![]() |
LP LOCATIONName:
The univariate measurement model (or location model) is
![]()
where The least power (Lp) provides a broad class of location estimators. This class includes the mean, the median, and the mid-range as special cases. The Lp norm (for p >= 1) is defined as
![]()
For p = 1, 2, and
![]() ![]() ![]()
The Lp norm estimation is based on the minimization of the
Lp norm of a suitable residual vector. Specifically, the
Lp estimator of
![]()
where arg min means the argument of the minimum. That is, the value
of The Lp estimate is the solution of the equation
![]() The special cases mentioned above correspond to
Values of p between 1 and 2 are of most interest as these have efficiency and robustness properties between the median (p = 1) and the mean (p = 2).
The Pennecchi and Callegaro paper provides the following guidelines
for choosing a suitable value for p. Compute the sample
kurtosis,
Pennecchi and Callegaro propose the following as an estimate of the asymptotic variance
![]() where
![]()
<SUBSET/EXCEPT/FOR qualification> where <y> is the response variable; <par> is a parameter where the computed lp location value is saved; and where the <SUBSET/EXCEPT/FOR qualification> is optional. Use this syntax to compute the Lp location estimate.
<SUBSET/EXCEPT/FOR qualification> where <y> is the response variable; <par> is a parameter where the computed variance of the lp location value is saved; and where the <SUBSET/EXCEPT/FOR qualification> is optional. Use this syntax to compute the variance of the Lp location estimate.
<SUBSET/EXCEPT/FOR qualification> where <y> is the response variable; <par> is a parameter where the computed sd of the lp location value is saved; and where the <SUBSET/EXCEPT/FOR qualification> is optional. Use this syntax to compute the standard deviation of the Lp location estimate.
LET ALOC = LP LOCATION Y LET AVAR = LP VARIANCE Y LET ASD = LP SD Y LET ALOC = LP LOCATION Y SUBSET Y > 0
SD LP LOCATION is a synonym for SD OF LP LOCATION VARIANCE LP LOCATION is a synonym for VARIANCE OF LP LOCATION
LET Y1 = NORMAL RANDOM NUMBERS FOR I = 1 1 50 LET Y2 = LAPLACE RANDOM NUMBERS FOR I = 1 1 50 LET Y3 = UNIFORM RANDOM NUMBERS FOR I = 1 1 50 LET Y4 = SLASH RANDOM NUMBERS FOR I = 1 1 50 LET Y X = STACKED Y1 Y2 Y3 Y4 . MULTIPLOT SCALE FACTOR 2 MULTIPLOT CORNER COORDINATES 5 5 95 95 LABEL CASE ASIS TIC MARK LABEL CASE ASIS TITLE CASE ASIS TITLE OFFSET 2 Y1LABEL DISPLACEMENT 15 XLIMITS 1 4 MAJOR XTIC MARK NUMBER 4 MINOR XTIC MARK NUMBER 0 X1TIC MARK LABEL FORMAT ALPHA X1TIC MARK LABEL CONTENT Normal Laplace Uniform Slash TIC MARK OFFSET UNITS DATA X1TIC MARK OFFSET 0.5 0.5 CHARACTER X BLANK LINE BLANK SOLID . MULTIPLOT 2 2 LET P = 1 Y1LABEL L(1) Location LP LOCATION PLOT Y X LET P = 1.5 Y1LABEL L(1.5) Location LP LOCATION PLOT Y X LET P = 2 Y1LABEL L(2) Location LP LOCATION PLOT Y X LET P = 100 Y1LABEL L(100) Location LP LOCATION PLOT Y X END OF MULTIPLOT . SET WRITE DECIMALS 4 SET LET CROSS TABULATE COLLAPSE LET P = 1.5 LET XGROUP = CROSS TABULATE GROUP ONE X LET YMEAN = CROSS TABULATE LP LOCATION Y X LET YSD = CROSS TABULATE SD OF LP LOCATION Y X PRINT XGROUP YMEAN YSDThe following output is generated
Date created: 07/14/2011 |