LSNPDF

LSNPDF (LET)

Library Function

Compute the log-skew-normal probability density function.

The log-skew-normal distribution can be defined in terms of the skew-normal distribution as follows:

with denoting the skewness parameter, sd denoting the standard deviation of the corresponding normal distribution, and denoting the probability density of the skew normal distribution.

This is analogous to how the lognormal distribution is defined in terms of the normal distribution. If = 0, the log-skew-normal distribution reduces to the lognormal distributiion.

The standard log-skew-normal distribution can be generalized with location and scale parameters in the usual way.

LET <y> = LSNPDF(<x>,<lambda>,<sd>,<loc>,<scale>)
                        <SUBSET/EXCEPT/FOR qualification>
where <x> is a variable or a parameter;
            <lambda> is a number of parameter that specifies the value of the skewness shape parameter;
            <sd> is a number of parameter that specifies the value of the sd shape parameter;
            <loc> is a number of parameter that specifies the value of the location parameter;
            <scale> is a number of parameter that specifies the value of the scale parameter;
            <y> is a variable or a parameter (depending on what <x> is) where the computed log-skew-normal pdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

Note that the location and scale parameters are optional.

LET A = LSNPDF(3,1,1)
LET A = LSNPDF(A1,LAMBDA,SD)
LET X2 = LSNPDF(X1,0.5,1.5)

Log-skew-normal random numbers, probability plots, and goodness of fit tests can be generated with the commands:
LET LAMBDA = <value>
LET SD = <value>
LET Y = LOG SKEW NORMAL RANDOM NUMBERS FOR I = 1 1 N
LOG SKEW NORMAL PROBABILITY PLOT Y
LOG SKEW NORMAL KOLMOGOROV SMIRNOV GOODNESS OF FIT Y
LOG SKEW NORMAL CHI-SQUARE GOODNESS OF FIT Y


The following commands can be used to estimate the shape parameters for the log-skew-normal distribution:

LET LAMBDA1 = <value>
LET LAMBDA2 = <value>
LET SD1 = <value>
LET SD2 = <value>
LOG SKEW NORMAL PPCC PLOT Y
LOG SKEW NORMAL KS PLOT Y

None

None

LSNCDF	= Compute the log-skew-normal cumulative distribution function.
LSNPPF	= Compute the log-skew-normal percent point function.
SNPDF	= Compute the skew-normal probability density function.
NORPDF	= Compute the normal density function.
LGNPDF	= Compute the lognormal density function.
CHIPDF	= Compute the chi probability density function.
CHSPDF	= Compute the chi-square probability density function.
WEIPDF	= Compute the Weibull probability density function.

"Log-Skew-Normal and Log-Skew-t Distributions as Models for Family Income Data", Azzalini, Dal Cappello, and Kotz, Journal of Income Distribution, Vol. 11, No. 3-4, 2003, pp. 12-20.

"A Class of Distributions Which Includes the Normal Ones", Azzalini, Scandinavian Journal of Statistics, 12, 171-178.

"Continuous Univariate Distributions: Volume I", Second Edition, Johnson, Kotz, and Balakrishnan, Wiley, 1994, p. 454.

Distributional Modeling

3/2004

 
Y1LABEL Probability
X1LABEL X
LABEL CASE ASIS
X1LABEL DISPLACEMENT 12
Y1LABEL DISPLACEMENT 12
MULTIPLOT 2 2
MULTIPLOT CORNER COORDINATES 0 0 100 100
TITLE LOG-SKEW-NORMAL: LAMBDA = 0
PLOT LSNPDF(X,0) FOR X = 0.01  0.01  5
TITLE LOG-SKEW-NORMAL: LAMBDA = 1
PLOT LSNPDF(X,1) FOR X = 0.01  0.01  5
TITLE LOG-SKEW-NORMAL: LAMBDA = 5
PLOT LSNPDF(X,5) FOR X = 0.01  0.01  5
TITLE LOG-SKEW-NORMAL: LAMBDA = 10
PLOT LSNPDF(X,10) FOR X = 0.01  0.01  5
END OF MULTIPLOT