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BNOPDFName:
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 - )/ .
<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 X2 = BNOPDF(X1,0.1,0.1) PLOT BNOPDF(X,0.1,0.1) FOR X = -10 0.01 10
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 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.
Eugene, Lee, and Famoye (2002). "Beta-Normal Distribution and Its Applications", Communications in Statistics-Theory and Methods, 31, pp. 497-512.
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
Date created: 3/27/2006 |