
GHPDFName:
with Z_{p} denoting the normal percent point function of p. When g = 0 and h = 0, the gandh distribution reduces to a standard normal distribution. The gandh probability density function is computed by taking the numerical derivative of the cumulative distribution function (which is turn computed by numerically inverting the percent point function using a bisection method). The value of g controls the degree of skewness. For g = 0, the distribution is symmetric. As the absolute value of g increases, the amount of the skewness increases. The sign of g controls the direction of the skewness (but not the amount). Positive values of g skew the distribution to the right tail while negative values of g skew the distribution to the left tail. Values for g are typically in the range (1,1). The value of h controls the elongation, or how heavy the tails are, of the distribution. For h = 0, the elongation is equivalent to that of a normal distribution. For h = 1, the elongation is equivalent to that of a Cauchy distribution. Values of h are typically in the range (0,1). Specifying values for both g and h gives this distribution great flexibility in modeling data.
<SUBSET/EXCEPT/FOR qualification> where <x> is a variable, number, or parameter; <g> is a number or parameter that specifies the skewness shape parameter; <h> is a number or parameter that specifies the elongation shape parameter; <y> is a variable or a parameter (depending on what <x> is) where the computed gandh pdf value is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional. Note that the location and scale parameters are optional.
LET Y = GHPDF(X,0.5,0.2) PLOT GHPDF(X,0.5,0.2) FOR X = 5 0.1 5
LET H = <value> LET Y = GH RANDOM NUMBERS FOR I = 1 1 N GH PROBABILITY PLOT Y GH KOLMOGOROV SMIRNOV GOODNESS OF FIT Y GH CHISQUARE GOODNESS OF FIT Y The following commands can be used to estimate the shape parameters for the gandh distribution:
LET G2 = <value> LET H1 = <value> LET H2 = <value> GH PPCC PLOT Y GH KS PLOT Y The default values for G1 and G2 are 1 and 1. The default values of H1 and H2 are H1 and H2.
MULTIPLOT 2 2 MULTIPLOT SCALE FACTOR 2 MULTIPLOT CORNER COORDINATES 0 0 100 95 Y1LABEL X X1LABEL PROBABILITY X1LABEL DISPLACEMENT 12 Y1LABEL DISPLACEMENT 12 TITLE G = 0.2, H = 0.2 PLOT GHPDF(X,0.5,0.5) FOR X = 10 0.1 10 TITLE G = 0.5, H = 0.2 PLOT GHPDF(X,0.5,0.2) FOR X = 10 0.1 10 TITLE G = 0.2, H = 0.5 PLOT GHPDF(X,0.2,0.5) FOR X = 10 0.1 10 TITLE G = 0.5, H = 0.5 PLOT GHPDF(X,0.5,0.5) FOR X = 10 0.1 10 END OF MULTIPLOT MOVE 50 97 JUSTIFICATION CENTER TEXT GANDH DISTRIBUTIONS
Date created: 7/7/2004 