Construction of Survival Distributions

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 distributions C. 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 Weibull distribution are often required to prescribe the nonmonotonic nature of the empirical hazard rates. We also briefly outline some of the 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) distributions 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 Gompertz–Makeham) distributions are possibly the earliest ageing models used for smoothing mortality tables, which were of considerable interest to actuaries. Several 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. Let T be the lifetime random variable with f .t /, F .t / being its probability density function and cumulative distribution function (CDF), respectively. The reliability or survival function is given by FN .t / D 1  F .t /. The hazard (failure) rate function is defined as h.t / D

f .t / f .t / I D 1  F .t / FN .t /

(1)

h.t /t gives (approximately) the probability of failure in .t; t C t  given the ‘unit’ has survived until time t .

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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]

C. D. LAI

The cumulative hazard rate function is defined as Z

t

H.t / D

h.x/dx:

(2)

0

It is easy to show that the reliability function can be represented as FN .t / D e 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/ D 0; and (iii) lim H.t / D 1. t !1 Because the reliability function FN .t / and the hazard rate function h.t / can be uniquely determined from each other, a new 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 R1 FN .x/dx .t / D E.T  t j T > t / D t ; (5) FN .t / summarizing the entire residual life of a unit at age t . The hazard rate function h.t / and the survival function FN .t / may be obtained from .t / through the relationships: h.t / D

1 C 0 .t / ; t > 0; .t /

(6)

and   Z t  1 N F .t / D .x/ dx ; t > 0 exp  .t / 0

(7)

with  D E.T / D .0/. If h.t / is unimodal (inverted bathtub), then either .t / is increasing (if h.0/ > 1=) or it has a bathtub shape. Furthermore, 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 let G denote the base (underlying) distribution from which a new distribution F is constructed from. Unless otherwise specified, we assume F and G have density functions f and g, 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?

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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. Component 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 /. Copyright © 2012 John Wiley & Sons, Ltd.

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With the inclusion of assembly errors, h.t / emerges as bathtub shaped. This is because the failures resulting from the assembly errors can be viewed as a new mode of failure and the hazard rate associated with this new distribution is generally decreasing with time t . 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 large t , and thus a bathtub shape results. With the inclusion of noncompliance components, h.t / may result in an N (modified bathtub) shape. In the presence of both assembly errors and noncompliance components, h.t / may end up with a W (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  > 0 to the existing hazard rate: h D h.t / C :

(8)

Adding a ‘lifting’ factor , although not altering the shape of a hazard rate curve, may be required in the presence of a 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 X by (i) linear transformation; (ii) power transformation (e.g. the Weibull is obtained from the exponential); (iii) non-linear transformation (e.g. the lognormal 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 Let G./ be the original CDF and F ./ be the CDF of the new ageing distribution derived from G./ by exponentiating the following: 

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F .t / D ŒG.t /˛ . For example, the generalized modified Weibull of Carrasco et al. [9], the exponentiated Erlang distribution 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 / D 1  Œ1  G.t /ˇ . 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 Hjorth’s [12] model and Xie and Lai’s [13] additive Weibull model are the prime examples. The important Gompertz–Makeham 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 of two survival functions FN1 and FN2 , then the survival function and the hazard rate of the mixture are given respectively as FN .t / D p FN1 .t / C .1  p/FN2 .t /I 0 < p < 1;

(9)

and h.t / D

pf1 .t / C .1  p/f2 .t / : p FN1 .t / C .1  p/FN2 .t /

(10)

The preceding equation may also be written as h.t / D w.t /h1 .t / C .1  w.t //h2 .t /;

(11)

where w.t / D p FN1 .t /=FN .t /; 0 6 w.t / 6 1. For a recent example, see the finite mixture of Burr type XII distribution and its reciprocal by Ahmad et 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 / C ˇh2 .t /; ˛; ˇ > 0 can result in a bimodal (a double-bathtub shaped) hazard rate if the two turning points are suitably far away from each other. The resulting survival function can be expressed as   Z t Z t FN .t / D exp ˛ h1 .t /dt  ˇ h2 .t /dt I ˛; ˇ > 0: 0

0

3.8. Generalized mixtures We say F .t / is a generalized mixture distribution if it can be written as F .t / D pF1 .t / C .1  p/F2 .t /;

(12)

1 1 6p6 : 1ˇ 1˛

(13)

where p is a real number such that

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Here, ˛ D inft >0 .f2 .t /=f1 .t // 6 1 and ˇ D inft >0 .f2 .t /=f1 .t // > 1. The condition (13) is imposed to ensure F to be a proper 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 and h2 .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 U shape) 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 or greater than 1. Copyright © 2012 John Wiley & Sons, Ltd.

