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CHAPTER 5
Generalizations of Gompertz distribution
and their Applications
5.1 Introduction
The Gompertz distribution plays an important role in modeling survival times, human mor-
tality and actuarial data. According to the literature, the Gompertz distribution was for-
mulated by Gompertz (1825) to fit mortality tables. Johnson et al. (1995) and Garg et
al. (1970) studied the properties of the Gompertz distribution and obtained the maximum
likelihood estimates for the parameters. Johnson et al. (1994) note that the Gompertz dis-
tribution is a truncated extreme value distribution. Gompertz distributions can be viewed
as extensions of the exponential distributions because exponential distributions are limits
of sequences of Gompertz distributions. Makeham (1860) examined the fit to actuarial
data provided by the Gompertz distribution and observed with specific examples that he
could improve the fit with the modification now known as the Gompertz-Makeham distri-
bution. Burga et al. (2009) discussed the stress-strength reliability problem in Gompertz
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CHAPTER 5. GENERALIZATIONS OF GOMPERTZ DISTRIBUTION AND THEIR APPLICATIONS
case. Marshall and Olkin (1997) introduced a method of obtaining an extended family of
distributions including one more parameter. The new parameter results in added flexibility
of distributions and influence the reliability properties. Reliability properties of extended
Makeham distributions is studied by Bassiouny and Abdo (2009).
In this chapter we discuss some generalizations of Gompertz distribution, particularly
properties of Marshall-Olkin Gomperz distribution, and also develop a three parameter AR
(1) model. Consider an AR (1) structure given by
Xn =
εn w.p. q
Xn−1 w.p. p(1− q)
min(pXn−1, εn) w.p. (1− p)(1− q)
, n ≥ 1.
where w.p. means with probability. If q = 0 we get the ordinary process, where 0 ≤p ≤ 1, {εn} is a sequence of independent and identically distributed random variables
independent of {Xn−1, Xn−2, . . . }.
This chapter is organized as follows. In section 5.2 we consider generalizations and
characteristic properties. Section 5.3 deals with autoregressive minification processes.
Odds function and reverse hazard rate are discussed in section 5.4. Estimation of reliability
is done in section 5.5. Simulation study is conducted in section 5.6. Conclusions are given
in section 5.7.
5.2 Generalizations and Characteristic Properties
A random variable X is said to have a Gompertz distribution if its pdf is
fG(x) = βeαxe−βα
(eαx−1); x ≥ 0, α, β > 0
Corresponding survival function is
FG(x) = e−βα
(eαx−1).
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CHAPTER 5. GENERALIZATIONS OF GOMPERTZ DISTRIBUTION AND THEIR APPLICATIONS
Hazard rate function of Gompertz distribution is
hG(x) = βeαx.
Hence the hazard rate increases exponentially over time. when α → 0, the Gompertz dis-
tribution will tend to an exponential distribution with constant hazard rate (Wu et al.(2003)).
The two-parameter Gompertz model is a commonly used survival time distribution in actu-
arial science and reliability and life testing, and is discussed by Ananda et al.(1996). There
are several forms for the Gompertz distribution given in the literature. Some of these are
given in Johnson et al.(1994). The Gompertz distribution is unimodal. It has positive skew-
ness and an increasing hazard rate function. In addition, the Gompertz distribution can be
interpreted as a truncated extreme value type-I distribution. According to Jaheen (2003)
the Gompertz distribution has been used as a growth model, especially in epidemiological
and biomedical studies. The Gompertz distribution represents another extension of the
exponential distribution. Like the Weibull, the Gompertz distribution is characterized by
two parameters. In the pdf of Gompertz distribution (GD) when α < 0(> 0), the hazard
function decreases (increases) from exp(α), and when α = 0, it reduces to the con-
stant hazard function of an exponential distribution. Applying the Marshall-Olkin (1997)
technique to Gompertz distribution, we get the distribution function of the Marshall- Olkin
Gompertz distribution (MOGD) as
GG(x) =1− e−
βα
(eαx−1)
1 + (p− 1)e−βα
(eαx−1).
Corresponding pdf is obtained as,
gG(x) =pβe−
βα
(eαx−1)+αx
(1 + (p− 1)e−βα
(eαx−1))2.
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CHAPTER 5. GENERALIZATIONS OF GOMPERTZ DISTRIBUTION AND THEIR APPLICATIONS
Figure 5.1: Graphs of pdf and cdf of Gompertz distribution
pdf of GD, when β = .03 and α = .9, α = .8, α = .7 cdf of GD, when β = .9 and α = .09, α = .8, α = 5
Hazard rate function of Marshall-Olkin Gompertz distribution is
hG(x) =βeαx
1 + (p− 1)e−βα
(eαx−1).
