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The expected time lost due to an extra risk
Reliability Engineering and System Safety 82 (2003) 225–228 www.elsevier.com/locate/ress
Short communication
The expected time lost due to an extra risk M.S. Finkelstein Department of Mathematical Statistics, University of the Free State, P.O. Box 339, 9300 Bloemfontein, South Africa Received 10 April 2003; accepted 28 May 2003
Abstract The difference between the mean remaining lifetime in a baseline and a more risky environment is investigated. Two models for a more severe environment are considered. The additive hazards model describes an additional independent risk, whereas the proportional hazards model does not require the assumption of independence. It is shown that the expected time lost due to an extra risk decreases with time for heterogeneous population. The origin of this interesting effect is in the fact that ‘the weakest populations are dying out first’. Simple examples are considered. q 2003 Elsevier Ltd. All rights reserved. Keywords: Mean residual life function; Failure rate; Mixture of distributions; Random environment; Additive hazards; Proportional hazards model
1. Introduction Let X be a lifetime random variable with a Cdf FðxÞ; x [ ½0; 1Þ and a failure rate lðxÞ; x $ 0: Along with the failure rate, the mean residual life (MRL) function plays an important role in reliability theory, survival analysis, risk analysis and demography. General results on the properties of the MRL function can be found in Refs. [7,9,12] to name a few. As usual, define the remaining lifetime variable Xt by the following survival function ðtþx Ft ðxÞ ¼ P{Xt . x} ¼ Fðt þ xÞ ¼ exp 2 lðuÞdu ð1Þ FðtÞ t and the MRL function as the expectation of Xt : ð1 ðxþt FðuÞdu ð1 t ¼ exp 2 lðuÞdu dx mðtÞ ; E½Xt ¼ FðtÞ 0 t
where lðtÞ is the lifetable probability of surviving from birth to age t [11]. Let X; Y be two lifetime random variables with Cdf’s FðtÞ; GðtÞ and failure rates lF ðtÞ; lG ðtÞ; respectively. Let X be the lifetime of an object in some baseline or reference conditions (environment) whereas Y be the lifetime of the same object in a more risky (hazardous) conditions. As the failure rate of an object is a measure of the instantaneous risk, then clearly the following hazard rate ordering X $hr Y holds [13]:
lF ðtÞ # lG ðtÞ;
It is worth noting that in actuarial and demographic literature the failure rate is called the ‘force of mortality’ or the ‘mortality rate’, whereas mðtÞ is usually denoted as eðtÞ and called the ‘remaining life expectancy’. Relation (2) in the corresponding notation is then written as: ð1 lðuÞdu ð3Þ eðtÞ ¼ t lðtÞ E-mail address: [email protected] (M.S. Finkelstein). 0951-8320/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0951-8320(03)00138-8
ð4Þ
which, in accordance with definition (2), implies the MRL ordering X $MRL Y : mF ðtÞ $ mG ðtÞ;
ð2Þ
;t [ ½0; 1Þ
;t [ ½0; 1Þ
ð5Þ
If the goal of the study is to assess quantitatively the impact of additional risks on the MRL function, then it is reasonable to use the following measure to be called the DMRL -distance: DMRL ðtÞ ¼ mF ðtÞ 2 mG ðtÞ;
;t [ ½0; 1Þ
ð6Þ
In some instances the corresponding relative characteristic can be of interest: m ðtÞ RDMRL ¼ 1 2 G ð7Þ mF ðtÞ In fact, we need to obtain the distance DMRL ðtÞ to address the topic of the paper formulated in its title, as DMRL ðtÞ; ;t $ 0 shows the expected remaining lifetime lost due to the object’s exposure to an additional risk. This problem was addressed in
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M.S. Finkelstein / Reliability Engineering and System Safety 82 (2003) 225–228
the literature on demographic issues [9,11] but only for oversimplified settings.
