Some sources in the literature use the parameterization
b = a2 = 0.52
The shape parameters and can be expressed in terms of a1 and a2 as
The probability mass function for the Hermite distribution is:
with denoting the modified Hermite polynomial:
with [n/2] denoting the integer part of (n/2).
The first few terms of the Hermite probability mass function are:
A general recuurence relation is:
For x < 26, Dataplot uses the above recurrence relation to compute the probabilities. For x > 25, Dataplot uses an asymptotic formula due to Patel (see Reference section below) to compute the probabilities.
where <x> is a non-negative integer variable, number, or parameter;
<alpha> is a number or parameter that specifies the first shape parameter;
<beta> is a number or parameter that specifies the second shape parameter;
<y> is a variable or a parameter (depending on what <x> is) where the computed Hermite cdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET X2 = HERCDF(X1,ALPHA,BETA)
PLOT HERCDF(X,0.8,1.4) FOR X = 0 1 20
"An Asymptotic Expression for Cumulative Sum of Probabilities of the Hermite Distribution", Y. C. Patel, Communications in Statistics--Theory and Methods, 14, pp. 2233-2241.
"Some Properties of the Hermite Distribution", Kemp and Kemp, Biometrika (1965), 52, 3 and 4, P. 381.
"Even Point Estimation and Moment Estimation in Hermite Distributions", Y. C. Patel, Biometrics, 32, December, 1976, pp. 865-873.
MULTIPLOT 2 2 MULTIPLOT CORNER COORDINATES 0 0 100 100 XTIC OFFSET 0.5 0.5 LINE BLANK SPIKE ON TITLE AUTOMATIC X1LABEL X Y1LABEL PROBABILITY X1LABEL DISPLACEMENT 12 Y1LABEL DISPLACEMENT 12 TITLE SIZE 3 PLOT HERCDF(X,0.5,2) FOR X = 0 1 50 PLOT HERCDF(X,2,0.5) FOR X = 0 1 50 PLOT HERCDF(X,0.5,0.5) FOR X = 0 1 50 PLOT HERCDF(X,2,2) FOR X = 0 1 50 END OF MULTIPLOT
Date created: 7/7/2004