Theorem 5.2.1. Marshall-Olkin Gompertz distribution(MOGD) is geometric extreme
stable.
Theorem 5.2.2. Let {Xi, i ≥ 1} be a sequence of independent and identically dis-
tributed random variables with common survival function F (x) and N be a geometric
random variable with parameter p and P (N = n) = pqn−1; n = 1, 2, ...; 0 < p < 1, q =
1 − p. which is independent of {Xi} for all i ≥ 1. Let UN = min1≤i≤N
Xi. Then {UN} is
distributed as MOGD if and only if {Xi} is distributed as Gompertz.
A random variable X is said to have a Gompertz-Makeham distribution (GM), if its pdf
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CHAPTER 5. GENERALIZATIONS OF GOMPERTZ DISTRIBUTION AND THEIR APPLICATIONS
Figure 5.2: Graphs of pdf and cdf of Marshall-Olkin Gompertz distribution
pdf of MOGD β = 1.2 and α = 0.6, p=2, 1.25, 0.8 cdf of MOGD β = 0.9 and α = 0.09, p=4, 0.9, 0.08
is of the form
fGM(x) = {βθ + βeax}e−βα
(eαx−1)−βθx, x ≥ 0 , α, β, θ > 0.
The corresponding cdf is
FGM(x) = 1− e−βα
(eαx−1)−βθx.
The survival function is
FGM(x) = e−βα
(eαx−1)−βθx.
Hazard rate function of Gompertz-Makeham distribution is
hGM(x) = {βθ + βeax}.
Survival function of Marshall-Olkin Gompertz-Makeham distribution (MOGM) is
GMOGM(x) =p.e−
βα
(eαx−1)−βθx
1− (1− p)e−βα
(eαx−1)−βθx.
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CHAPTER 5. GENERALIZATIONS OF GOMPERTZ DISTRIBUTION AND THEIR APPLICATIONS
The pdf of Marshall-Olkin Gompertz-Makeham distribution is
gMOGM(x) =pβ(eαx + θ)e−
βα
(eαx−1)−βθx
(1− (1− p)e−βα
(eαx−1)−βθx)2.
Now we have the following results.
Theorem 5.2.3. Marshall-Olkin Gompertz-Makeham distribution (MOGM) is geo-
metric extreme stable.
Theorem 5.2.4. Let {Xi, i ≥ 1} be a sequence of independent and identically dis-
tributed random variables with common survival function F (x) and N be a geometric
random variable with parameter p and P (N = n) = pqn−1; n = 1, 2, ...; 0 < p < 1, q =
1 − p. which is independent of {Xi} for all i ≥ 1. Let UN = min1≤i≤N
Xi. Then {UN} is
distributed as MOGM if and only if {Xi} is distributed as GM.
5.3 Autoregressive Minification Processes
Theorem 5.3.1. Consider an AR(1) structure given by
Xn =
εn with probability p
min(Xn−1, εn) with probability 1− p
where 0 ≤ p ≤ 1, {εn} is a sequence of independent and identically distributed random
variables independent of {Xn−1, Xn−2, . . . }. Then {Xn} is a stationary Markovian
AR(1) process with MOG marginals if and only if {εn} is distributed as Gompertz
distribution.
Theorem 5.3.2. Consider an AR(1) structure given by
Xn =
εn with probability p
min(Xn−1, εn) with probability 1− p
where 0 ≤ p ≤ 1, {εn} is a sequence of independent and identically distributed ran-
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CHAPTER 5. GENERALIZATIONS OF GOMPERTZ DISTRIBUTION AND THEIR APPLICATIONS
dom variables independent of {Xn−1, Xn−2, . . . }. Then {Xn} is a stationary Markovian
AR(1) process with MOGM marginals if and only if {εn} is distributed as GM distri-
bution.
Theorem 5.3.3. Consider an AR (1) structure given by
Xn =
εn w.p. q
Xn−1 w.p. p(1− q)
min(pXn−1, εn) w.p. (1− p)(1− q)
, n ≥ 1.
where w.p. means with probability, also 0 ≤ p ≤ 1, {εn} is a sequence of independent
and identically distributed random variables independent of {Xn−1, Xn−2, . . . }. Then
{Xn} is a stationary Markovian AR (1) process with MOG marginals if and only if
{εn} is distributed as Gompertz distribution.
