Construction of Survival Distributions

7/24/2019 Construction of Survival Distributions

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Special Issue Paper

Received 4 December 2011, Revised 12 December 2011, Accepted 13 December 2011 Published online 28 Nov. 2012 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/asmb.948

Constructions and applications of

lifetime distributionsC. D. Lai*

Lifetime (ageing) distributions play a fundamental role in reliability. We present a semi-unified approach in constructing them,and show that most of the existing distributions may arise from one of these methods. Generalizations/modifications of the Weibulldistribution are often required to prescribe the nonmonotonic nature of the empirical hazard rates. We also briefly outline some ofthe known applications of lifetime distributions in diverse disciplines. Copyright 2012 John Wiley & Sons, Ltd.

Keywords: ageing; constructions; distribution; hazard rate; lifetime; survival function; Weibull

1. Introduction

Broadly speaking, any probability distribution defined on the positive real line can be considered as a lifetime distribution.

Of course, not all such distributions are meaningful for prescribing an ageing phenomenon. Many ageing (lifetime) distri-

butions have been constructed with a view for applications in various disciplines, in particular, in reliability engineering,

survival analysis, demography, actuarial study and others.

Historically speaking, the Gompertz and Makeham (also known as GompertzMakeham) distributions are possibly

the earliest ageing models used for smoothing mortality tables, which were of considerable interest to actuaries. Sev-

eral extensions of the two models were subsequently derived to improve model flexibility. In the reliability engineering

front, distributions such as the exponential, gamma, Weibull, Pareto and inverted beta (related to F-distribution) are often

used. Generalizations of the Pareto such as the Lomax, log-logistic and Burr XII are also popular in reliability arenas.Of course, the lognormal and the inverse Gaussian distributions are long-standing ageing distributions among the social

scientists when considering the hazards of social events; see [1] or [2] for discussions on these ageing distributions.

In this paper, we outline some common methods for constructing lifetime distributions with the aim to provide some

insights on general construction mechanisms. Examples are given to provide the readers a possible source of ideas to draw

upon. Applications of lifetime distributions in reliability engineering, insurance, survival analysis and mortality studies

are briefly discussed.

2. Measures of ageing

Statistical analysis of lifetime data is an important topic in biomedical science, reliability engineering, social sciences and

others. Typically, lifetime refers to human life length, the life span of a device before it fails, the survival time of a patient

with serious disease from the date of diagnosis or major treatment or the duration of a social event such as marriage.

LetTbe the lifetime random variable withf.t/,F.t/being its probability density function and cumulative distributionfunction (CDF), respectively. The reliability or survival function is given by NF.t/D1 F.t/.

The hazard (failure) rate function is defined as

h.t/D f.t/

1 F.t/D

f.t/

NF.t/I (1)

h.t/tgives (approximately) the probability of failure in .t; tC t given the unit has survived until timet .

Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand

*Correspondence to: C. D. Lai, Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand.E-mail: [email protected]

Copyright 2012 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2013,29 127140

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The cumulative hazard rate function is defined as

H.t/D

Z t0

h.x/dx: (2)

It is easy to show that the reliability function can be represented as

NF.t/De

H.t/

: (3)

Obviously, the cumulative hazard function completely determines the lifetime (ageing) distribution and it must satisfy the

following three conditions to yield a proper lifetime (ageing) distribution: (i) H.t/ is nondecreasing for all t > 0; (ii)H.0/D0; and (iii) lim

t!1H.t/D 1.

Because the reliability function NF.t/and the hazard rate function h.t/can be uniquely determined from each other, anew ageing distribution can therefore be derived by constructing one of them first.

