GompertzMakehamDistribution
✖
GompertzMakehamDistribution
represents a Gompertz distribution with scale parameter λ and frailty parameter ξ.
represents a Gompertz–Makeham distribution with parameters λ, ξ, θ, and α.
Details
- The hazard function for value in a Gompertz distribution is given by for , and is zero for .
- The hazard function for value in a Gompertz–Makeham distribution is given by for and is zero for .
- GompertzMakehamDistribution allows λ and ξ to be any positive real numbers and θ and α any non–negative real numbers.
- GompertzMakehamDistribution allows λ to be a quantity of any unit dimension and ξ, θ, and α to be dimensionless quantities. »
- GompertzMakehamDistribution can be used with such functions as Mean, CDF, and RandomVariate.
Background & Context
- GompertzMakehamDistribution[λ,ξ,θ,α] represents a continuous statistical distribution defined over the interval and parametrized by two non-negative real numbers θ and α and two positive real numbers λ and ξ called a "scale parameter" and a "frailty parameter", respectively. The overall behavior of the probability density function (PDF) of a Gompertz–Makeham distribution is determined by the values of the parameters λ, ξ, θ, and α, and in particular the PDF may be either monotonically decreasing with a potential singularity approaching the lower boundary of its domain or unimodal. In addition, depending on its parameters, the tails of the PDF may be either "fat" or "thin", in the sense that the PDF may decrease either algebraically or exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The two-parameter version of the distribution GompertzMakehamDistribution[λ,ξ] is often referred to as the Gompertz distribution and is equivalent to GompertzMakehamDistribution[λ,ξ,0,0].
- The Gompertz–Makeham distribution was introduced in the 1890s when English mathematician W. M. Makeham generalized a distribution originally studied by British mathematician Benjamin Gompertz in the early 1820s. Gompertz's original distribution was constructed as an attempt to smoothly model human mortality, subject to the assumption that death is due only to deterioration, while Makeham's work was the result of generalizing this model to account for death stemming from either deterioration or random causes. The Gompertz–Makeham distribution is the basis for the so-called Gompertz–Makeham law of mortality, which states that the human death rate is composed of two components: one age dependent and the other age independent. This law is used in a number of fields, including computer science, actuarial science, gerontology, demography, biology, and reliability theory.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Gompertz–Makeham distribution. Distributed[x,GompertzMakehamDistribution[λ,ξ,θ,α]], written more concisely as xGompertzMakehamDistribution[λ,ξ,θ,α], can be used to assert that a random variable x is distributed according to a Gompertz–Makeham distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
- The probability density and cumulative distribution functions may be given using PDF[GompertzMakehamDistribution[λ,ξ,θ,α],x] and CDF[GompertzMakehamDistribution[λ,ξ,θ,α],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
- DistributionFitTest can be used to test if a given dataset is consistent with a Gompertz–Makeham distribution, EstimatedDistribution to estimate a Gompertz–Makeham parametric distribution from given data, and FindDistributionParameters to fit data to a Gompertz–Makeham distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Gompertz–Makeham distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Gompertz–Makeham distribution.
- TransformedDistribution can be used to represent a transformed Gompertz–Makeham distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a Gompertz–Makeham distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Gompertz–Makeham distributions.
- The Gompertz–Makeham distribution is related to several other distributions. For example, GompertzMakehamDistribution is related to ExponentialDistribution, in the sense that the hazard function (see HazardFunction) of GompertzMakehamDistribution[λ,θ/λ] tends to that of ExponentialDistribution[θ] as λ tends to zero. In addition, GompertzMakehamDistribution is a truncated GumbelDistribution (i.e. GumbelDistribution[a,b] restricted to [ is the same as GompertzMakehamDistribution[1/b,Exp[-a/b]]). It is related to a truncated WeibullDistribution and therefore also to the other "extreme value distributions", namely to FrechetDistribution and to ExtremeValueDistribution. GompertzMakehamDistribution is also related to GammaDistribution, ExpGammaDistribution, RayleighDistribution, and StudentTDistribution.