Proof. From theorem, it follows that
FXn(x) = qFXn−1(x) + (1− p)(1− q)FXn−1(x)F εn(x) + p(1− q)F εn(x) (5.3.1)
under stationarity
F̄X(x) =pF̄ε(x)
1− (1− p)F̄ε(x),
if we take
F̄ε(x) = e−βα
(eαx−1).
Then
F̄X(x) =pe−
βα
(eαx−1)
1− (1− p)(e−βα
(eαx−1))
which is the survival function of the Marshall-Olkin Gompertz distribution. Conversely if we
take
F̄X(x) =pe−
βα
(eαx−1)
1− (1− p)(e−βα
(eαx−1)),
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CHAPTER 5. GENERALIZATIONS OF GOMPERTZ DISTRIBUTION AND THEIR APPLICATIONS
then we get
F̄ε(x) = e−βα
(eαx−1).
Theorem 5.3.4. Consider an AR (1) structure given by
Xn =
εn w.p. q
Xn−1 w.p. p(1− q)
min(pXn−1, εn) w.p. (1− p)(1− q)
, n ≥ 1.
where w.p. means with probability. where 0 ≤ p ≤ 1, {εn} is a sequence of independent
and identically distributed random variables independent of {Xn−1, Xn−2, . . . }. Then
{Xn} is a stationary Markovian AR (1) process with MOGM marginals if and only if
{εn} is distributed as GM distribution.
Proof is similar to theorem 5.3.3.
5.3.1 Sample Path Properties
In order to study the behavior of the processes, we simulate the sample paths for various
values of n and p. In particular we take in Fig 5.3a, n = 300, p = 0.8, β = 0.06 and
α = 0.07 and in Fig 5.3b, n = 200, p = 0.6, β = 0.6 and α = 0.6. Fig 5.4a and Fig 5.5a
depict the plots of acf (auto correlation function) and Fig 5.4b and Fig 5.5b depict the plots
of pacf (partial auto correlation function) of the simulated series respectively.
5.4 Odds Function and Reverse hazard rate
Sankaran and Jayakumar (2008) considered the relevance of Marshall-Olkin distribution
with respect to proportional hazards odds model in reliability theory. Let X be a random
variable and F (x) be the distribution function of it. Then the family of distributions H(x)
given in Marshall-Olkin (1997) is characterized by the identity
βH(x) =1
pβF (x)
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CHAPTER 5. GENERALIZATIONS OF GOMPERTZ DISTRIBUTION AND THEIR APPLICATIONS
Figure 5.3: Sample path for various values of n and p
Fig 5.3a, n = 300, p = 0.8, β = 0.06 , α = 0.07 Fig 5.3b, n = 200, p = 0.6, β = 0.6 and α = 0.6
Figure 5.4: ACF and PACF of the simulated sample paths when n = 300, p = 0.8,β = 0.06 and α = 0.07
Fig 5.4a, n = 300, p = 0.8, β = 0.06 , α = 0.07 Fig 5.4b, n = 300, p = 0.8, β = 0.06 and α = 0.07
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CHAPTER 5. GENERALIZATIONS OF GOMPERTZ DISTRIBUTION AND THEIR APPLICATIONS
Figure 5.5: ACF and PACF of the simulated sample paths when n = 200, p = 0.6,β = 0.6 and α = 0.6
Fig 5.5a, n = 200, p = 0.6, β = 0.6 , α = 0.6 Fig 5.5b, n = 200, p = 0.6, β = 0.6 and α = 0.6
where
H(x) =pF (x)
1− (1− p)F (x),
and p is the Marshall-Olkin tilt parameter. Then
βF (x) =F̄ (x)
F (x)
is known as the Odds function of a random variable X. It measures the ratio of the proba-
bility that the unit will survive beyond x to the probability that it will fail before x.
In Gompertz distribution
βF (x) =1
eβα
(eαx−1) − 1
In Marshall-Olkin Gompertz distribution
βG(x) =p
eβα
(eαx−1) − 1
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CHAPTER 5. GENERALIZATIONS OF GOMPERTZ DISTRIBUTION AND THEIR APPLICATIONS
In Gompertz - Makeham distribution
βF (x) =1
eβα
(eαx−1)−βθx − (1− p)
In Marshall -Olkin Gompertz - Makeham distribution
βG(x) =p
eβα
(eαx−1)−βθx − (1− p)
Let X is a random variable, f(x) be its pdf and F (x) be the cdf, the reversed hazard
function is defined as
r(x) =f(x)
F (x)
In Gompertz distribution, the reversed hazard function is
r(x) =eαxβe−
βα
(eαx−1)
1− e−βα
(eαx−1)
Marshall Olkin Gompertz distribution the reversed hazard function is
rMOG(x) =1− e−
βα
(eαx−1)pβe−βα
(eαx−1)+αx
1 + (p− 1)e−βα
(eαx−1).