life expectancy mð0Þ: As ð1 ð1 exp{ 2 lr u}FðuÞdu FðuÞdu; mG ð0Þ ¼ mF ð0Þ ¼ 0
0
2. Modelling a more risky environment
the expected lifetime loss due to this risk can be defined as
The most natural way to model additional independent risks is via the additive hazards model
DMRL ð0Þ ¼ mF ð0Þ 2 mG ð0Þ
lG ðtÞ ¼ lF ðtÞ þ lr ðtÞ;
t$0
ð8Þ
where lr ðtÞ is the failure rate which accounts for additional independent risks. It is clear also that Eq. (8) can be also interpreted via the notion of independent competing risks. We shall consider the practical realization of this model in Section 3. The other popular model is the proportional hazards (PH) model:
lG ðtÞ ¼ zlF ðtÞ;
z . 1; t $ 0
ð9Þ
where z is some parameter (in regression models usually a ¼ exp{xT y}; where x ¼ ðx1 ; …; xn Þ is a vector of timeindependent covariates and y ¼ ðy1 ; …; y2 Þ is a vector of unknown parameters). In a more general form Eq. (9) can be written with a time-dependent zðtÞ :
lG ðtÞ ¼ zðtÞlF ðtÞ;
zðtÞ . 1; t $ 0
ð10Þ
It is worth noting, that whereas the additive hazards model is mostly useful for modelling additional independent risks, the PH model usually accounts for a more general situation and under certain restrictions can describe an impact of a more severe environment without assuming independence. The third approach, which is often used for modelling additional risks mostly in engineering applications, is the accelerated life model (ALM): GðtÞ ¼ FðztÞ ðGðtÞ ¼ FðzðtÞÞÞ: It is clear that for z . 1 ðzðtÞ . 1Þ the random variable Y will be stochastically smaller than X : Y #st X; which means that GðtÞ $ FðtÞ: This ordering is weaker than the corresponding hazard rate ordering. The failure rate for the described ALM is defined as
lG ðtÞ ¼ zlF ðztÞ; lG ðtÞ ¼ z0 ðtÞlF ðzðtÞÞ;
z . 1; t $ 0;
ð11Þ
zðtÞ . 1; t $ 0
Thus, it is not necessary that lG ðtÞ $ lF ðtÞ for the ALM as for the previous models. In fact, we need additional assumptions for describing properties of DMRL ðtÞ for the ALM and this topic needs a special study. Therefore, we shall focus on the first two models.
¼
ð1 0
ð1 1 12 exp{ 2 lr u}f ðuÞdu FðuÞdu 2 lr 0 ð12Þ
where f ðtÞ ¼ F 0 ðtÞ: For exponential Cdf FðtÞ ¼ 1 2 exp{ 2 lr t} the right hand side of relation (12) is equal to lr =lðl þ lr Þ: This simple setting can be generalized in many ways. Firstly, it is obvious that relation (12) definitely follows from the additive model (9), where lr ðtÞ ; lr : This specific case allows for simple computations for some other concrete types of FðtÞ as well (e.g. Erlangian and inverse Gaussian). On the other hand, in accordance with definition (2), it is easy to obtain the following general formal equation: DMRL ðtÞ ¼ mF ðtÞ 2 mG ðtÞ ðxþt ð1 ¼ exp 2 lF ðuÞdu dxðtÞ 0
t
ðxþt ð1 exp 2 ðlF ðuÞ þ lr ðuÞÞdu dx 2 0
ð13Þ
t
If lF ðtÞ and lG ðtÞ are known, DMRL ðtÞ can be calculated at least numerically. It is natural and important for application to generalize a single event case to the point process of recurrent events. Let Pt ; t $ 0 denotes the non-homogeneous Poisson process of additional harmful events with rate lP ðtÞ; which potentially can lead to failure (death). Let Pt ; t $ 0 be independent of the lifetime random variable X and assume that the event occurred at time t results in failure with probability uðtÞ and is survived (without any consequences to the object) with probability 1 2 uðtÞ: Let T be the lifetime random variable when Pt ; t $ 0 is the only cause of death of an object. Denote by Fr ðtÞ; lr ðtÞ the corresponding Cdf and the failure rate, respectively. ¼ FðtÞ F r ðtÞ Thus, due to assumption of independence: GðtÞ and the additive hazards model (8) describes this setting. It can be proved [3,8] that ðt ð14Þ Fr ðtÞ ¼ 1 2 exp 2 uðuÞlP du 0
3. Independent additional risks A simple specific case of one additional hazard, which strikes an object at an exponentially distributed (with parameter lr Þ time and leads to failure (death) with probability 1, was considered in Ref. [10] for obtaining
which means that the failure rate lr ðtÞ is defined as
lr ðtÞ ¼ uðtÞlP ðtÞ
ð15Þ
and relations (14) and (15) can be now used in formula (13) for obtaining the expected time lost due to the impact of the Poisson process of additional harmful events.
