ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = ^A4" PRINT "MEAN ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = ^B4" PRINT "MEDIAN ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = ^C4"Dataplot generates the following output: BIWEIGHT LOCATION ESTIMATE FOR NORMAL RANDOM NUMBERS = 0.001006 MEAN ESTIMATE FOR NORMAL RANDOM NUMBERS = 0.005167 MEDIAN ESTIMATE FOR NORMAL RANDOM NUMBERS = 0.01028 BIWEIGHT LOCATION ESTIMATE FOR LOGISTIC RANDOM NUMBERS = 0.0074 MEAN ESTIMATE FOR LOGISTIC RANDOM NUMBERS = 0.000867 MEDIAN ESTIMATE FOR LOGISTIC RANDOM NUMBERS = 0.016679 BIWEIGHT LOCATION ESTIMATE FOR CAUCHY RANDOM NUMBERS = 0.00439 MEAN ESTIMATE FOR CAUCHY RANDOM NUMBERS = 3.70155 MEDIAN ESTIMATE FOR CAUCHY RANDOM NUMBERS = 0.01582 BIWEIGHT LOCATION ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = 0.00203 MEAN ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = 0.00723 MEDIAN ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = 0.00557Program 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 LOCATION OF DIAMETER BIWEIGHT LOCATION 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 LOCATION PLOT Y X1LABEL B025 = ^B025, B975 = ^B975 TITLE BOOTSTRAP OF BIWEIGHT LOCATION: CAUCHY RANDOM NUMBERS HISTOGRAM YPLOT END OF MULTIPLOT
 
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Date created: 11/20/2001 