In Gompertz-Makeham distribution the reversed hazard function is
rGM(x) =βθ + β(eαx)
eβα
(eαx−1)−βθx − 1.
In Marshall-Olkin Gompertz-Makeham distribution the reversed hazard function is
rMOGM(x) =pβ(eαx + θ)e−
βα
(eαx−1)−βθx
(1− (1− p)e−βα
(eαx−1)−βθx)(1− e−βα
(eαx−1)−βθx)
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CHAPTER 5. GENERALIZATIONS OF GOMPERTZ DISTRIBUTION AND THEIR APPLICATIONS
5.5 Estimation of reliability
Let X and Y be two independent random variables following Marshall Olkin Gompertz
distribution with parameters α1, β, p and α2, β, p respectively. Then according to Gupta et
al. (2009) the reliability of the system given by P (X < Y ) where X is the stress and Y is
the strength is given by
R = P (X < Y ) =
∫ ∞
−∞P (Y > X/X = x)gX(x)dx
=
∫ ∞
0
α1βe−βα
(eαx−1)+αx
(1 + (α1 − 1)e−βα
(eαx−1))2
α2e− β
α(eαx−1)
1− (1− α2)e− β
α(eαx−1)
dx
=α1
α2
(α1
α2− 1)2
[− ln
α1
α2
+α1
α2
− 1
]
Let (x1, . . . , xm) and (y1, . . . , yn) be two independent random samples of sizes m and
n from Marshall-Olkin Gompertz distribution with Marshall-Olkin parameters α1 and α2,
respectively, and common unknown parameters β and p. L is the log likelihood function,
then maximum likelihood estimates of the unknown parameters α1, α2 are the solutions of
the non-linear equations ∂L∂α1
= 0 and ∂L∂α2
= 0 respectively. The elements of information
matrix are
I11 = −E
(∂2L
∂α21
)=
m
3α21
Similarly,
I22 = −E
(∂2L
∂α22
)=
n
3α22
I12= I21 = −E
(∂2L
∂α1∂α2
)= 0.
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CHAPTER 5. GENERALIZATIONS OF GOMPERTZ DISTRIBUTION AND THEIR APPLICATIONS
By the property of m.l.e for m →∞, n →∞, we obtain that
(√
m(α̂1 − α1),√
n(α̂2 − α2))T d→ N2
(0, diag{a−1
11 , a−122 }),
where a11 = limm,n→∞
1
mI11 =
1
3α21
and a22 = limm,n→∞
1
nI22 =
1
3α22
. The 95% confidence
interval for R is given by
R̂∓ 1.96 α̂1b1(α̂1, α̂2)
√3
m+
3
n,
where R̂ = R(α̂1, α̂2) is the estimator of R and
b1(α1, α2) =∂R
∂α1
=α2
(α1 − α2)3
[2(α1 − α2) + (α1 + α2) log
α2
α1
].
5.6 Simulation Study
We generate N = 1000 sets of X-samples and Y -samples from Marshall-Olkin Gompertz
distribution with parameters α1, β, p and α2, β, p respectively. The combinations of sam-
ples of sizes m = 20, 30, 40 and n = 20, 30, 40 are considered. The estimates of α1 and
α2 are then obtained from each sample to obtain R̂. The validity of the estimate of R is
discussed by the measures:
1) Average bias of the simulated N estimates of R:
1
N
N∑i=1
(R̂i −R)
2) Average mean square error of the simulated N estimates of R:
1
N
N∑i=1
(R̂i −R)2
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CHAPTER 5. GENERALIZATIONS OF GOMPERTZ DISTRIBUTION AND THEIR APPLICATIONS
Table 5.1: Average bias and average mean square error of the simulated estimates ofR for β = 2, p = 2.5
(α1, α2)Average bias (b) Average Mean Square Error (AMSE)
(m,n) (0.5,2) (0.7,0.9) (1.5,1.8) (1.7,0.7) (0.5,2) (0.7,0.9) (1.5,1.8) (1.7,0.7)
(20,20) 0.1500 0.0294 0.0210 -0.1019 0.0232 0.0016 0.0010 0.0111(30,20) 0.1530 0.0302 0.0236 -0.1016 0.0239 0.0016 0.0011 0.0109(20,30) 0.1515 0.0300 0.0208 -0.1022 0.0236 0.0015 0.0009 0.0110(20,40) 0.1508 0.0300 0.0217 -0.1035 0.0233 0.0015 0.0009 0.0112(40,20) 0.1515 0.0301 0.0225 -0.1010 0.0234 0.0015 0.0009 0.0109
Table 5.2: Average confidence length and coverage probability of the simulated esti-mates of R for β = 2, p = 0.5
(α1, α2)Average confidence length Coverage probability
(m,n) (0.5,2) (0.7,0.9) (1.5,1.8) (1.7,0.7) (0.5,2) (0.7,0.9) (1.5,1.8) (1.7,0.7)
(20,20) 0.3511 0.3567 0.3570 0.3541 0.8530 1 1 0.9960(30,20) 0.3212 0.3257 0.3260 0.3238 0.6460 1 1 0.9920(20,30) 0.3209 0.3258 0.3259 0.3246 0.6470 1 1 0.9930(20,40) 0.3043 0.3091 0.3092 0.3075 0.5250 1 1 0.9910(40,20) 0.3045 0.3091 0.3093 0.3072 0.5170 1 1 0.9880
3) Average length of the asymptotic 95% confidence intervals of R:
1
N
N∑i=1
2(1.96)α̂1i b1i(α̂α1i, α̂α2i)
√3
m+
3
n
4) The coverage pprobability of the N simulated confidence intervals given by the pro-
portion of such interval that include the parameter R.