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3.9. Convolutions A simple technique in constructing a life distribution is by the convolutions of simple distributions. Let T D X1 C X2 C    C Xn ; where the Xi0 s are independent lifetime random variables. For example, if Xi are independent and identically distributed exponential random variables, then T has the Erlang distribution (gamma distribution with the shape parameter being an integer). 3.10. Compound (infinite mixture) distributions Let F D fF j 2 ‚g be a family of distributions and G be the distribution of ‚. Then, Z F .t / D F .x/dG./;

(14)

‚

is a compound distribution, that is, F ./ is the continuous (infinite) mixture of F ./ with the mixing distribution G./. If G is 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: Z FN .t / D FN .x/dG./: ‚

Other forms of compounding include T D min.X1 ; X2 ; : : :; XN / or T D max.X1 ; X2 ; : : :; XN /, where N is a discrete positive random variable . For example, if Xi is exponential and N is geometric [17], then T D max.X1 ; X2 ; : : :; XN / is the so-called exponentialgeometric 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 X 0 s represent their lifetimes, and each defect can be detected only after causing failure, in which case it is repaired perfectly. Then, T represents 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) densities g1 ./ and g2 ./ with the latter having support on the unit interval, then a new ageing distribution may be obtained by f .t / D g2 .G1 .t //g1 .t /;

(15)

where G1 ./ is the CDF of g1 ./. For example, Wahed et al. [19] constructed the beta-Weibull with G1 .t / being the Weibull CDF and g2 ./ the beta density. Equation (15) is equivalent to F .t / D G2 .G1 .t //. Another possibility is FN .t / D G2 .GN 1 .t //; GN 1 .t / D 1  G1 .t /: For example, Marshall and Olkin [16] constructed a family of distributions with G2 .x/ D

ˇx ; ˇ > 0 and G1 .t / 1  .1  ˇ/x

is either the exponential or the Weibull. 3.12. Beta-G distributions

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This is in fact a special case of the forgoing subsection in which G2 .t / has a beta distribution with parameters a and b and G1 .t / D G.t / denotes any ageing distribution function. Then, the beta-G distribution is defined as Z G.t / 1 x a1 .1  x/b1 dxI a; b > 0; (16) F .t / D B.a; b/ 0

C. D. LAI

where B.a; b/ is the beta function. Several examples are listed in the following table: G.t /

Reference

Normal Fréchet Gumbel Exponential Weibull Modified Weibull

Eugene et al. [20] Nadarajah and Gupta [21] Nadarajah and Kotz [22] Nadarajah and Kotz [23] Wahed et al. [19] Silva et al. [24]

3.13. Probability-generating function induced G distributions Consider a zero-truncated discrete distribution with probability-generating function: P .s/ D

n X

pi s i ; 1 6 s 6 1:

i D1

Let G.t / be the CDF of a lifetime distribution. We can generate two new distributions as follows: 1. F .t / D P .G.t //; and N //: 2. FN .t / D P .G.t Special case. The geometric distribution with support f1; 2; : : :; g: ps pi D q i 1 p; 1 < p < 1; q D 1  p; i D 1; 2; : : :; so P .s/ D : 1  qs Case 1

F .t / D Case 2 FN .t / D

N / p G.t N /. 1q G.t

pG.t / 1  qG.t /

(17)

It follows that F .t / D 1  FN .t / D

N /  p G.t N / G.t / 1  q G.t D : N / N / 1  q G.t 1  q G.t

(18)

Combining (17) and (18) to form a single parametric family: F .t / D

 G.t / ; N D 1  ; 0 <  < 1: 1  N G.t /

(19)

In (17), 0 <  D p 6 1, and in (18),  D 1=p > 1 (see [2]) for details). F is called the exponential geometric when G is exponential [25] and Weibull geometric when G is Weibull [26]. 3.14. Laplace transform method R1 Let .s/ D 0 e sx 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: Z 1 1 N e sx dG.x/: (20) F .t js/ D .s/ t The resulting density function has a simple form (see, [2, p.260]): f .t js/ D

e st g.t / : .s/

(21)

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We note that a new bivariate lifetime distribution may also obtained from another bivariate distribution in the same manner. Copyright © 2012 John Wiley & Sons, Ltd.