The reversed hazard rate, defined as the ratio of the density to the distribution function,

r.t/D f.t/

F.t/; (4)

had attracted the attention of researchers only relatively recently. It also characterizes an ageing distribution although its

importance in reliability is yet to be established.Another important measure of ageing is the mean residual (remaining) life defined by

.t/DE.T t j T > t/D

R1

tNF.x/dx

NF.t/; (5)

summarizing the entire residual life of a unit at aget .The hazard rate functionh.t/and the survival function NF.t/may be obtained from .t/through the relationships:

h.t/D1 C 0.t /

.t/ ; t > 0; (6)

and

NF.t/D

.t/exp

Z t0

.x/1dx

; t > 0 (7)

withDE.T /D.0/.Ifh.t/ is unimodal (inverted bathtub), then either .t/ is increasing (ifh.0/ > 1=) or it has a bathtub shape. Fur-

thermore, limt!1 .t/D 1= limt!1 h.t/ provided that the latter limit exists and is finite [3]. Thus, if a unimodal h.t/decreases asymptotically to a finite value, then.t/increases asymptotically to a finite value.

It should be noted that several other measures of ageing have been proposed in the literature, but we devote on those

discussed previously as they are the major ones in reliability practice.

3. Constructions of lifetime distributions

In what follows, we letG denote the base (underlying) distribution from which a new distribution F is constructed from.Unless otherwise specified, we assume F andG have density functions f andg , respectively. A subscript i (i D 1; 2)may be given to these functions for obvious reasons.

3.1. Why do we need so many lifetime distributions?

Hazard rates (also known as mortality curves) of lifetime variables may exhibit various forms and shapes depending on

many factors.

In reliability engineering, in addition to the ageing effect, there is also the effect of quality variations in production on

product reliability [4]. Two common causes for variations are component nonconformance and assembly errors. Compo-

nent nonconformance results in some of the items produced not conforming to the design reliability. For example, suppose

an item has been designed to give an increasing h.t/.

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With the inclusion of assembly errors, h.t/emerges as bathtub shaped. This is because the failures resulting from theassembly errors can be viewed as a new mode of failure and the hazard rate associated with this new distribution is

generally decreasing with timet . With this decreasing hazard rate being added to the designed increasing hazard rate,the hazard rate of the items produced with assembly errors will be decreasing for small t and increasing for larget ,and thus a bathtub shape results.

With the inclusion of noncompliance components,h.t/may result in anN(modified bathtub) shape. In the presence of both assembly errors and noncompliance components, h.t/may end up with aW(double bathtub)

shape.

Within the context of reliability of electronic products, Wong [5] provided nine critical factors that influence reliability

and contended that essentially all failures are caused by the interactions of built-in flaws, failure mechanisms and stresses.

He further suggested that these three ingredients contribute to form the failure distributions that have roller-coaster-shaped

hazard rate functions.

Besides, mixtures of nonhomogeneous populations is another cause of variation in hazard rate shapes. This is

comprehensively reviewed in [6]; see also [7].

In disciplines such as demography, medical studies and social sciences, there are different causes for having a variety

of hazard shapes.

Because different shapes of ageing distributions are required for fitting various types of lifetime data, numerous lifetime

models were proposed and tested. It is hoped that this summary of methods of constructions will be instrumental to those

dealing with hazard analysis.

3.2. Some simple inelegant techniques

Several elementary techniques could be employed to form a new distribution from an existing lifetime distribution by

doing the following:

adding a location, scale or a shape parameter to enhance the flexibility of the original distribution; and

adding a constant > 0to the existing hazard rate:

hD h.t/ C : (8)

Adding a lifting factor, although not altering the shape of a hazard rate curve, may be required in the presence ofa constant competing risk (see, e.g. [8]). Situations corresponding to (8) arise, for example, in systems whose failures

are governed by two competing risks: perhaps one comes from natural (endogenous) deterioration and the other from

a constant (exogenous) hazard.

A new lifetime distribution may also arise from truncation such as the truncated normal or by removing zero from

discrete lifetime distributions. A zero-inflated distribution such as the zero-inflated Poisson distribution may be used when

dealing with count data with many zeros.

Algebra (sum, product and ratio) of random variables may be formed from two or more lifetime random variables for

some lifetime analysis.