5.7 Conclusions
As a generalization of the Gompertz distribution, Marshall-Olkin Gompertz distribution is
considered and Marshall-Olkin Gompertz-Makeham distribution is introduced. A three pa-
rameter AR(1) process is also considered, and sample path is drawn for Marshall-Olkin
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CHAPTER 5. GENERALIZATIONS OF GOMPERTZ DISTRIBUTION AND THEIR APPLICATIONS
Gompertz process. When X and Y are two independent random variables following Mar-
shall Olkin Gompertz distribution, then average bias ,average mean square error, average
confidence length and coverage probability of the of the simulated estimates of reliability
R are computed.
References
Ananda, M.M., Dalpatadu, R.J., Singh, A.K. (1996) Adaptive Bayes Estimators for Pa-
rameters of the Gompertz Survival Model, Applied Mathematics and Computation,
75, 2, 167-177.
Burga, S., Mehmet, F.K., Abd-Elfattah, A.M. (2009) Comparison of estimators for stress-
strength reliability in Gompertz case, Hacettepe Journal of Mathematics and Statis-
tics, 38 (3), 339-349.
EL-Bassiouny, A.H., Abdo, N.F. (2009) Reliability Properties of Extended Makeham Dis-
tributions, Computational methods in science and technology,15(2), 143-149.
Garg, M.L., Rao, B.R., Redmond, C.K. (1970) Maximum Likelihood Estimation of the
Parameters of the Gompertz Survival Function, Journal of the Royal Statistical
Society , 19, 152-159.
Gompertz, B. (1825) On the Nature of the Function Expressive of the Law of Human
Mortality and on the New Mode of Determining the Value of Life Contingencies,
Phil. Trans. R. Soc. A., 115, 513-580.
Gupta, R.C., Ghitany, M.E., Al- Mutairi, D.K. (2009) Estimation of reliability from Marshall-
Olkin extended Lomax distributions, Journal of Statistical Computation and Sim-
ulation, 1-11.
Jaheen, Z.F. (2003) Bayesian Prediction under a Mixture of Two-Component Gompertz
Lifetime Model, Sociedad Espanola de Estadistica e Investigacion Operativa Test,
12, 2, 413-426.
98
CHAPTER 5. GENERALIZATIONS OF GOMPERTZ DISTRIBUTION AND THEIR APPLICATIONS
Johnson, N.L., Kotz, S., Balakrishnan, N. (1994) Continuous Univariate Distributions, Vol.
1, 2nd ed., John Wiley and Sons.
Johnson, N.L., Kotz, S., Balakrishnan, N. (1995) Continuous Univariate Distributions, Vol.
2, 2nd ed., John Wiley and Sons.
Makeham, W.M. (1860) On the Law of Mortality and the Construction of Annuity Tables,
The Assurance Magazine and Journal of the Institute of Actuaries VIII 301-310.
Marshall, A.W., Olkin, I. (1997) A new method for adding a parameter to a family of dis-
tributions with application to the exponential and weibull families, Biometrica 84(3),
641-652.
Sankaran, P.G., Jayakumar, K. (2008) On proportional odds models, Stat. Papers, 49,
779-789.
Wu, J.W., Hung, W.L., Tsai, C.H. (2003) Estimation of the Parameters of the Gompertz
Distribution under the First Failure-Censored Sampling Plan, Statistics, 37(6), 517-
525.
99