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3.15. Transformations from the normal distribution function Let g./ be an increasing function of t , then a new ageing distribution can be obtained by setting F .t / D ˆ.g.t //, where ˆ./ denotes the standardized normal distribution function. 



The lognormal distribution is constructed from the normal:   log t   ; 1 <  < 1; > 0; F .t / D ˆ

(22)

where g.t / D .log t  /= : The Birnbaum–Saunders distribution is given by   F .t / D ˆ ˇ 1 g.t =˛/ ; ˛; ˇ > 0;

(23)

where g.t / D t 1=2  t 1=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. Gurvich et al. [27] proposed a method of construction using FN .t / D expfG.t /g;  > 0;

(24)

where G.t / is an increasing non-negative function of t . This formulation essentially restates the relationship between H.t / and FN .t / as given in (3) via setting H.t / D G.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, the hazard function H.t / of a modified Weibull contains .t /˛ or t ˛ . 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 construct a distribution from .t / that has a specific shape. For example, Shen et al. [28] constructed a life model that has an upside-down bathtub-shaped mean residual life through (6), which expresses h.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 of Chapter 7]. Let F .t / be a distribution function with hazard rate function h.t /. Suppose a new distribution F .j / is defined in terms of F by the formula FN .t j / D ŒFN .t / ; FN D 1  F I > 0:

(25)

Then, is called the frailty parameter and fF .j /; > 0g is a proportional hazards family with underlying distribution F . 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 and hazard rate that correspond to F .t j / are respectively given by f .t j / D f .t /ŒFN .t /1 and h.t j / D h.t /:

(26)

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h.t / is often known as the baseline hazard rate in survival analysis and the frailty parameter is a function of k covariates ´1 ; ´2 ; : : :; ´k usually in the form D expfˇ1 ´1 C ˇ2 ´2 C    C ˇk ´k g . Finkelstein [6] argued that the frailty variable may be considered as the mixing random variable for the heterogeneous population, 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 F to the power of , we obtain a resilience parameter family of distributions. The distribution given by F .t j / D F .t / is also known as the proportional reversed hazard rate model, because the reversed hazard rate of F .t j / is proportional to the reversed hazard rate of F .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 Fréchet (inverse Weibull) and other distributions. 3.20. Adding a tilt parameter The idea of adding a tilt parameter is discussed in [2, Section F of Chapter 7]. Suppose that F .j/ is defined in terms of the underlying distribution F by the formula F .t j/ 1 F .t / D ;  FN .t / FN .t j/

 > 0; t > 0:

(27)

It follows that FN .t j/ D

 FN .t /  FN .t / D ; N D 1  ;  > 0: N F .t / C  F .t / 1  N FN .t /

(28)

The resulting density and hazard rates are respectively given by f .t j/ D

f .t / Œ1  N FN .t /2

and

h.t j/ D

h.t / : Œ1  N FN .t /

Marshall and Olkin [16] constructed two tilt parameter families with F being the exponential and Weibull, whereas Ghitany et al. [34] constructed the Lomax tilt family. 3.21. Discretizations Let Y D ŒX  or Y D ŒX  C 1, where [ ] denotes the integer part of a continuous lifetime variable X with CDF F . Then, a discrete lifetime random variable Y may be defined by p.x/ D PrŒY D x D PrŒx 6 X < x C 1 D F .x C 1/  F .x/I x D 0; 1; 2: : :

(29)

p.x/ D Pr.Y D x/ D PrŒx  1 6 X < x D F .x/  F .x  1/; x D 1; 2: : :

(30)

or

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

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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 analysis 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 Copyright © 2012 John Wiley & Sons, Ltd.

