PERCENTAGE BEND MIDVARIANCE
Name:
PERCENTAGE BEND MIDVARIANCE (LET)
Type:
Purpose:
Compute the percentage bend midvariance for a variable.
Description:
Mosteller and Tukey (see Reference section below) define
two types of robustness:
- resistance means that changing a small part, even by a
large amount, of the data does not cause a large change
in the estimate
- 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 know what
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 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:
- Set m = [(1 -
n
+ 0.5]. This is the value of (1 -
n
+ 0.5 rounded down to the nearest integer.
- Let Wi = |Xi - M|
for i = 1, ..., n where M is the
median of X.
- Sort the Wi in ascending order.
= Wm (i. e., the mth order
statistic). Wm is the estimate of the
(1 -
)
quantile of W.
The value of
is selected
between 0 and 0.5. Higher values of
is selected
result in a higher breakdown point at the expense of lower
efficiency.
Syntax:
LET <par> = PERCENTAGE BEND MIDVARIANCE <y>
<SUBSET/EXCEPT/FOR qualification>
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.
Examples:
LET A = PERCENTAGE BEND MIDVARIANCE Y1
LET A = PERCENTAGE BEND MIDVARIANCE Y1 SUBSET TAG > 2
Note:
Note:
Support for the percentage bend midvariance has been added to the
following plots and commands:
PERCENTAGE BEND MIDVARIANCE PLOT
CROSS TABULATE PERCENTAGE BEND MIDVARIANCE PLOT
BOOTSTRAP PERCENTAGE BEND MIDVARIANCE PLOT
JACKNIFE PERCENTAGE BEND MIDVARIANCE PLOT
DEX PERCENTAGE BEND MIDVARIANCE PLOT
PERCENTAGE BEND MIDVARIANCE INFLUENCE CURVE
PERCENTAGE BEND MIDVARIANCE INTERACTION STATISTIC PLOT
Default:
Synonyms:
Related Commands:
BIWEIGHT MIDVARIANCE
|
= Compute a biweight midvariance estimate of a variable.
|
BIWEIGHT SCALE
|
= Compute a biweight scale estimate of a variable.
|
BIWEIGHT LOCATION
|
= Compute a biweight location estimate of a variable.
|
BIWEIGHT CONFIDENCE LIMITS
|
= Compute a biweight based confidence interval.
|
WINSORIZED VARIANCE
|
= Compute the Winsorized variance of a variable.
|
MEDIAN ABSOLUTE DEVIATION
|
= Compute the median absolute deviation of a variable.
|
VARIANCE
|
= Compute the variance of a variable.
|
STATISTIC PLOT
|
= Generate a statistic versus group plot for a given
statistic.
|
CROSS TABULATE PLOT
|
= Generate a statistic versus group plot for a given
statistic and two group variables.
|
BOOTSTRAP PLOTi
|
= Generate a bootstrap plot for a given statistic.
|
DEX PLOT
|
= Generate various types of design of experiment plots.
|
INFLUENCE CURVE
|
= Generate an influence curve for a given statistic.
|
INTERACTION STATISTIC PLOT
|
= Generate an interaction plot for a given statistic.
|
References:
"Robust Estimates of and Tests for the One- and Two-Sample
Scale Models", Shoemaker and Hettmansperger, Biometrika 69,
1982, pp. 47-54.
"Introduction to Robust Estimation and Hypothesis Testing",
Rand Wilcox, Academic Press, 1997.
"Data Analysis and Regression: A Second Course in Statistics",
Mosteller and Tukey, Addison-Wesley, 1977, pp. 203-209.
Applications:
Implementation Date:
Program 1:
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 Y4
Program 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