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:
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.
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.
LET X2 = HERPPF(P,ALPHA,BETA)
PLOT HERPPF(P,0.8,1.4) FOR P = 0 0.01 0.99
"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.
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