M.S. Finkelstein / Reliability Engineering and System Safety 82 (2003) 225–228
The next step of generalization is to consider instead of the Poisson process of harmful events the corresponding doubly stochastic Poisson process of harmful events P^ t ; t $ 0 [4]. Let the rate of this process lP ðt; ZÞ be indexed by a random variable Z with support in ð0; 1Þ and the probability density function pðzÞ: Therefore, the rate of the Poisson process is now random itself. It is worth noting that the doubly stochastic Poisson process describes a more flexible approach and effectively models the diversity among subpopulations. Assume for simplicity that uðtÞ does not depend on Z; although the forthcoming formulas can be written without this assumption as well. It is clear, that for the fixed Z ¼ z relations (14) and (15) are valid: ðt ð16Þ Fr ðt;zÞ ; Fr ðtlZ ¼ zÞ ¼ 1 2 exp 2 uðuÞlP ðu;zÞdu 0
lr ðt;zÞ ¼ uðtÞlP ðt;zÞ
ð17Þ
In order to obtain Fr ðtÞ for this case the operator of expectation with respect to Z should be applied to Fr ðt;ZÞ : ðt Fr ðtÞ ¼ 1 2 E exp 2 uðuÞlP ðu;ZÞdu ð18Þ 0
As the failure rate is a conditional characteristic, lr ðtÞ should be obtained via the conditional expectation [6,14]:
lr ðtÞ ¼ uðtÞE½lP ðt;ZÞlT . t
ð19Þ
where, as previously, T is the lifetime when P^ t ; t $ 0 is the only cause of failure. We can also interpret Fr ðtÞ as the Cdf of a mixture (with pðzÞ as a mixing probability density function) whereas lr ðtÞ is a mixture failure rate. There can be different models for lP ðt;ZÞ; the multiplicative being the simplest and the most appealing one: lP ðt;ZÞ ¼ Z lPB ðtÞ; where lPB ðtÞ; as usually in PH-type models, plays the role of a baseline (reference) rate. Therefore, the model under consideration is a combination of the proportional and the additive models. The PH model with a random Z are usually called the frailty models [15]. Eq. (19) for this specific case turns to
lr ðtÞ ¼ uðtÞlPB ðtÞE½ZlT . t
ð20Þ
In accordance with Refs. [5,6]: ð1 lr ðtÞ ¼ zuðtÞlPB ðtÞpðzltÞdz ¼ uðtÞlPB ðtÞE½Zlt
ð21Þ
0
where pðzltÞ is a conditional probability density function of Z given T $ t and E½Zlt is the corresponding conditional expectation: pðzÞF r ðt;zÞ pðzltÞ ¼ ð1 F r ðt;zÞpðzÞdz
ð22Þ
0
and Fr ðt;zÞ for the multiplicative model denotes the Cdf defined by the failure rate zuðtÞlPB ðtÞ: Thus, we can plug in relation (21) for lr ðtÞ into Eq. (13) and obtain the expected time lost due to an extra risk DMRL ðtÞ:
227
Example 1. Consider the case of the homogeneous doubly stochastic Poisson process P^ t ; t $ 0 with constant in time multiplicative rate zlPB and probability u: Therefore, ulPB is also a constant. It is well known [1,2] that the failure rate of a mixture of distributions with decreasing (or constant) failure rates is also decreasing. Thus, lr ðtÞ ðlr ð0Þ ¼ ulPB E½ZÞ is decreasing and tends to 0 as t ! 1; which can be clearly illustrated by the specific case of exponential Z with parameter q: Using definitions (21) and (22):
lr ðtÞ ¼
ul 1 ! ! 0; ult þ q t
t!1
ð23Þ
It can be easily shown that the same asymptotic result is valid when pðzÞ is a gamma density. This also follows from general asymptotic considerations, as the failure rate of a gamma distribution tends to a constant as t ! 1: In fact, relation (23) means that the influence of additional hazard for the heterogeneous setting is fading out in time! Thus, in accordance with definitions (2) and (6): DMRL ðtÞ ¼ mF ðtÞ 2 mG ðtÞ is decreasing and DMRL ðtÞ ! 0 as t ! 1: It is clear that the latter statement holds with an additional assumption that limt!0 lF ðtÞ – 0: If limt!0 lF ðtÞ ¼ 0; then some obvious additional assumptions should be formulated. The similar result is valid for the relative distance RDMRL ðtÞ defined by relation (7). It is clear that RDMRL ð0Þ ¼ 1 2 mG ð0Þ=mF ð0Þ and then this function asymptotically decreases to 0 as t ! 1: Example 1 shows that for the constant in time rate of the doubly stochastic Poisson process the expected time lost due to the extra risk decreases in t and eventually tends to 0 : the older the population, the less the statistical effect of additional risks. The same result trivially holds for the decreasing baseline failure rate lBP ðtÞ: What is more, and this is really surprising, it takes place, at least ultimately, for increasing lBP ðtÞ: It can be easily shown [6] that if, for instance, lBP ðtÞ is an increasing failure rate of the Weibull Cdf and Z has a gamma distribution, then the function ulBP ðtÞE½ZlT . t initially increases, reaches a maximum and then decreases asymptotically to 0 as t ! 1; thus exhibiting the dramatically different from lBP ðtÞ shape of the observed hazard rate lr ðtÞ:
4. Proportional hazards As it was stated in Section 2, the PH model (9) and (10) can be used for modelling a more severe environment without assumption of independence of additional risks. There is not much to say about homogeneous case (9), as DMRL ðtÞ can be formally obtained via relation (13), where lF ðtÞ þ lr ðtÞ should be substituted by zlF ðtÞ: We can deal with the heterogeneous case similar to the approach of Section 3.