3.3. Transformations of variables

New distributions arise from a transformation of an existing lifetime random variable Xby (i) linear transformation; (ii)

power transformation (e.g. the Weibull is obtained from the exponential); (iii) non-linear transformation (e.g. the lognor-mal from the normal); (iv) log transformation (e.g. the log Weibull, also known as the type 1 extreme value distribution);

and (v) inverse transformation (e.g. the inverse Weibull and the inverse gamma).

3.4. Transformations of distribution/reliability function

LetG./be the original CDF andF ./be the CDF of the new ageing distribution derived from G./by exponentiating thefollowing:

F.t/ D G.t/. For example, the generalized modified Weibull of Carrasco et al. [9], the exponentiated Erlangdistribution of Lai [10] and the exponentiated Weibull of Mudholkar and Srivastava [11]. This is in fact a reversed

proportional hazards model to be discussed in Section 3.19.

F.t/D1 1 G.t/ . The Lomax is obtained from the Pareto in this way.


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3.5. Competing risk approach

The competing risk problem encompasses the study of any failure process in which there is more than one distinct cause

or type of failure. The resulting hazard rate is the sum of the individual hazard rates. The Hjorths [12] model and Xie and

Lais [13] additive Weibull model are the prime examples. The important GompertzMakeham distribution is derived by

allowing the Gompertz distribution to compete with the exponential. The competing model has a similar effect as mixtures

in that the resulting hazard rate would often lead to a bathtub shape.

3.6. Mixtures of two or more lifetime distributions

Mixtures arise from two or more inhomogeneous populations being mixed together. Let p be the mixing proportion oftwo survival functions NF1and NF2, then the survival function and the hazard rate of the mixture are given respectively as

NF.t/Dp NF1.t / C .1 p/ NF2.t /I 0 < p < 1; (9)

and

h.t/D pf1.t / C .1 p/f2.t/

p NF1.t/ C .1 p/ NF2.t/: (10)

The preceding equation may also be written as

h.t/Dw.t/h1.t/ C .1 w.t//h2.t/; (11)

wherew.t/Dp NF1.t/= NF.t/;0 6 w.t/ 6 1.For a recent example, see the finite mixture of Burr type XII distribution and its reciprocal by Ahmadet al. [14]. The

hazard rate of a mixture of two distributions may lead to a bathtub shape (e.g. mixture of two gammas).

3.7. Linear combination of two hazard rate functions

A mixture or a linear combination of two unimodal (bathtub) hazard rate functions of the form h.t/ D h1.t / Ch2.t/; ; > 0 can result in a bimodal (a double-bathtub shaped) hazard rate if the two turning points are suitablyfar away from each other.

The resulting survival function can be expressed as

NF.t/Dexp

Z t0

h1.t /dt

Z t0

h2.t/dt

I ; > 0:

3.8. Generalized mixtures

We sayF.t/is a generalized mixture distribution if it can be written as

F.t/DpF1.t/ C .1 p/F2.t/; (12)

wherep is a real number such that

1

1 6 p 6

1

1 : (13)

Here, D inft >0.f2.t/=f1.t// 6 1and D inft >0.f2.t/=f1.t// > 1. The condition (13) is imposed to ensure Fto be aproper distribution function.

Thus, we see that the generalized mixtures include both the usual mixtures and mixtures with some negative coefficients.

Although the generalized mixtures have been considered in various contexts in the past, Navarro et al. [15] provided a

comprehensive study of their reliability properties. In particular, they considered the cases when h1.t/ D 0 andh2.t/is either linear or the hazard rate of the extension of the exponential-geometric distribution defined in [16]. It was shown

that the resulting hazard rate curve can achieve various shapes such as increasing, decreasing, bathtub shape (U shape),inverted bathtub (upside-down bathtub or unimodal), modified bathtub (first increasing then Ushape) or reflected N shape(first decreasing then an inverted bathtub).

Note that expression (11) continues to hold for generalized mixtures, but in this case, wp can be negative orgreater than 1.

