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Dataplot Vol 2 Vol 1

MAKHAZ

Name:
    MAKHAZ (LET)
Type:
    Library Function
Purpose:
    Compute the Gompertz-Makeham hazard function.
Description:
    The Gompertz-Makeham distribution is effectively a smallest extreme value distribution truncated at zero.

    There are a number of parameterizations of the Gompertz-Makeham distribution in the literature. Dataplot supports several different parameterizations.

    1. The Digital Library of Mathematical Functions (DLMF) uses the following parameterization of the hazard function:

        h(x;xi,lambda,theta) = xi*theta*lambda + xi*lambda*EXP(lambda*x) x > 0; lambda, xi, theta > 0

      Note that this definition uses three shape parameters.

    2. Meeker and Escobar use a parameterization based on the parameters gamma, k, and lambda. This can be translated to the DLMF definition as follows:

        xi(DLMF) = gamma(Meeker)/k(Meeker)

        lambda(DLMF) = k(Meeker)

        theta(DLMF) = lambda(Meeker)/gamma(Meeker)

      After making the above substitutions, it is equivalent to the DLMF definition. The hazard function can be written as

        h(x;gamma,k,lambda) = lambda + gamma*EXP(k*x) x > 0; gamma, k > 0, lambda >= 0

      Note that Meeker and Escobar parameterization can be expressed in terms of the DLMF parameterization:

        lambda(Meeker) = theta(DLMF)*lambda(DLMF)*xi(DLMF)

        gamma(Meeker) = lambda(DLMF)*xi(DLMF)

        k(Meeker) = lambda(DLMF)

    3. Given the three shape parameters definition of Meeker and Escobar, they reparameterize the distribution in the following way:

        theta = (1/k)

        eta = LOG(k/gamma)

        zeta = lambda/k

      An attractive feature of this parameterization is that it reduces the three shape parameters to two shape parameters (eta and zeta) and a scale parameter (theta). The hazard function is:

        h(x;zeta,eta) = eta + EXP(-zeta)*EXP(x) x > 0; eta >= 0

    The default parameterization is the Meeker and Escobar definition with two shape parameters.

    To specify the DLMF parameterization, enter the command

      SET GOMPERTZ MAKEHAM DEFINITION DLMF

    To specify the Meeker and Escobar parameterization with three shape parameters, enter the command

      SET GOMPERTZ MAKEHAM DEFINITION MEEKER

    To reset the default Meeker and Escobar definition with two shape parameters, enter

      SET GOMPERTZ MAKEHAM DEFINITION REPARAMETERIZED MEEKER

    The Gompertz-Makeham distribution can be generalized with location and scale parameters in the usual way. Simply replace x with (x-loc)/scale in the above equations.

Syntax 1:
    LET <y> = MAKHAZ(<x>,<eta>,<zeta>,<loc>,<scale>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a number, parameter, or variable;
                <eta> is a number, parameter, or variable that specifies the first shape parameter;
                <zeta> is a non-negative number, parameter, or variable that specifies the second shape parameter;
                <loc> is a positive number, parameter, or variable that specifies the location parameter;
                <scale> is a positive number, parameter, or variable that specifies the scale parameter;
                <y> is a variable or a parameter (depending on what <x> is) where the computed Gompertz-Makeham hazard value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    Note that the location and scale parameters are optional.

    This syntax implements the definition as given by Meeker and Escobar that reparameterizes the distribution to have only two shape parameters.

Syntax 2:
    LET <y> = MAKHAZ(<x>,<xi>,<lambda>,<theta>,<loc>,<scale>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a number, parameter, or variable;
                <xi> is a positive number, parameter, or variable that specifies the XI shape parameter;
                <lambda> is a positive number, parameter, or variable that specifies the LAMBDA shape parameter;
                <theta> is a positive number, parameter, or variable that specifies the THETA shape parameter;
                <loc> is a positive number, parameter, or variable that specifies the location parameter;
                <scale> is a positive number, parameter, or variable that specifies the scale parameter;
                <y> is a variable or a parameter (depending on what <x> is) where the computed Gompertz-Makeham hazard value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    Note that the location and scale parameters are optional.

    This syntax implements the DLMF definition.

Syntax 3:
    LET <y> = MAKHAZ(<x>,<gamma>,<k>,<lambda>,<loc>,<scale>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a number, parameter, or variable;
                <gamma> is a positive number, parameter, or variable that specifies the gamma shape parameter;
                <k> is a positive number, parameter, or variable that specifies the k shape parameter;
                <lambda> is a positive number, parameter, or variable that specifies the lambda shape parameter;
                <loc> is a positive number, parameter, or variable that specifies the location parameter;
                <scale> is a positive number, parameter, or variable that specifies the scale parameter;
                <y> is a variable or a parameter (depending on what <x> is) where the computed Gompertz-Makeham hazard value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    Note that the location and scale parameters are optional.

    This syntax implements the definition for three shape parameters as given by Meeker and Escobar.

Examples:
    SET GOMPERTZ MAKEHAM DEFINITION REPARAMETERIZED MEEKER
    LET A = MAKHAZ(0.3,0.5,2)
    LET A = MAKHAZ(X,ETA,ZETA)
    PLOT MAKHAZ(X,ETA,ZETA) FOR X = 0.01 0.01 5

    SET GOMPERTZ MAKEHAM DEFINITION DLMF
    LET A = MAKHAZ(0.3,0.5,2,1.4)
    LET A = MAKHAZ(X,XI,LAMBDA,THETA)
    PLOT MAKHAZ(X,XI,LAMBDA,THETA) FOR X = 0.01 0.01 5

Default:
    None
Synonyms:
    None
Related Commands:
    MAKCDF = Compute the Gompertz-Makeham cumulative distribution function.
    MAKPDF = Compute the Gompertz-Makeham probability density function.
    MAKPPF = Compute the Gompertz-Makeham percent point function.
    MAKCHAZ = Compute the Gompertz-Makeham cumulative hazard function.
    GOMPDF = Compute the Gompertz probability density function.
    EXPPDF = Compute the exponential probability density function.
    WEIPDF = Compute the Weibull probability density function.
    EV1PDF = Compute the extreme value type I probability density function.
    EV2PDF = Compute the extreme value type II probability density function.
Reference:
    "Statistical Methods for Reliability Data", Meeker and Escobar, Wiley, 2000, pp. 108-109.
Applications:
    Survival Analysis, Distributional Modeling
Implementation Date:
    2003/12: Original implementation (using the DLMF definition)
    2004/7: Added support for alternate parameterizations
Program: 
     
Y1LABEL Hazard
X1LABEL X
LABEL CASE ASIS
Y1LABEL DISPLACEMENT 12
X1LABEL DISPLACEMENT 12
TITLE DISPLACEMENT 2
.
MULTIPLOT 2 3
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT SCALE FACTOR 2
TITLE ZETA = 0.5, ETA = 0.2
PLOT MAKHAZ(X,0.5,0.2) FOR X = 0.01  0.01  3
TITLE ZETA = 0.5, ETA = 2
PLOT MAKHAZ(X,0.5,2) FOR X = 0.01  0.01  3
TITLE ZETA = 3, ETA = 0.2
PLOT MAKHAZ(X,3,0.2) FOR X = 0.01  0.01  3
TITLE ZETA = 3, ETA = 2
PLOT MAKHAZ(X,3,2) FOR X = 0.01  0.01  3
END OF MULTIPLOT
.
JUSTIFICATION CENTER
MOVE 50 97
TEXT Gompertz-Makeham Hazard
    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.