Preservation of ageing classes in deterioration models with independent increments
© Sangüesa et al.; licensee Springer. 2014
Received: 21 February 2014
Accepted: 5 May 2014
Published: 20 May 2014
In the present paper we consider ageing properties in a deterioration model in which the stochastic process measuring deterioration is a process with independent increments. Preservation of increasing and decreasing failure rates, as well as decreasing reversed hazard rate, is considered. We also take into account the preservation of log-concave and log-convex densities. Our main results are based on technical results concerning preservation of log-concave and log-convex functions by positive linear operators, and they include the study of stochastic ordering properties among the random variables in the process.
MSC:60G51, 60E15, 60K10, 26A51.
Deterioration models belong to the topics of interest in reliability theory. They aim to describe how a mechanism deteriorates with age. A convenient way in modeling the uncertainty in time-dependent deterioration is by regarding it as a stochastic process. That is, deterioration in time of a device is described by a stochastic process , in which each represents the degree of deterioration at an instant t. Gamma processes have been mainly considered to model degradation in time [1–3]. Also, the so-called shock models are appropriate if deterioration is caused due to external shocks occurring at certain instants in time (see, for instance, ). Although it seems more realistic to consider processes with non-negative increments in order to measure deterioration, Brownian motion has also been considered (geometric, with drift, or alone as an additional term measuring errors; see [5, 6], and the references therein). General Markovian processes have also been considered (see  and the references therein).
The analytical form of previous functions is usually not easy to deal with, although several expressions are known for specific models (see  for the geometric Brownian motion, as well as for the gamma process when we have a fixed threshold). Then it seems natural to study under which conditions ρ inherits from Y the common reliability properties studied in the literature. In reliability theory, the principal ageing properties considered for a random variable involve the study of the log-concavity (positive ageing) or log-convexity (negative ageing) of a certain function (which usually is the distribution function, survival function or density function). For instance, if is log-concave, the random variable is said to be increasing failure rate (IFR), whereas if it is log-convex, we have the decreasing failure rate property (DFR). Moreover, Y is said to be the decreasing reversed hazard rate (DRHR) if is log-concave. We will recall the properties we are going to use in Definition 2.3, although a more detailed discussion can be found in [, Ch. 2], for instance. In the context of deterioration models, preservation of common ageing properties for a fixed threshold has been studied, for instance, in  in a context of pure-jump processes. As far as we know, the problem of a random threshold was firstly considered by Esary et al. in a context of shock models  and by Abdel-Hameed in several papers [1, 8, 12]. In  a gamma wear process was considered, whereas in [8, 12] results are obtained for a pure-jump wear processes. See also  for a recent review. Our aim in this paper is to address this question for processes with independent increments, thus including Lévy processes. To this end, we use the representation given in (1) and apply techniques based on the preservation of log-convexity and log-concavity by positive linear operators (see [14, 15]). These techniques involve the study of stochastic order properties of the random variables in the process. This approach is different from that used in  for pure-jump Lévy processes, which is based on the underlying Lévy measure of the process. Our results generalize previous ones for a compound Poisson process (see Remark 3.5), as well as for a gamma process (see Remark 3.8 and Remark 3.12). On the other hand, it is usual that preservation results of positive ageing properties (IFR, for instance) hold true under more restrictive assumptions than their analogous negative ageing properties (DFR). This can be seen in , Theorem 2.3, in which for the preservation of the IFR property the requirement is a log-concave density for the Lévy measure, whereas for the DFR property no assumption on this density is needed. Our approach also gives different conditions for the preservation of the IFR and DFR property (see Proposition 3.1(a) and Corollary 3.11, respectively, in Section 3). We also include preservation results concerning the DRHR property (Proposition 3.1(b)). This property is of recent interest and has not been dealt with in the afore-mentioned papers. It should be pointed out that the flexibility of our approach allows us to add deterministic trends without making an extra effort. This approach also allows us to prove stronger results, having to do with the log-convexity or log-concavity of the density function (Section 4).
As mentioned in the Introduction, the concept of log-concavity will play an important role in our results. We first recall this concept.
or equivalently logf is concave (in the interval in which f is strictly positive).
Remark 2.2 If the inequality in the previous definition is reversed, we obtain the dual concept of log-convexity.
Log-concavity is an important concept in reliability theory. Actually the principal ageing classes considered in the literature can be defined in terms of log-concavity or log-convexity (see, for instance, [, Ch. 2]). We recall the definitions of the main ageing classes to be used along the paper.
an increasing failure rate (IFR) if is log-concave on ℝ;
a decreasing reversed hazard rate (DRHR) if is log-concave on ℝ;
log-concave if X is absolutely continuous and its density is log-concave on .
If in parts (a) and (c) log-concavity is replaced by log-convexity we have the decreasing failure rate (DFR) and log-convex ageing classes, respectively. For the DFR property the log-convexity has to be restricted to .
Remark 2.4 It is interesting to point out that X log-concave ⇒ X IFR and DRHR, and that X log-convex ⇒ X DFR ⇒ X DRHR (see [, p.181]).
