generalizations of gompertz distribution and their...

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

84

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).

85


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.

86


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

87


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.

88


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-

89


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)),

90


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)

91


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


92


Figure 5.5: ACF and PACF of the simulated sample paths when 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

93


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)

94


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.

95


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

96


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

97


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


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

generalizations of gompertz distribution and their...

Documents