BNOPDF

BNOPDF (LET)

Library Function

Compute the beta-normal probability density function with shape parameters and .

There are a class of distributions defined by the following cumulative distribution function:

with G(x) denoting a cumulative distribution function. That is, this is the beta cumulative distribution function, but the upper limit of integration is defined by another cumulative distribution function.

The case where G(x) denotes the standard normal cumulative distribution function results in the beta-normal distribution. The probability density function for this distribution is:

with , , B, , and denoting the two shape parameters, the Beta function, the standard normal cumulative distribution function, and the standard normal probability density function, respectively.

This distribution can be extended with location and scale parameters by replacing the standard normal distribution with a normal distribution with location parameter, , and scale parameter, . The and are also the location and scale parameter of the beta-normal distribution. The above formula for the pdf can be modified by replacing x with (x - )/ .

LET <y> = BNOPDF(<x>,<alpha>,<beta>,<loc>,<scale>)
                        <SUBSET/EXCEPT/FOR qualification>
where <x> is a number, parameter, or variable;
            <y> is a variable or a parameter (depending on what <x> is) where the computed beta-normal pdf value is stored;
            <alpha> is a number, parameter, or variable that specifies the first shape parameter;
            <beta> is a number, parameter, or variable that specifies the second shape parameter;
            <loc> is a number, parameter, or variable that specifies the location parameter;
            <scale> is a number, parameter, or variable that specifies the scale parameter;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

The <loc> and <scale> parameters are optional.

LET A = BNOPDF(0.3,0.1,0.1)
LET X2 = BNOPDF(X1,0.1,0.1)
PLOT BNOPDF(X,0.1,0.1) FOR X = -10 0.01 10

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


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

LET ALPHA1 = <value>
LET ALPHA2 = <value>
LET BETA1 = <value>
LET BETA2 = <value>
BETA NORMAL PPCC PLOT Y
BETA NORMAL KS PLOT Y


The default values for ALPHA1 and ALPHA2 are 0.5 and 5. The default values for BETA1 and BETA2 are 0.5 and 5.

None

None

BNOCDF	= Compute the beta-normal cumulative distribution function.
BNOPPF	= Compute the beta-normal percent point function.
BETPDF	= Compute the beta probability density function.
NORPDF	= Compute the normal probability density function.
GAMPDF	= Compute the gamma probability density function.

"Handbook of Beta Distribution and Its Applications", Edited by Arjun Gupta and Saralees Nadarajah, Marcel Dekker Inc., 2004, pp. 146-152.

Eugene, Lee, and Famoye (2002). "Beta-Normal Distribution and Its Applications", Communications in Statistics-Theory and Methods, 31, pp. 497-512.

Distributional Modeling

2006/3

 
MULTIPLOT 3 3
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT SCALE FACTOR 3
LABEL CASE ASIS
X1LABEL X
Y1LABEL Probability Density
Y1TIC MARK LABEL DECIMAL 2
X1LABEL DISPLACEMENT 14
Y1LABEL DISPLACEMENT 15
TITLE DISPLACEMENT 2
.
LET A = DATA 0.1  1  5
LET B = DATA 0.1  1  5
LOOP FOR K = 1 1 3
    LET ALPHA = A(K)
    LOOP FOR L = 1 1 3
        LET BETA = B(L)
        TITLE ALPHA = ^ALPHA, BETA = ^BETA
        PLOT BNOPDF(X,ALPHA,BETA) FOR X = -10  0.01  10
    END OF LOOP
END OF LOOP
.
END OF MULTIPLOT
JUSTIFICATION CENTER
MOVE 50 97
CASE ASIS
TEXT PDF's For Beta-Normal Distribution