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3.9. Convolutions

A simple technique in constructing a life distribution is by the convolutions of simple distributions. Let

T DX1C X2C C Xn;

where theX0i s are independent lifetime random variables. For example, ifXi are independent and identically distributedexponential random variables, then Thas the Erlang distribution (gamma distribution with the shape parameter beingan integer).

3.10. Compound (infinite mixture) distributions

Let FD fF j2 gbe a family of distributions and G be the distribution of. Then,

F.t/D

Z

F .x/dG./; (14)

is a compound distribution, that is,F ./is the continuous (infinite) mixture ofF ./with the mixing distributionG./. IfGis discrete and finite, then a compound distribution reduces to a finite mixture distribution as given in (9). For example,if the parameter of the exponential distribution has a gamma distribution, then the resulting distribution is a Pareto,which is a compound distribution.

Equation (14) may be expressed in terms of survival functions instead:

NF.t/D

Z

NF .x/dG./:

Other forms of compounding include T D min.X1; X2; : : : ; X N/ or T D max.X1; X2; : : : ; X N/, whereN is a discretepositive random variable .

For example, ifXi is exponential and Nis geometric [17], then T Dmax.X1; X2; : : : ; X N/ is the so-called exponential-geometric distribution. We may think of a situation where failure of a device occurs because of the presence of an unknown

number, N, of initial defects of the same kind. The X0s represent their lifetimes, and each defect can be detected onlyafter causing failure, in which case it is repaired perfectly. Then,Trepresents the time to the first failure.

In a Bayesian nonparametric context, there are many works considering infinite mixtures with a mixing distribution

chosen by a process (e.g. Dirichlet process) on the space of all probability measures (see, e.g. [18]).

3.11. Probability integral transforms

The idea is to incorporate a distribution into a larger family through an application of the probability integral transform.

Specifically, given two lifetime (ageing) densitiesg1./andg2./with the latter having support on the unit interval, then anew ageing distribution may be obtained by

f.t/Dg2.G1.t//g1.t/; (15)

where G1./ is the CDF ofg1./. For example, Wahedet al.[19] constructed the beta-Weibull with G1.t / being the WeibullCDF andg2./the beta density.

Equation (15) is equivalent to F.t/DG2.G1.t//. Another possibility is

NF.t/DG2

. NG1

.t//; NG1

.t/D1 G1

.t/:

For example, Marshall and Olkin [16] constructed a family of distributions withG2.x/ D x

1 .1 /x; > 0andG1.t/

is either the exponential or the Weibull.

3.12. Beta-G distributions

This is in fact a special case of the forgoing subsection in which G2.t/has a beta distribution with parametersaandb andG1.t/DG.t/denotes any ageing distribution function. Then, the beta-G distribution is defined as

F.t/D 1

B.a; b/

Z G.t/0

xa1.1 x/b1dxI a; b > 0; (16)


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whereB.a; b/is the beta function. Several examples are listed in the following table:

G.t/ Reference

Normal Eugeneet al.[20]

Frchet Nadarajah and Gupta [21]

Gumbel Nadarajah and Kotz [22]

Exponential Nadarajah and Kotz [23]Weibull Wahedet al.[19]

Modified Weibull Silvaet al.[24]

3.13. Probability-generating function induced G distributions

Consider a zero-truncated discrete distribution with probability-generating function:

P.s/D

nXiD1

pi si ; 1 6 s 6 1:

LetG.t/be the CDF of a lifetime distribution. We can generate two new distributions as follows:

1. F.t/DP .G.t//; and2. NF.t/DP . NG.t//:

Special case.The geometric distribution with support f1 ; 2 ; : : : ; g:

pi D qi1p; 1 < p < 1; qD1 p; i D 1; 2 ; : : : ; soP.s/D

ps

1 qs:

Case1

F.t/D pG.t/

1 qG.t/ (17)

Case2NF.t/D p

NG.t/

1q N

G.t/

. It follows that

F.t/D1 NF.t/D1 q NG.t/ p NG.t/

1 q NG.t/D

G.t/

1 q NG.t/: (18)

Combining (17) and (18) to form a single parametric family:

F.t/D G.t/

1 NG.t/; ND 1 ; 0 < 1(see [2]) for details).Fis called the exponential geometric whenG is exponential [25] and Weibull geometric when G is Weibull [26].