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M.S. Finkelstein / Reliability Engineering and System Safety 82 (2003) 225–228
Therefore:
lG ðt; zÞ ¼ zlF ðtÞ; t$0 ðb lG ðtÞ ¼ zlF ðtÞpðzltÞdz ¼ lF ðtÞE½Zlt
ð24Þ
fact, that E½ulx 2 a should decrease sharper than lF ðxÞ increases. Some sufficient conditions for this are obtained in Ref. [6].
ð25Þ
a
where the interval of support for Z; modelling a more severe environment, is now ½a; b; 1 # a , b # 1: It is proved in Ref. [5] that E½Zlt is a decreasing function. Thus, as previously, the effect of heterogeneity decreases the influence of a severe environment with time. The difference with the additive model is that in Eq. (8) only the term lr ðtÞ is decreasing (at least for substantially large t), leaving the first term lF ðtÞ unchanged. In model (25) the heterogeneous environment is ‘bending down’ the baseline lF ðtÞ itself. The extent of this impact depends on pðzltÞ and, specifically, on the interval of support ½a; b: Example 2. Let again FðtÞ be an exponential Cdf with parameter l: Therefore, mF ðtÞ ¼ 1=l: As it was mentioned, the mixture failure rate in this specific case is decreasing, monotonically converging to the failure rate of the strongest population [2]: lim lG ðtÞ ¼ al
ð26Þ
t!1
This means that asymptotically the expected time lost due to additional risk can be obtained as DMRL ðtÞ ¼ mF ðtÞ 2 mG ðtÞ !
1 1 a21 2 ¼ al al l
ð27Þ
Relation (25) can be illustrated explicitly by the specific case of the uniform in ½a; b mixing distribution: pðzÞ ¼ 1=ðb 2 aÞ: Direct integration results in: a exp{2 lta}2b exp{2 ltb} 1 lG ðtÞ ¼ l þ !al exp{2 lta}2exp{2 ltb} t as t !1 Example 2 and foregoing considerations show that DMRL ðtÞ is decreasing in time and the corresponding effect is usually more substantial than in the case of additive hazards. On the other hand, analysing relation (27), one can arrive at the following setting. Assume that a , 1 and E½Z . 1; therefore the environment for Y can be still considered as a more severe one, but in expectation. It turns out that under these assumptions DMRL ðtÞ in Eq. (27) can be negative for sufficiently large t, which paradoxically means that the more severe in expectation environment for old populations is, in fact, the less severe. This result holds for the general case under the condition that the observed (mixture) failure rate asymptotically tends to the failure rate of the strongest population as in Eq. (26):
lG ðxÞ ! alF ðxÞ as x ! 1
ð28Þ
This condition is not ‘very strong’ and can be often met in practical situations (e.g. for the Weibull Cdf). It means, in
5. Concluding remarks The impact of the more severe conditions on the MRL function of a lifetime distribution was studied. We generalized a simple setting with only one additional harmful event [10] to 1. The Poisson process of harmful events 2. Heterogeneous population Two models were considered. It was proved that in the additive hazards model for heterogeneous population the additional hazard lr ðtÞ decreases and eventually fades out as t ! 1: The PH model also showed the similar behaviour: the impact of a more severe environment decreases with age. Surprisingly, for old populations the more severe environment in expectation can eventually result in the less severe one. The reason for these interesting effects is that ‘the weakest populations are dying out first’. Technically this is due to the fact that the failure rate is a conditional characteristic.
