 Dataplot Vol 2 Vol 1

# MAKPDF

Name:
MAKPDF (LET)
Type:
Library Function
Purpose:
Compute the Gompertz-Makeham probability density 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 probability density function: Note that this definition uses three shape parameters.

2. Meeker and Escobar use a parameterization based on the parameters , k, and . This can be translated to the DLMF definition as follows:   After making the above substitutions, the DLMF definition can be used.

Note that Meeker and Escobar parameterization can be expressed in terms of the DLMF parameterization:   3. Given the three shape parameters definition of Meeker and Escobar, they reparameterize the distribution in the following way:   An attractive feature of this parameterization is that it reduces the three shape parameters to two shape parameters ( and ) and a scale parameter ( ). The probability density function is: 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 MEEKER REPARAMETERIZED

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> = MAKPDF(<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 pdf 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> = MAKPDF(<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 pdf 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> = MAKPDF(<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 pdf 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 MEEKER REPARAMETERIZED
LET A = MAKPDF(0.3,0.5,2)
LET A = MAKPDF(X,ETA,ZETA)
PLOT MAKPDF(X,ETA,ZETA) FOR X = 0.01 0.01 5

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

Note:
Gompertz-Makeham random numbers, probability plots, and goodness of fit tests can be generated with the following commands:

If the DLMF definition is used, define the parameters with the commands:

LET XI = <value>
LET LAMBDA = <value>
LET THETA = <value>

If the three shape parameters Meeker and Escobar definition is used, define the parameters with the commands:

LET GAMMA = <value>
LET K = <value>
LET LAMBDA = <value>

If the two shape parameters Meeker and Escobar definition is used, define the parameters with the commands:

LET ETA = <value>
LET ZETA = <value>

Then use the commands

LET Y = GOMPERTZ MAKEHAM RANDOM NUMBERS FOR I = 1 1 N
GOMPERTZ MAKEHAM PROBABILITY PLOT Y
GOMPERTZ MAKEHAM KOLMOGOROV SMIRNOV ...
GOODNESS OF FIT Y
GOMPERTZ MAKEHAM CHI-SQUARE GOODNESS OF FIT Y

If the two shape parameters Meeker and Escobar definition is used, the shape parameters can be estimated with the ks plot or ppcc plot:

LET ETA1 = <value>
LET ETA2 = <value>
LET ZETA1 = <value>
LET ZETA2 = <value>
GOMPERTZ MAKEHAM KS PLOT Y
GOMPERTZ MAKEHAM PPCC PLOT Y
Default:
None
Synonyms:
None
Related Commands:
 MAKCDF = Compute the Gompertz-Makeham cumulative distribution function. MAKPPF = Compute the Gompertz-Makeham percent point function. MAKHAZ = Compute the Gompertz-Makeham hazard 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 Probability
X1LABEL X
LABEL CASE ASIS
Y1LABEL DISPLACEMENT 12
X1LABEL DISPLACEMENT 12
TITLE DISPLACEMENT 2
YLIMITS 0 2.5
.
MULTIPLOT 2 2
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT SCALE FACTOR 2
TITLE ETA = 0.2, ZETA = 0.5
PLOT MAKPDF(X,0.5,0.2) FOR X = 0.01  0.01  3
TITLE ETA = 2, ZETA = 0.5
PLOT MAKPDF(X,0.5,2) FOR X = 0.01  0.01  3
TITLE ETA = 0.2, ZETA = 3
PLOT MAKPDF(X,3,0.2) FOR X = 0.01  0.01  3
TITLE ETA = 2, ZETA = 3
PLOT MAKPDF(X,3,2) FOR X = 0.01  0.01  3
END OF MULTIPLOT
.
JUSTIFICATION CENTER
MOVE 50 97
TEXT Gompertz-Makeham PDF
``` Date created: 7/7/2004
Last updated: 7/7/2004