3.14. Laplace transform method

Let.s/ DR1

0 esx dG.x/; s > 0 be the Laplace transform of the distribution function G . Then, a family of survival

functions can be constructed by incorporating the Laplace transform parameter s as an additional distribution parameter:

NF .t js/D 1

.s/

Z 1

t

esx dG.x/: (20)

The resulting density function has a simple form (see, [2, p.260]):

f .t js/Dest g.t/

.s/ : (21)

We note that a new bivariate lifetime distribution may also obtained from another bivariate distribution in the same manner.

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3.15. Transformations from the normal distribution function

Letg./be an increasing function oft , then a new ageing distribution can be obtained by settingF.t/D .g.t//, where./denotes the standardized normal distribution function.

The lognormal distribution is constructed from the normal:

F.t/D log t

; 1< < 1; > 0; (22)whereg.t/D.log t /=:

The BirnbaumSaunders distribution is given by

F.t/D

1g.t=/

; ; > 0; (23)

whereg.t/Dt 1=2 t1=2:

The inverse Gaussian can also be derived analogously albeit less easily.

3.16. Ageing distributions constructed from hazard function

Equation (3) provides a convenient platform to construct Weibull type and other types of ageing distributions.

Gurvichet al.[27] proposed a method of construction using

NF.t/DexpfG.t/g; > 0; (24)

where G.t/ is an increasing non-negative function oft . This formulation essentially restates the relationship between H.t/and NF.t/as given in (3) via setting H.t/DG.t /.

Lai and Xie [1, Chapter 3] gave an account of Weibull-related distributions that are largely derived by this method.

Thus, F.t/ may be generated by assigning a non-negative and increasing function to its hazard function. Typically, thehazard functionH.t/of a modified Weibull contains .t/ ort .

3.17. Life distributions arising from mean residual life specifications

In Section 2, we see that the mean residual life .t/completely characterizes a lifetime distribution. Thus, one can con-struct a distribution from.t/that has a specific shape. For example, Shen et al.[28] constructed a life model that has anupside-down bathtub-shaped mean residual life through (6), which expressesh.t/in terms of.t/.

3.18. Adding frailty and resilience parameters

This method of constructing a frailty parameter family (a proportional hazards family) is disused in [2, Section E ofChapter 7].

LetF.t/be a distribution function with hazard rate function h.t/. Suppose a new distributionF .j/is defined in termsofFby the formula

NF .t j/D NF.t/ ; NF D1 FI > 0: (25)

Then,is called the frailty parameter andfF .j/; > 0gis a proportional hazards family with underlying distributionF. Clearly, F .j1/ D F ./. Models in which is regarded as a random variable are much used in survival analysis and

are often called Cox proportional hazards models (often shortened as Cox models) or frailty models. The density andhazard rate that correspond to F .t j/are respectively given by

f .t j/Df .t/ NF.t/1

and

h.tj/D h.t/: (26)

h.t/is often known as the baseline hazard rate in survival analysis and the frailty parameter is a function ofk covariates1; 2; : : : ; k usually in the formD expf11C 22C C k kg.

Finkelstein [6] argued that the frailty variable may be considered as the mixing random variable for the heterogeneouspopulation, so the population (observed) hazard rate is given via mixtures instead of the conditional hazard rate as in (26).


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Accelerated life model.In survival analysis, an accelerated life model is a parametric model that provides an alternative

to the proportional hazards models. Whereas a proportional hazards model assumes that the effect of a covariate is to

multiply the baseline hazard by some constant, an accelerated life model is to multiply the predicted event time.

For a distribution to be used in an accelerated model, it must have a parameterization that includes a scale parameter.