References [1] Barlow R, Proschan F. Statistical theory of reliability and life testing probability models. New York: Holt, Rinehart and Winston; 1975. [2] Barlow RE. A Bayes explanation of an apparent failure rate paradox. IEEE Trans Reliab 1985;R-34:107 –8. [3] Block HW, Savits TH, Borges W. Age dependent minimal repair. J Appl Prob 1985;22:370–86. [4] Cox DR, Isham V. Point processes. London: Chapman & Hall; 1980. [5] Finkelstein MS, Esaulova V. Why the failure rate decreases? Reliab Engng Syst Saf 2001;71:173–7. [6] Finkelstein MS, Esaulova V. Modeling a failure rate for a mixture of distribution functions. Prob Engng Inf Sci 2001;15:383–400. [7] Finkelstein MS. On the shape of the mean residual life function. Appl Stochastic Models Business Ind 2002;18:135–46. [8] Finkelstein MS. Simple bounds for terminating Poisson and renewal processes. J Stat Planning Inference 2003;113:541–8. [9] Gupta RC, Akman HO. Mean residual life function for certain types of non-monotonic aging. Stochastic Models 1995;11:219 –25. [10] Hutchinson TP. Calculation of the expected lifetime lost due to an extra risk. Math Popul Stud 2001;9:209–16. [11] Keyfitz N. Applied mathematical demography. New York: Springer; 1985. [12] Mi J. Bathtub failure rate and upside-down bathtub mean residual life. IEEE Trans Reliab 1995;44:388– 91. [13] Shaked M, Shantikhumar J. Stochastic orders and their applications. Boston: Academic Press; 1993. [14] Yashin AI, Manton KG. Effects of unobserved and partially observed covariate processes on system failure: a review of models and estimation strategies. Stat Sci 1997;12:20–34. [15] Vaupel JW, Manton KG, Stallard E. The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 1979; 16:439–54.
Short communication
The expected time lost due to an extra risk M.S. Finkelstein Department of Mathematical Statistics, University of the Free State, P.O. Box 339, 9300 Bloemfontein, South Africa Received 10 April 2003; accepted 28 May 2003
Abstract The difference between the mean remaining lifetime in a baseline and a more risky environment is investigated. Two models for a more severe environment are considered. The additive hazards model describes an additional independent risk, whereas the proportional hazards model does not require the assumption of independence. It is shown that the expected time lost due to an extra risk decreases with time for heterogeneous population. The origin of this interesting effect is in the fact that ‘the weakest populations are dying out first’. Simple examples are considered. q 2003 Elsevier Ltd. All rights reserved. Keywords: Mean residual life function; Failure rate; Mixture of distributions; Random environment; Additive hazards; Proportional hazards model
1. Introduction Let X be a lifetime random variable with a Cdf FðxÞ; x [ ½0; 1Þ and a failure rate lðxÞ; x $ 0: Along with the failure rate, the mean residual life (MRL) function plays an important role in reliability theory, survival analysis, risk analysis and demography. General results on the properties of the MRL function can be found in Refs. [7,9,12] to name a few. As usual, define the remaining lifetime variable Xt by the following survival function ðtþx Ft ðxÞ ¼ P{Xt . x} ¼ Fðt þ xÞ ¼ exp 2 lðuÞdu ð1Þ FðtÞ t and the MRL function as the expectation of Xt : ð1 ðxþt FðuÞdu ð1 t ¼ exp 2 lðuÞdu dx mðtÞ ; E½Xt ¼ FðtÞ 0 t
where lðtÞ is the lifetable probability of surviving from birth to age t [11]. Let X; Y be two lifetime random variables with Cdf’s FðtÞ; GðtÞ and failure rates lF ðtÞ; lG ðtÞ; respectively. Let X be the lifetime of an object in some baseline or reference conditions (environment) whereas Y be the lifetime of the same object in a more risky (hazardous) conditions. As the failure rate of an object is a measure of the instantaneous risk, then clearly the following hazard rate ordering X $hr Y holds [13]:
lF ðtÞ # lG ðtÞ;
It is worth noting that in actuarial and demographic literature the failure rate is called the ‘force of mortality’ or the ‘mortality rate’, whereas mðtÞ is usually denoted as eðtÞ and called the ‘remaining life expectancy’. Relation (2) in the corresponding notation is then written as: ð1 lðuÞdu ð3Þ eðtÞ ¼ t lðtÞ E-mail address: [email protected] (M.S. Finkelstein). 0951-8320/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0951-8320(03)00138-8
ð4Þ
which, in accordance with definition (2), implies the MRL ordering X $MRL Y : mF ðtÞ $ mG ðtÞ;
ð2Þ
;t [ ½0; 1Þ
;t [ ½0; 1Þ
ð5Þ
If the goal of the study is to assess quantitatively the impact of additional risks on the MRL function, then it is reasonable to use the following measure to be called the DMRL -distance: DMRL ðtÞ ¼ mF ðtÞ 2 mG ðtÞ;
;t [ ½0; 1Þ
ð6Þ
In some instances the corresponding relative characteristic can be of interest: m ðtÞ RDMRL ¼ 1 2 G ð7Þ mF ðtÞ In fact, we need to obtain the distance DMRL ðtÞ to address the topic of the paper formulated in its title, as DMRL ðtÞ; ;t $ 0 shows the expected remaining lifetime lost due to the object’s exposure to an additional risk. This problem was addressed in
226
M.S. Finkelstein / Reliability Engineering and System Safety 82 (2003) 225–228
the literature on demographic issues [9,11] but only for oversimplified settings.