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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 FN .t / D exp .t ˛ / ; ; ˛ > 0I t > 0:

(31)

It follows from (3) that H.t / D t ˛ . By a simple differentiation, we obtain the hazard rate function h.t / D ˛t ˛1 , which is increasing (decreasing) if ˛ > 1 .˛ < 1/. We now see that despite its many applications, the Weibull distribution lacks flexibility 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: "   ˛ 1= # t N F .t / D 1  1  1   ; ˛; ˇ > 0; (32) ˇ   where the support of F is .0; 1/ if  6 0 and 0; ˇ=1=˛ if  > 0. As  ! 0, FN .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 multiplicative models; and (iv) n-fold sectional models; see [36] for further details. Some of the generalizations also involve mixtures of two different generalized Weibulls, for example, that of Bebbington et 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 / or FN .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 Lai et al. Lai et al. [39] introduced a generalized Weibull n o FN .t / D exp at ˛ e t ;  > 0; ˛; a > 0I t > 0;

(33)

which reduces to (31) when  D 0. For moments of the aforementioned distribution, see [40]. 4.3.2. Generalized modified Weibull family. The survival function of the distribution studied by Carrasco et al. [7] is  n o ˇ ;  > 0; ˛; ˇ; a > 0I t > 0: FN .t / D 1  1  exp at ˛ e t

(34)

Clearly, it is a simple extension of the modified Weibull distribution of Lai et al. [39] because (34) reduces to (31) when ˇ D 1. In fact, it includes several other distributions such as type 1 extreme value, the exponentiated Weibull of Mudholkar 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 Weibull–Gompertz distribution. Nadarajah and Kotz [41] proposed a generalization of Weibull with four parameters having survival function given as: n o  d R.t / D exp at b e ct  1 ; a; d > 0I b; c > 0I t > 0: (35)

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Because (35) includes the Gompertz (or Gompertz–Makeham) as its special case when b D 0, we may refer it as the generalized Weibull–Gompertz 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: n   o FN .t / D exp 1  1 C .t =ˇ/˛ ; t > 0I ˛; ˇ;  > 0: (36) Its hazard rate function h.t / can give rise to increasing, decreasing, bathtub or upside-down bathtub shapes. Of course, the case  D 1 reduces it to the Weibull distribution. 4.3.5. Flexible Weibull distribution. Bebbington et al. [44] obtained a generalization of the Weibull distribution having a simple and yet flexible cumulative failure rate function H : n  o FN .t / D exp  e ˛t ˇ =t I ˛; ˇ > 0I t > 0:

(37)

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

 FN .t / D 1  1  exp .t =ˇ/˛ ; t > 0I ˛; ˇ > 0;  > 0:

(38)

The special case  D 1 reduces (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. Let T be the lifetime variable that follows a certain distribution F, say, the Weibull. Then, the odds that an individual will die at time t is F .t /=FN .t /. Let this odds of death be denoted by y, and it can be considered as a random variable Y , so we can write   F .t / Pr.Y 6 y/ D G.y/ D G : FN .t / 1  ˛ Now, suppose F .t / D 1  e .t =ˇ / and Y has a log-logistic distribution with G.y/ D 1  1 C y  ;  > 0. Then, the corrected (resulting) distribution of T is given by    1 ˛ I ˛; ˇ;  > 0: F  .t / D 1  1 C e .t =ˇ /  1

(39)

˛

If F has the inverse Weibull distribution given by F .t / D e .t =ˇ / ; ˛ < 0; ˇ > 0, then    1 ˛ F  .t / D 1  1 C e .t =ˇ /  1 I ˛ < 0; ˇ;  > 0:

(40)

Combining (39) and (40) and letting  D ˙;  > 0, we obtain the CDF of the odd Weibull family as    1 ˛ F  .t / D 1  1 C e .t =ˇ /  1 I ˇ > 0; ˛ > 0:

(41)

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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. Copyright © 2012 John Wiley & Sons, Ltd.

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5. Generalizations of Gompertz distribution Gompertz and its extension Gompertz–Makeham distributions are well known in insurance, mortality and population studies. The original Gompertz distribution has mortality (hazard) rate function given by h.t / D e t ; ;  > 0;

(42)

The Gompertz–Makeham distribution was constructed to improve the fit of the actuarial data provided by the Gompertz distribution [46]. The modified mortality (hazard) rate function is h.t / D e t C c; ; ; c > 0:

(43)

It follows from (42) or (43) that, in the Gompertz–Makeham model, the mortality rate is positively accelerating with age t . 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 Gompertz–Makeham model overestimates the senility at advanced ages. For a recent discussion of this issue as related to the Gompertz–Makeham 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, methods of graduation of the type of Gompertz’s 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  I s; ;  > 0;  1 C s  e t  1