The logarithm of the scale parameter is then modelled as a linear function of the covariates (see, e.g. [29]).

3.19. Exponentiated type distributions

By raising the distribution function Fto the power of, we obtain a resilience parameter family of distributions. Thedistribution given byF .t j/D F.t/ is also known as the proportional reversed hazard rate model, because the reversedhazard rate ofF .t j/ is proportional to the reversed hazard rate ofF.t/. For a proportional reversed hazards model,see [30,31].

The distribution obtained via exponentiating may be called the exponentiated distribution. The generalized exponential

(also known as the exponentiated exponential) distribution proposed by Gupta and Kundu [32] is a prime example.

Nadarajah and Kotz [33] also constructed several exponentiated-type distributions that generalize the standard gamma,

standard Weibull, standard Frchet (inverse Weibull) and other distributions.

3.20. Adding a tilt parameter

The idea of adding a tilt parameter is discussed in [2, Section Fof Chapter 7]. Suppose that F .j/is defined in terms of

the underlying distributionFby the formula

F .t j /

NF .t j /D

1

F.t/

NF.t/; > 0; t > 0: (27)

It follows that

NF .t j /D NF.t/

F.t/ C NF.t/D

NF.t/

1 N NF.t/; ND 1 ; > 0: (28)

The resulting density and hazard rates are respectively given by

f .t j/D f.t/

1 N NF.t/

2 and h.t j /D

h.t/

1 N NF.t/

:

Marshall and Olkin [16] constructed two tilt parameter families with Fbeing the exponential and Weibull, whereasGhitany et al.[34] constructed the Lomax tilt family.

3.21. Discretizations

LetY DX or Y DX C 1, where [ ] denotes the integer part of a continuous lifetime variableXwith CDFF. Then, adiscrete lifetime random variableYmay be defined by

p.x/DPrY Dx DPrx 6 X < xC 1DF .xC 1/ F.x/I x D0; 1;2:: : (29)

or

p.x/DPr.Y Dx/ DPrx 1 6 X < xDF.x/ F .x 1/; xD1; 2 : : : (30)

The discrete Weibull, discrete inverse Weibull, discrete Burr and Pareto distributions are all constructed in this manner

(see, e.g. [35]).

4. Constructions of generalized Weibulls

4.1. Standard Weibull distribution

The Weibull distribution has been found very useful in fitting reliability, survival and warranty data, and thus it is one of

the most important continuous distributions in applications. A drawback of the Weibull distribution as far as lifetime anal-

ysis is concerned is the monotonic behaviour of its hazard (failure) rate function. In real life applications, empirical hazard

rate curves often exhibit nonmonotonic shapes such as a bathtub, upside-down bathtub (unimodal) and others. Thus, there

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is a genuine desire to search for some generalizations or modifications of the Weibull distribution that can provide more

flexibility in lifetime modelling.

The standard Weibull distribution is given by

NF.t/Dexp .t / ; ; > 0I t > 0: (31)

It follows from (3) thatH.t/Dt . By a simple differentiation, we obtain the hazard rate function h.t/Dt 1, which

is increasing (decreasing) if > 1 . < 1/. We now see that despite its many applications, the Weibull distribution lacksflexibility for many reliability applications. For other properties of the Weibull distribution, we refer our readers to [36]

for details.

4.2. Generalizations and modifications

There are many generalizations or extensions of the Weibull in the literature. In a strict sense, a generalized Weibull can

be reduced to (31) by setting one of their parameters to zero or letting it converge to zero.

Consider the generalized Weibull of Mudholkar et al.[37] defined by the survival function:

NF.t/D1

"1

1

t

1=#; ; > 0; (32)

where the support ofF is.0; 1/if 6 0and

0;=1=

if > 0. As !0, NF.t/tends to (31).A generalized Weibull may involve two or more Weibull distributions through the following: (i) finite mixtures; (ii)

n-fold competing risk (equivalent to independent components being arranged in a series structure); (iii) n-fold multiplica-tive models; and (iv) n-fold sectional models; see [36] for further details. Some of the generalizations also involve mixturesof two different generalized Weibulls, for example, that of Bebbingtonet al.[38].