life expectancy mð0Þ: As ð1 ð1 exp{ 2 lr u}FðuÞdu FðuÞdu; mG ð0Þ ¼ mF ð0Þ ¼ 0
0
2. Modelling a more risky environment
the expected lifetime loss due to this risk can be defined as
The most natural way to model additional independent risks is via the additive hazards model
DMRL ð0Þ ¼ mF ð0Þ 2 mG ð0Þ
lG ðtÞ ¼ lF ðtÞ þ lr ðtÞ;
t$0
ð8Þ
where lr ðtÞ is the failure rate which accounts for additional independent risks. It is clear also that Eq. (8) can be also interpreted via the notion of independent competing risks. We shall consider the practical realization of this model in Section 3. The other popular model is the proportional hazards (PH) model:
lG ðtÞ ¼ zlF ðtÞ;
z . 1; t $ 0
ð9Þ
where z is some parameter (in regression models usually a ¼ exp{xT y}; where x ¼ ðx1 ; …; xn Þ is a vector of timeindependent covariates and y ¼ ðy1 ; …; y2 Þ is a vector of unknown parameters). In a more general form Eq. (9) can be written with a time-dependent zðtÞ :
lG ðtÞ ¼ zðtÞlF ðtÞ;
zðtÞ . 1; t $ 0
ð10Þ
It is worth noting, that whereas the additive hazards model is mostly useful for modelling additional independent risks, the PH model usually accounts for a more general situation and under certain restrictions can describe an impact of a more severe environment without assuming independence. The third approach, which is often used for modelling additional risks mostly in engineering applications, is the accelerated life model (ALM): GðtÞ ¼ FðztÞ ðGðtÞ ¼ FðzðtÞÞÞ: It is clear that for z . 1 ðzðtÞ . 1Þ the random variable Y will be stochastically smaller than X : Y #st X; which means that GðtÞ $ FðtÞ: This ordering is weaker than the corresponding hazard rate ordering. The failure rate for the described ALM is defined as
lG ðtÞ ¼ zlF ðztÞ; lG ðtÞ ¼ z0 ðtÞlF ðzðtÞÞ;
z . 1; t $ 0;
ð11Þ
zðtÞ . 1; t $ 0
Thus, it is not necessary that lG ðtÞ $ lF ðtÞ for the ALM as for the previous models. In fact, we need additional assumptions for describing properties of DMRL ðtÞ for the ALM and this topic needs a special study. Therefore, we shall focus on the first two models.
¼
ð1 0
ð1 1 12 exp{ 2 lr u}f ðuÞdu FðuÞdu 2 lr 0 ð12Þ
where f ðtÞ ¼ F 0 ðtÞ: For exponential Cdf FðtÞ ¼ 1 2 exp{ 2 lr t} the right hand side of relation (12) is equal to lr =lðl þ lr Þ: This simple setting can be generalized in many ways. Firstly, it is obvious that relation (12) definitely follows from the additive model (9), where lr ðtÞ ; lr : This specific case allows for simple computations for some other concrete types of FðtÞ as well (e.g. Erlangian and inverse Gaussian). On the other hand, in accordance with definition (2), it is easy to obtain the following general formal equation: DMRL ðtÞ ¼ mF ðtÞ 2 mG ðtÞ ðxþt ð1 ¼ exp 2 lF ðuÞdu dxðtÞ 0
t
ðxþt ð1 exp 2 ðlF ðuÞ þ lr ðuÞÞdu dx 2 0
ð13Þ
t
If lF ðtÞ and lG ðtÞ are known, DMRL ðtÞ can be calculated at least numerically. It is natural and important for application to generalize a single event case to the point process of recurrent events. Let Pt ; t $ 0 denotes the non-homogeneous Poisson process of additional harmful events with rate lP ðtÞ; which potentially can lead to failure (death). Let Pt ; t $ 0 be independent of the lifetime random variable X and assume that the event occurred at time t results in failure with probability uðtÞ and is survived (without any consequences to the object) with probability 1 2 uðtÞ: Let T be the lifetime random variable when Pt ; t $ 0 is the only cause of death of an object. Denote by Fr ðtÞ; lr ðtÞ the corresponding Cdf and the failure rate, respectively. ¼ FðtÞ F r ðtÞ Thus, due to assumption of independence: GðtÞ and the additive hazards model (8) describes this setting. It can be proved [3,8] that ðt ð14Þ Fr ðtÞ ¼ 1 2 exp 2 uðuÞlP du 0
3. Independent additional risks A simple specific case of one additional hazard, which strikes an object at an exponentially distributed (with parameter lr Þ time and leads to failure (death) with probability 1, was considered in Ref. [10] for obtaining
which means that the failure rate lr ðtÞ is defined as
lr ðtÞ ¼ uðtÞlP ðtÞ
ð15Þ
and relations (14) and (15) can be now used in formula (13) for obtaining the expected time lost due to the impact of the Poisson process of additional harmful events.
