Dataplot Vol 2 Vol 1

# HERCDF

Name:
HERCDF (LET)
Type:
Library Function
Purpose:
Compute the Hermite cumulative distribution function.
Description:
If X1 and X2 are independent Poisson random variables with shape parameters and (1/2)2, respectively, then X1 + 2X2 follows a Hermite distribution with shape parameters and .

Some sources in the literature use the parameterization

a = a1 =
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:

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.

Syntax:
LET <y> = HERCDF(<x>,<alpha>,<beta>)             <SUBSET/EXCEPT/FOR qualification>
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.
Examples:
LET A = HERCDF(3,0.5,2)
LET X2 = HERCDF(X1,ALPHA,BETA)
PLOT HERCDF(X,0.8,1.4) FOR X = 0 1 20
Default:
None
Synonyms:
None
Related Commands:
 HERPDF = Compute the Hermite probability density function. HERPPF = Compute the Hermite percent point function. POIPDF = Compute the Poisson cumulative distribution function. BINPDF = Compute the binomial probability density function. NBPDF = Compute the negative binomial probability density function. GEOPDF = Compute the geometric probability density function.
Reference:
"Discrete Univariate Distributions" Second Edition, Johnson, and Kotz, and Kemp, Wiley, 1992, pp. 357-364.

"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.

Applications:
Distributional Modeling
Implementation Date:
2004/4
Program:
```
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
Last updated: 7/7/2004
Please email comments on this WWW page to alan.heckert@nist.gov.