4.3. Some important generalized Weibull families

Several generalized Weibull families that are constructed through generalizing H.t/ or exponentiating either F.t/ orNF.t/ of the Weibull distribution are given in the following discussion. A salient feature of these families is that their

survival and hazard rate functions are also quite simple because of the manner of constructions. In addition, they can give

rise to nonmonotonic hazard rate functions of various shapes such as a bathtub, upside-down bathtub (unimodal) or a

modified bathtub.

4.3.1. Modified Weibull of Laiet al. Laiet al.[39] introduced a generalized Weibull

NF.t/Dexpn

at eto

; > 0; ; a > 0I t > 0; (33)

which reduces to (31) whenD0. For moments of the aforementioned distribution, see [40].

4.3.2. Generalized modified Weibull family. The survival function of the distribution studied by Carrascoet al.[7] is

NF.t/D1

1 expn

at eto

; > 0; ;; a > 0I t > 0: (34)

Clearly, it is a simple extension of the modified Weibull distribution of Lai et al.[39] because (34) reduces to (31) when

D 1. In fact, it includes several other distributions such as type 1 extreme value, the exponentiated Weibull of Mud-holkar and Srivastava [11] as given in (38) and others. An important feature of this lifetime (ageing) distribution is its

considerable flexibility in providing hazard rates of various shapes.

4.3.3. Generalized WeibullGompertz distribution. Nadarajah and Kotz [41] proposed a generalization of Weibull with

four parameters having survival function given as:

R.t/Dexpn

at b

ectd

1o

; a; d > 0I b; c > 0I t > 0: (35)

Because (35) includes the Gompertz (or GompertzMakeham) as its special case when b D 0, we may refer it as thegeneralized WeibullGompertz distribution. Clearly, it contains several distributions listed in [42, Table 1]. Again, it can

prescribe increasing, decreasing or bathtub-shaped hazard rate functions.


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4.3.4. Generalized power Weibull family. Nikulin and Haghighi [43] proposed a three-parameter family of ageing

distributions:

NF.t/Dexpn

1

1 C .t=/o

; t > 0I ; ; > 0: (36)

Its hazard rate functionh.t/can give rise to increasing, decreasing, bathtub or upside-down bathtub shapes. Of course, the

caseD 1reduces it to the Weibull distribution.

4.3.5. Flexible Weibull distribution. Bebbingtonet al.[44] obtained a generalization of the Weibull distribution having a

simple and yet flexible cumulative failure rate function H:

NF.t/Dexpn

et= to

I ; > 0I t > 0: (37)

It was shown that the distribution has an increasing hazard rate if > 27=64 and a modified bathtub (N orroller-coaster shape) hazard rate if 6 27=64. Note that there are few generalized Weibull distributions that havethis shape.

4.3.6. Exponentiated Weibull family. Mudholkar and Srivastava [11] proposed a simple generalization of Weibull

distribution by simply raising the CDF of the Weibull to the power of giving

NF.t/D1

1 exp .t=/

; t > 0I ; > 0; > 0: (38)

The special caseD 1reduces (38) to the standard Weibull distribution.The distribution is found to be very flexible for reliability modelling as it can model increasing (decreasing),

bathtub-shaped (upside-down) hazard rate distributions.