(44)

where s 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 a slackening rate as age advances if s < . 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/ D T .Xr /=.X1 C X2 C    C Xn /, derived from R F 1 .p/ the total time on test transform of F given by HF1 .p/ D 0 FN .x/dx. Here, T .Xr / D nX1 C .n  1/.X2  X1 / C    C .n  r C 1/.Xr  Xr1 /, r D 1; 2; : : :; n, where n is the total number of failure times ordered as X1 < X2 <    < Xn . The shape characteristics of the scaled total time on test plot .r=n; .r=n// are related to the shape of the empirical hazard rate, which in turn will assist us to select a model. One may, of course, plot the empirical hazard rate directly, particularly when the data are grouped and uncensored. (ii) The empirical cumulative hazard function plot using the Nelson–Aalen estimate defined by X dj nj

(45)

j Wtj 6t

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HO .t / D

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is another useful tool to identify the shape of a hazard rate function. Here, t1 < t2 <    < tk represent the distinct times at which failures are observed (the possibility of more than one failure/death at tj is allowed); dj represents the number of failures/deaths at tj (see, e.g. [53, p. 85]) Plots of HO .t / give useful information about the shape of the hazard rate function. For example, h.t / is constant if H.t / is linear, monotonic if H.t / is either convex or concave and nonmonotonic if H.t / is polynomial. (iii) If these plots show evidence of bimodal or multiple change points, a mixture model could be selected as the starting point. (iv) A flexible model with fewer parameters, say three or less, is generally preferred. (v) Lastly, our choice could be at least partly based on the nature of data sources (subject disciplines). For example, if one is interested in the reliability of electronics, electromechanical and mechanical products, one of the models that gives rise to a bathtub-shaped hazard rate may be appropriate, whereas for social science duration of an event, a unimodal hazard model often provides a good fit. 6.2. Selection of a model from statistical points of view Statisticians generally apply one or more goodness-of-fit tests to see how well a selected distribution fits to the lifetime data. The general procedure consists of defining a test statistic, which is some function of the data measuring the distance between the hypothesis and the data, and then calculating the probability (P -value) of obtaining data, which have a still larger value of this test statistic than the value observed. Several well-known tests such as the Kolmogorov–Smirnov, Anderson–Darling and Chi-squared tests are often applied to judge the model validation. Besides, if the proposed lifetime distributions satisfy the regularity condition for maximum likelihood estimation, then a simple information criterion such as the Akaike information criterion becomes a powerful tool to select the best model.

7. Applications of ageing distributions 7.1. Reliability engineering Ageing distributions are an integral part of reliability engineering both at the component and system levels. Their applications include (but not limited to) lifetime analysis, accelerated life test, product burn-in before field use, replacement and maintenance policies and many others. The monograph by Meeker and Escobar [54] is a highly recommended text for this subject. Most ageing distributions have arisen from reliability contexts. 7.2. Survival analysis in biomedical sciences Survival analysis is an indispensable part of the biomedical science as all biological organisms end up with a ‘death’. In medical studies concerning potentially life-threatening diseases such as cancer, one is interested in the survival time of individuals with the disease measured from the date of diagnosis or a major treatment. It is common to compare medical treatments for a disease in terms of the survival time distributions for patients receiving the different treatments. References [7, 16, 40] are prime examples of applications of ageing distributions in medical sciences. 7.3. Mortality study and insurance The hazard rate is also known as the force of mortality in insurance industries. The applicability of Gompertz’s law to life insurance mortality data is well known. The Gompertz law has been extended in various ways to improve its fit to the whole human life span as well as for various diverse population characteristics. For example, Bebbington et al. [34] have successfully used a mixture of two modified Weibull distributions to fit a human mortality data set. 7.4. Social sciences

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In social sciences, the time spent (duration) in a state before exiting is an important consideration for managers and planners. For example, a social scientist may be concerned with the recidivism in criminal justice, the length of time to complete a PhD degree, the duration an individual remains unemployed, the duration an individual stays in an employment, the duration an individual remains married, the durations of coalitions, the time until announcement of support of a bill, the length a leader stays in power, the duration of a war and others. Surprisingly, many such ‘duration’ variables have a unimodal hazard rate that can be fitted by the inverse Weibull, inverse Gaussian, lognormal, exponentiated or generalized power Weibull family. Copyright © 2012 John Wiley & Sons, Ltd.

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

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