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 < 11, Dataplot uses the above recurrence relation to compute the probabilities. For x > 10, 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 pdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET X2 = HERPDF(X1,ALPHA,BETA)
PLOT HERPDF(X,0.8,1.4) FOR X = 0 1 20
LET BETA = <value>
LET Y = HERMITE RANDOM NUMBERS FOR I = 1 1 N
HERMITE PROBABILITY PLOT Y
HERMITE CHI-SQUARE GOODNESS OF FIT Y
HERMITE PPCC PLOT Y
HERMITE MAXIMUM LIKELIHOOD Y
The HERMITE MAXIMUM LIKELIHOOD command estimates the parameters of the Hermite distribution using the following methods:
Patel discusses the various estimators for the Hermite distribution and gives some guidance on when the various estimators are most appropriate. He also gives formulas for the variances and covariances of the estimators. Default:
"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 X1LABEL X Y1LABEL PROBABILITY X1LABEL DISPLACEMENT 12 Y1LABEL DISPLACEMENT 12 TITLE SIZE 3 TITLE HERPDF(X,0.5,2) PLOT HERPDF(X,0.5,2) FOR X = 0 1 50 TITLE HERPDF(X,2,0.5) PLOT HERPDF(X,2,0.5) FOR X = 0 1 50 TITLE HERPDF(X,0.5,0.5) PLOT HERPDF(X,0.5,0.5) FOR X = 0 1 50 TITLE HERPDF(X,2,2) PLOT HERPDF(X,2,2) FOR X = 0 1 50 END OF MULTIPLOT
Date created: 5/4/2004