4.3.7. The odd Weibull family. Cooray [44] has constructed a generalization of the Weibull family called the odd Weibull

family. LetTbe the lifetime variable that follows a certain distribution F, say, the Weibull. Then, the odds that an individ-ual will die at time t isF.t/= NF.t/. Let this odds of death be denoted by y , and it can be considered as a random variableY, so we can write

Pr.Y 6 y/ DG.y/DG

F.t/

NF.t/

:

Now, supposeF.t/D1 e.t=/

andYhas a log-logistic distribution with G.y/D 1

1 C y1

; > 0. Then, the

corrected (resulting) distribution ofTis given by

F.t /D1

1 C

e.t=/

11

I ; ; > 0: (39)

IfFhas the inverse Weibull distribution given byF.t/De.t=/

;

< 0; > 0, then

F.t/D1

1 C

e.t=/

11

I < 0; ; > 0: (40)

Combining (39) and (40) and lettingD ; > 0, we obtain the CDF of the odd Weibull family as

F.t/D1

1 C

e.t=/

11

I > 0; > 0: (41)

Cooray [45] has shown that the odd Weibull family can model various hazard shapes (increasing, decreasing, bathtub and

unimodal); thus the family is proved to be flexible for fitting reliability and survival data.

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5. Generalizations of Gompertz distribution

Gompertz and its extension GompertzMakeham distributions are well known in insurance, mortality and population

studies. The original Gompertz distribution has mortality (hazard) rate function given by

h.t/Det ; ; > 0; (42)

The GompertzMakeham distribution was constructed to improve the fit of the actuarial data provided by the Gompertzdistribution [46]. The modified mortality (hazard) rate function is

h.t/Det C c; ;; c > 0: (43)

It follows from (42) or (43) that, in the GompertzMakeham model, the mortality rate is positively accelerating with

aget . This property does not seem to be consistent with the real-life phenomenon. It has been reported in many studies,although the human mortality rate increases during the late life phase, it nevertheless levels off to a finite value as age

advances. It is now generally accepted that the GompertzMakeham model overestimates the senility at advanced ages.

For a recent discussion of this issue as related to the GompertzMakeham distribution, see [47].

The first important observation of mortality levelling off in humans was given by Greenwood and Irwin [48, p. 14].

They stated that the increase of mortality rate with age advances at a slackening rate, that nearly all, perhaps all, meth-

ods of graduation of the type of Gompertzs formula overstate senile mortality. They also suggested the possibility that,with advancing age, the rate of mortality asymptotes to a finite value. See also a recent discussion on late-life mortality

deceleration phenomenon in humans by Vaupel [49].

Economos [50] was one of the first to study mortality levelling phenomenon in animals and manufactured products. He

demonstrated mortality levelling off at advanced ages for invertebrates (including fruit flies and house flies), rodents and

several manufactured products.

The first mathematical model for mortality levelling-off phenomenon was proposed by the British actuary Robert

Beard ([51, 52]). See also the discussion in [6, Section 6].

The logistic frailty model given by

h.t/D e t

1 C s e

t 1I s; ; > 0; (44)

wheres may be considered as a deceleration parameter, is an extension of the Gompertz distribution. Equation (44)reduces to (42) when s D 0. It is easy to show that in the logistic frailty model, the mortality rate h.t/ increases with aslackening rate as age advances ifs < . In fact, it asymptotes to a finite value =s.

Many other extensions and modifications of the Gompertz distribution were constructed to meet different population

species and characteristics; some of them are given in [2, Chapter 10].

6. Selection of ageing models

Besides the ageing models discussed previously, there exist many other feasible models. With such a plethora of

candidates, a prospective analyst is faced with a dilemma of choosing a right model to fit their lifetime (ageing) data.

6.1. Some suggestions in reliability engineering

(i) We often see the use of the scaled total time on test statistic.r=n/DT .Xr /=.X1C X2C C Xn/, derived from

the total time on test transform ofFgiven byH1F .p/DRF1.p/

0NF.x/dx.

Here,T .Xr /D nX1C .n 1/.X2 X1/ C C .n rC 1/.Xr Xr1/,r D1; 2 ; : : : ; n, wheren is the totalnumber of failure times ordered as X1 < X2


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is another useful tool to identify the shape of a hazard rate function. Here,t1 < t2


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C. D. LAI

Acknowledgement

The author is indebted to the referees and the editor for their careful reading of the manuscript. Their comments and suggestionshave led to significant improvements to the paper.

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