Dataplot Vol 2 Vol 1

# BIWEIGHT SCALE

Name:
BIWEIGHT SCALE (LET)
Type:
Let Subcommand
Purpose:
Compute a biweight based scale estimate for a variable.
Description:
Mosteller and Tukey (see Reference section below) define two types of robustness:

1. resistance means that changing a small part, even by a large amount, of the data does not cause a large change in the estimate

2. robustness of efficiency means that the statistic has high efficiency in a variety of situations rather than in any one situation. Efficiency means that the estimate is close to optimal estimate given that we distribution that the data comes from. A useful measure of efficiency is:

Efficiency = (lowest variance feasible)/ (actual variance)

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:

$$ns_{bi}^2 = \frac{n\sum_{i=1}^{n}{(y - y')^2(1 - u^2)^4}} {(\sum_{i=1}^{n}{(1 - u^2)(1 - 5u^2)})(-1 + \sum_{i=1}^{n}{(1 - u^2)(1 - 5u^2)})}$$

where the summation is restricted to $$u_{i}^2 \le 1$$ and

$$y' = \mbox{median } y$$

and

$$u_{i} = \frac{y_{i} - y'}{9*MAD} \hspace{0.5in} \mbox{for } (\frac{y_{i} - y*}{cS})^{2} < 1$$

where MAD is the median absolute deviation.

Syntax:
LET <par> = BIWEIGHT SCALE <y>
<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.
Examples:
LET A = BIWEIGHT SCALE Y1
LET A = BIWEIGHT SCALE Y1 SUBSET TAG > 2
Note:
Dataplot statistics can be used in a number of commands. For details, enter

Default:
None
Synonyms:
None
Related Commands:
 BIWEIGHT MIDVARIANCE = Compute a biweight midvariance estimate of a variable. BIWEIGHT LOCATION = Compute a biweight location estimate of a variable. BIWEIGHT MIDCOVARIANCE = Compute a biweight midcovariance estimate of two variables. BIWEIGHT MIDCORRELATION = Compute a biweight midcorrelation estimate of two variables. BIWEIGHT CONFIDENCE LIMITS = Compute a biweight based confidence interval. AVERAGE ABSOLUTE DEVIATION = Compute the average absolute deviation of a variable. MEDIAN ABSOLUTE DEVIATION = Compute the median absolute deviation of a variable. STANDARD DEVIATION = Compute the standard deviation of a variable. VARIANCE = Compute the variance of a variable. RANGE = Compute the range of a variable.
Reference:
Mosteller and Tukey (1977), "Data Analysis and Regression: A Second Course in Statistics," Addison-Wesley, pp. 203-209.
Applications:
Robust Data Analysis
Implementation Date:
2001/11
Program 1:

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
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 = 0.684497

Program 2:

SKIP 25
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
Last updated: 11/02/2015