Examples
open allclose allBasic Examples (6)Summary of the most common use cases
Probability density function of Gompertz distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-i934aj
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-skw54j
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-y920x
Cumulative distribution function of Gompertz distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-dlmih2
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-ff4y6e
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-l07fv0
Probability density function of Gompertz–Makeham distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-0hfpq7
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-gnqtm4
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-lotpxi
Cumulative distribution function of Gompertz–Makeham distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-6pti5o
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-di3ang
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-coa35d
Mean and variance of a Gompertz distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-d03s10
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-dcbx9l
Median of Gompertz distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-etj9le
Scope (10)Survey of the scope of standard use cases
Generate a sample of pseudorandom numbers from a Gompertz distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-fgvl8j
Compare its histogram to the PDF:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-bd2g3p
Distribution parameters estimation:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-45b7g2
Estimate the distribution parameters from sample data:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-epi747
Compare the density histogram of the sample with the PDF of the estimated distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-f8ui5o
Skewness of Gompertz distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-fmbbvv
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-huc9m5
Kurtosis of Gompertz distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-qbjzv5
Different moments of Gompertz distribution with closed forms as functions of parameters:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-js043h
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-rx074o
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-pknsqa
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-zg9ct4
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-9gzmth
Hazard function of Gompertz distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-bg1ibe
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-p480hj
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-kope
Hazard function of Gompertz–Makeham distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-698ymv
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-31zshy
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-6wd5jp
Quantile function of Gompertz distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-4flxwd
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-e24ven
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-k3knub
Quantile function of Gompertz–Makeham distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-oduw2c
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-ogwwt5
Consistent use of Quantity in parameters yields QuantityDistribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-fxm75i
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-xttyr
Applications (4)Sample problems that can be solved with this function
The lifetime of a device has a Gompertz distribution. Find the reliability of the device:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-4m8ue6
The hazard function increasing in time:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-xz8iwk
Find the reliability of two such devices in series:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-xfhpfa
Find the reliability of two such devices in parallel:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-7trpvs
Compare the reliability of both systems for and :
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-5euiud
A steel pipe with thickness θ is exposed to corrosion and fails if any of the n microscopic pits penetrates the surface. Assume the time to penetration at each pit is proportional to the remaining thickness with factor k. If the depths of the pits are initially random and the depth in time follows a right-truncated exponential distribution with parameter λ, then time to failure of the pipe follows a Gompertz distribution. Find the reliability of the pipe:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-1s0dsl
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-slgb50
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-8cw0s0
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-ff6tjp
Find the mean time to failure:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-25p5pf
The female mortality in 1900 according to the Society of Actuaries is given by the table:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-l8v0vx
Create a sample population to use maximum likelihood estimation:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-xlo60x
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-2i6xnv
Fit Gompertz–Makeham distribution to the data:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-iocsmb
Plot probability density function:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-huav12
Compare the mortality data with the survival function of the estimated distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-53m74g
Find the average female life length in 1900:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-2vk6kf
Compare the mean residual lifetime from the data with estimated distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-45ekd7
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-5w4w50
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-0yntw
Define expo-power distribution using Gompertz distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-17gbxw
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-5jig8q
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-imadeu
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-bhd48n
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-udw8rr
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-6j0obz
WeibullDistribution is a limiting case:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-hvk97l
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-lq38vp
Properties & Relations (9)Properties of the function, and connections to other functions
GompertzMakehamDistribution is closed under scaling by a positive factor:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-kc9qv2
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-cuc9i9
The family of Gompertz distributions is closed under a minimum:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-ed5cxa
For different frailty parameters:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-24xjm6
The family of Gompertz–Makeham distributions is closed under a minimum:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-n7dp40
For different frailty parameters:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-p1bk5e
Relationships to other distributions:
Gompertz distribution is related to exponential distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-l3x5fs
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-ffuehf
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-kjcqc9
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-2pvf9g
The Gompertz–Makeham distribution simplifies to the Gompertz distribution for θ=0 and α=0:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-87vw5
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-1bhcng
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-3gy7oa
Gompertz distribution is a truncated GumbelDistribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-domfm
Gompertz distribution is related to truncated WeibullDistribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-j61n0i
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-zyzzxf
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-xdaw2j
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-ua1z4y
WeibullDistribution is related to Gompertz distribution:
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-grf22l
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-1wxxnv
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-cdetbo
https://wolfram.com/xid/0bgvnzrnlt5yd5gq0h26u8r9sq-45m2iq
Wolfram Research (2010), GompertzMakehamDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/GompertzMakehamDistribution.html (updated 2016).
Text
Wolfram Research (2010), GompertzMakehamDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/GompertzMakehamDistribution.html (updated 2016).
Wolfram Research (2010), GompertzMakehamDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/GompertzMakehamDistribution.html (updated 2016).
CMS
Wolfram Language. 2010. "GompertzMakehamDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/GompertzMakehamDistribution.html.
Wolfram Language. 2010. "GompertzMakehamDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/GompertzMakehamDistribution.html.
APA
Wolfram Language. (2010). GompertzMakehamDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GompertzMakehamDistribution.html
Wolfram Language. (2010). GompertzMakehamDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GompertzMakehamDistribution.html
BibTeX
@misc{reference.wolfram_2024_gompertzmakehamdistribution, author="Wolfram Research", title="{GompertzMakehamDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/GompertzMakehamDistribution.html}", note=[Accessed: 08-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_gompertzmakehamdistribution, organization={Wolfram Research}, title={GompertzMakehamDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/GompertzMakehamDistribution.html}, note=[Accessed: 08-January-2025
]}