with Zp denoting the normal percent point function of p. When g = 0 and h = 0, the g-and-h distribution reduces to a standard normal distribution.
The g-and-h 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.
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 g-and-h 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 CHI-SQUARE GOODNESS OF FIT Y
The following commands can be used to estimate the shape parameters for the g-and-h 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 G-AND-H DISTRIBUTIONS
Date created: 7/7/2004