
HERCDFName:
Some sources in the literature use the parameterization
b = a_{2} = 0.5^{2} The shape parameters and can be expressed in terms of a_{1} and a_{2} as
The probability mass function for the Hermite distribution is:
where
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 nonnegative 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 StatisticsTheory and Methods, 14, pp. 22332241. "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. 865873.
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 