Dataplot Vol 2 Vol 1

# HERPPF

Name:
HERPPF (LET)
Type:
Library Function
Purpose:
Compute the Hermite percent point 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:

Dataplot computes the Hermite percent point function by numerically inverting the Hermite cumulative distribution function. 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> = HERPPF(<p>,<alpha>,<beta>)             <SUBSET/EXCEPT/FOR qualification>
where <p> is a variable, number, or parameter in the range (0,1];
<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 ppf value is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
LET A = HERPPF(0.95,0.5,2)
LET X2 = HERPPF(P,ALPHA,BETA)
PLOT HERPPF(P,0.8,1.4) FOR P = 0 0.01 0.99
Default:
None
Synonyms:
None
Related Commands:
 HERCDF = Compute the Hermite cumulative distribution function. HERPDF = Compute the Hermite probability density 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.

Applications:
Distributional Modeling
Implementation Date:
2004/4
Program:
```
MULTIPLOT 2 2
MULTIPLOT CORNER COORDINATES 0 0 100 100
XTIC OFFSET 0.1 0.1
LINE BLANK
SPIKE ON
X1LABEL X
Y1LABEL PROBABILITY
X1LABEL DISPLACEMENT 12
Y1LABEL DISPLACEMENT 12
TITLE SIZE 3
TITLE HERPPF(P,0.5,2)
PLOT HERPPF(P,0.5,2) FOR P = 0 0.01 0.99
TITLE HERPPF(P,2,0.5)
PLOT HERPPF(P,2,0.5) FOR P = 0 0.01 0.99
TITLE HERPPF(P,0.5,0.5)
PLOT HERPPF(P,0.5,0.5) FOR P = 0 0.01 0.99
TITLE HERPPF(P,2,2)
PLOT HERPPF(P,2,2) FOR P = 0 0.01 0.99
END OF MULTIPLOT
```

Date created: 7/7/2004
Last updated: 7/7/2004