M.S. Finkelstein / Reliability Engineering and System Safety 82 (2003) 225–228
The next step of generalization is to consider instead of the Poisson process of harmful events the corresponding doubly stochastic Poisson process of harmful events P^ t ; t $ 0 [4]. Let the rate of this process lP ðt; ZÞ be indexed by a random variable Z with support in ð0; 1Þ and the probability density function pðzÞ: Therefore, the rate of the Poisson process is now random itself. It is worth noting that the doubly stochastic Poisson process describes a more flexible approach and effectively models the diversity among subpopulations. Assume for simplicity that uðtÞ does not depend on Z; although the forthcoming formulas can be written without this assumption as well. It is clear, that for the fixed Z ¼ z relations (14) and (15) are valid: ðt ð16Þ Fr ðt;zÞ ; Fr ðtlZ ¼ zÞ ¼ 1 2 exp 2 uðuÞlP ðu;zÞdu 0
lr ðt;zÞ ¼ uðtÞlP ðt;zÞ
ð17Þ
In order to obtain Fr ðtÞ for this case the operator of expectation with respect to Z should be applied to Fr ðt;ZÞ : ðt Fr ðtÞ ¼ 1 2 E exp 2 uðuÞlP ðu;ZÞdu ð18Þ 0
As the failure rate is a conditional characteristic, lr ðtÞ should be obtained via the conditional expectation [6,14]:
lr ðtÞ ¼ uðtÞE½lP ðt;ZÞlT . t
ð19Þ
where, as previously, T is the lifetime when P^ t ; t $ 0 is the only cause of failure. We can also interpret Fr ðtÞ as the Cdf of a mixture (with pðzÞ as a mixing probability density function) whereas lr ðtÞ is a mixture failure rate. There can be different models for lP ðt;ZÞ; the multiplicative being the simplest and the most appealing one: lP ðt;ZÞ ¼ Z lPB ðtÞ; where lPB ðtÞ; as usually in PH-type models, plays the role of a baseline (reference) rate. Therefore, the model under consideration is a combination of the proportional and the additive models. The PH model with a random Z are usually called the frailty models [15]. Eq. (19) for this specific case turns to
lr ðtÞ ¼ uðtÞlPB ðtÞE½ZlT . t
ð20Þ
In accordance with Refs. [5,6]: ð1 lr ðtÞ ¼ zuðtÞlPB ðtÞpðzltÞdz ¼ uðtÞlPB ðtÞE½Zlt
ð21Þ
0
where pðzltÞ is a conditional probability density function of Z given T $ t and E½Zlt is the corresponding conditional expectation: pðzÞF r ðt;zÞ pðzltÞ ¼ ð1 F r ðt;zÞpðzÞdz
ð22Þ
0
and Fr ðt;zÞ for the multiplicative model denotes the Cdf defined by the failure rate zuðtÞlPB ðtÞ: Thus, we can plug in relation (21) for lr ðtÞ into Eq. (13) and obtain the expected time lost due to an extra risk DMRL ðtÞ:
227
Example 1. Consider the case of the homogeneous doubly stochastic Poisson process P^ t ; t $ 0 with constant in time multiplicative rate zlPB and probability u: Therefore, ulPB is also a constant. It is well known [1,2] that the failure rate of a mixture of distributions with decreasing (or constant) failure rates is also decreasing. Thus, lr ðtÞ ðlr ð0Þ ¼ ulPB E½ZÞ is decreasing and tends to 0 as t ! 1; which can be clearly illustrated by the specific case of exponential Z with parameter q: Using definitions (21) and (22):
lr ðtÞ ¼
ul 1 ! ! 0; ult þ q t
t!1
ð23Þ
It can be easily shown that the same asymptotic result is valid when pðzÞ is a gamma density. This also follows from general asymptotic considerations, as the failure rate of a gamma distribution tends to a constant as t ! 1: In fact, relation (23) means that the influence of additional hazard for the heterogeneous setting is fading out in time! Thus, in accordance with definitions (2) and (6): DMRL ðtÞ ¼ mF ðtÞ 2 mG ðtÞ is decreasing and DMRL ðtÞ ! 0 as t ! 1: It is clear that the latter statement holds with an additional assumption that limt!0 lF ðtÞ – 0: If limt!0 lF ðtÞ ¼ 0; then some obvious additional assumptions should be formulated. The similar result is valid for the relative distance RDMRL ðtÞ defined by relation (7). It is clear that RDMRL ð0Þ ¼ 1 2 mG ð0Þ=mF ð0Þ and then this function asymptotically decreases to 0 as t ! 1: Example 1 shows that for the constant in time rate of the doubly stochastic Poisson process the expected time lost due to the extra risk decreases in t and eventually tends to 0 : the older the population, the less the statistical effect of additional risks. The same result trivially holds for the decreasing baseline failure rate lBP ðtÞ: What is more, and this is really surprising, it takes place, at least ultimately, for increasing lBP ðtÞ: It can be easily shown [6] that if, for instance, lBP ðtÞ is an increasing failure rate of the Weibull Cdf and Z has a gamma distribution, then the function ulBP ðtÞE½ZlT . t initially increases, reaches a maximum and then decreases asymptotically to 0 as t ! 1; thus exhibiting the dramatically different from lBP ðtÞ shape of the observed hazard rate lr ðtÞ:
4. Proportional hazards As it was stated in Section 2, the PH model (9) and (10) can be used for modelling a more severe environment without assumption of independence of additional risks. There is not much to say about homogeneous case (9), as DMRL ðtÞ can be formally obtained via relation (13), where lF ðtÞ þ lr ðtÞ should be substituted by zlF ðtÞ: We can deal with the heterogeneous case similar to the approach of Section 3.
