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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 alpha beta and (1/2)alpha2, respectively, then X1 + 2X2 follows a Hermite distribution with shape parameters alpha and beta.

    Some sources in the literature use the parameterization

      a = a1 = alpha beta
      b = a2 = 0.5alpha2

    The shape parameters alpha and beta can be expressed in terms of a1 and a2 as

      alpha = SQRT(2*b)

      beta = a/SQRT(2*b)

    The probability mass function for the Hermite distribution is:

      p(x,alpha,beta) = (alpha**x*H*(x)(beta)/x!)*p(x=0,alpha,beta) 
x = 0, 1, 2, ...; alpha, beta > 0

      where

      p[x=0] = EXP[-alpha*beta - alpha**2/2]

    with H*(x)(beta) denoting the modified Hermite polynomial:

      H*(x)(beta) = SUM[j=0 to INT(N/2)][N!*X**(N-2*j)/((N-2(j)!j!2**j)]

    with [n/2] denoting the integer part of (n/2).

    The first few terms of the Hermite probability mass function are:

      p[x=0] = EXP[-alpha*beta - alpha**2/2]

      p[X=1] = alpha*beta*p[X=0]

      p[X=2] = (alpha**2*(beta**2+1)/2!)*p[X=0]

      p[X=3] = (alpha**3*(beta**3+3*beta)/3!)*p[X=0]

      p[X=4] = (alpha**4*(beta**4+6*beta**2+3)/4!)*p[X=0]

      p[X=5] = (alpha**5*(beta**5+10*beta**3+15*beta)/5!)*p[X=0]

    A general recuurence relation is:

      p[X=x+1] = (1/(x+1))*(alpha*beta*p[X=x] + alpha**2*p[X=x-1])

    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
        
    plot generated by sample program

Date created: 7/7/2004
Last updated: 7/7/2004
Please email comments on this WWW page to alan.heckert@nist.gov.