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Therefore:
lG ðt; zÞ ¼ zlF ðtÞ; t$0 ðb lG ðtÞ ¼ zlF ðtÞpðzltÞdz ¼ lF ðtÞE½Zlt
ð24Þ
fact, that E½ulx 2 a should decrease sharper than lF ðxÞ increases. Some sufficient conditions for this are obtained in Ref. [6].
ð25Þ
a
where the interval of support for Z; modelling a more severe environment, is now ½a; b; 1 # a , b # 1: It is proved in Ref. [5] that E½Zlt is a decreasing function. Thus, as previously, the effect of heterogeneity decreases the influence of a severe environment with time. The difference with the additive model is that in Eq. (8) only the term lr ðtÞ is decreasing (at least for substantially large t), leaving the first term lF ðtÞ unchanged. In model (25) the heterogeneous environment is ‘bending down’ the baseline lF ðtÞ itself. The extent of this impact depends on pðzltÞ and, specifically, on the interval of support ½a; b: Example 2. Let again FðtÞ be an exponential Cdf with parameter l: Therefore, mF ðtÞ ¼ 1=l: As it was mentioned, the mixture failure rate in this specific case is decreasing, monotonically converging to the failure rate of the strongest population [2]: lim lG ðtÞ ¼ al
ð26Þ
t!1
This means that asymptotically the expected time lost due to additional risk can be obtained as DMRL ðtÞ ¼ mF ðtÞ 2 mG ðtÞ !
1 1 a21 2 ¼ al al l
ð27Þ
Relation (25) can be illustrated explicitly by the specific case of the uniform in ½a; b mixing distribution: pðzÞ ¼ 1=ðb 2 aÞ: Direct integration results in: a exp{2 lta}2b exp{2 ltb} 1 lG ðtÞ ¼ l þ !al exp{2 lta}2exp{2 ltb} t as t !1 Example 2 and foregoing considerations show that DMRL ðtÞ is decreasing in time and the corresponding effect is usually more substantial than in the case of additive hazards. On the other hand, analysing relation (27), one can arrive at the following setting. Assume that a , 1 and E½Z . 1; therefore the environment for Y can be still considered as a more severe one, but in expectation. It turns out that under these assumptions DMRL ðtÞ in Eq. (27) can be negative for sufficiently large t, which paradoxically means that the more severe in expectation environment for old populations is, in fact, the less severe. This result holds for the general case under the condition that the observed (mixture) failure rate asymptotically tends to the failure rate of the strongest population as in Eq. (26):
lG ðxÞ ! alF ðxÞ as x ! 1
ð28Þ
This condition is not ‘very strong’ and can be often met in practical situations (e.g. for the Weibull Cdf). It means, in
5. Concluding remarks The impact of the more severe conditions on the MRL function of a lifetime distribution was studied. We generalized a simple setting with only one additional harmful event [10] to 1. The Poisson process of harmful events 2. Heterogeneous population Two models were considered. It was proved that in the additive hazards model for heterogeneous population the additional hazard lr ðtÞ decreases and eventually fades out as t ! 1: The PH model also showed the similar behaviour: the impact of a more severe environment decreases with age. Surprisingly, for old populations the more severe environment in expectation can eventually result in the less severe one. The reason for these interesting effects is that ‘the weakest populations are dying out first’. Technically this is due to the fact that the failure rate is a conditional characteristic.
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