It is reasonable to assume that, in a deterioration process, each is non-negative, and that the degree of deterioration increases with t (in a certain stochastic order). In the next definition we recall the different stochastic orders we are going to use in our deterioration models. For a more detailed discussion, see [16, 17], for instance.
the usual stochastic order (written as ) if , for all ;
the hazard rate order () if is increasing in t;
the reversed hazard rate order () if is increasing in t;
the likelihood ratio order () if X and Y are absolutely continuous with respect to some dominating measure μ, with respective densities and such that is increasing in t.
Remark 2.6 The relations among the previous stochastic orders are as follows (see [, p.61]):
For a given process , we will use the notation to indicate that is increasing in the ⋅ stochastic order, for all (and if it is decreasing). From now on, we will use the notation to indicate that two random variables X and Y have the same distribution. In next definition we will describe the properties we will assume for the process , which are slightly more general than the ones defining a non-negative Lévy process.
a.s., for ;
the process has independent increments, that is: given , the random variables are independent;
- 3.the increments of the process satisfy
is continuous in probability, that is, , for all .
we will say that the process belongs to the class (independent positive stationary increments).
Particular examples of processes satisfying the above properties, which will be used along the paper, are the following:
The standard Poisson process, which is a process in the class such that , and such that, for each , has Poisson distribution of mean t (see [, p.15]).
The standard gamma process, which is a process in the class such that , and such that, for each , has gamma density , (see, for instance, [, p.6]).
Remark 2.8 Note that the processes considered in Definition 2.7 admit always a representation with right-continuous paths [, p.63], so that the expressions given in (1) hold true. Note also that if , the class coincides with Lévy processes with non-negative increments (or subordinators [, p.137]). In fact, the processes we are going to deal with mainly (compound Poisson process, cf. [, p.18] and gamma process) belong to this class. Moreover, it is readily seen (we include the proof of this fact in Lemma 3.3) that for a given process in the class, the time-transformed process , with being an increasing and convex function, belongs to the class, whereas if is increasing and concave, the process belongs to the class. We will use this fact in order to obtain results concerning non-homogeneous Poisson processes (Proposition 3.4) and non-homogeneous gamma processes (Proposition 4.4), which are time-transformed versions of the standard Poisson and gamma processes, respectively.
Finally, we state a technical result, which can be found in  and will play an important role in our proofs.
Theorem 2.9 (, Thm. 3.8)
f is log-concave;
Tf is continuous on .
Further assume that f is decreasing. If is in the class and , then Tf is a log-concave and decreasing function on .
Further assume that f is increasing. If is in the class and then Tf is a log-concave and increasing function on .
If is in the class and , then Tf is a log-concave function on .
Remark 2.10 If Tf is continuous at the origin we can extend the log-concavity property to the interval , as (2) at 0 as endpoint can be deduced by taking the limit as .
Remark 2.11 In , Thm. 3.8(c) there is an additional condition. If, for , we call (which is an interval if f is log-concave), the additional condition was that the set had to be an interval. But the previous condition is always verified if , so that it does not need to be checked. In the next lemma we give the proof of this fact.
Lemma 2.12 Let be a stochastic process such that and let be an interval. Then is an interval.
Then (5) and (6) are contradictory with the fact that , and the conclusion follows. □
3 Preservation of IFR, DRHR, and DFR classes for wear processes with independent increments
Our first results, concerning to the classes IFR and DRHR, are based on the following.
If is in the class with , and Y is IFR, then ρ is IFR.
If is in the class, and Y is DRHR, then ρ is DRHR.
Proof Condition (7) and (1) ensures us that and are continuous functions on [, Lem. 2.5]. The fact that and are right-continuous and condition 4 in Definition 2.7 allow us to extend the continuity to .
To show part (a), the IFR condition for Y means that is log-concave. Thus, by (1), Theorem 2.9(a) and Remark 2.10 we find that is log-concave on . To extend this property to ℝ, note that an IFR distribution cannot have positive mass at 0 (see [, p.104]). The fact that guarantees this property for ρ, as by (1) . Thus, using this property, the log-concavity property for is extended to ℝ, thus showing part (a).
For part (b), the DRHR condition for Y means that is log-concave, and by (1), Theorem 2.9(b), and Remark 2.10 we find that is log-concave on . As , , the log-concavity property is trivially extended to ℝ. □
Remark 3.2 Recall that implies both and . So that , together with in the class and are sufficient conditions for the preservation of both the IFR and the DRHR property.
First of all we have the following.
If and are increasing and convex functions, then is in the class.
If and are increasing and concave functions, then is in the class.
so that condition 3 is proved, thus concluding part (a).
Part (b) is proved taking into account that the inequalities in (11) are reversed if , are concave functions. □
Using the two previous results we have the following result concerning a non-homogeneous compound Poisson process.
Assume that is a convex function with , are DRHR and Y is IFR. Then ρ is IFR.
Assume that is a concave function, are IFR and Y is DRHR. Then ρ is DRHR.
(see [, Thm. 1.C.12]). Hence, the hypotheses in Proposition 3.1(a) are satisfied. Proof of part (b) is similar, using Proposition 3.1(b), taking into account Lemma 3.3(b) and again [, Thm. 1.C.12]). □
Remark 3.5 As mentioned before, Abdel-Hameed gave general conditions for a Lévy process in order to preserve the IFR property (see [, Thm. 2.3(i)]). In particular for a compound Poisson process these conditions require that be log-concave (as the Lévy measure in the compound Poisson process is proportional to the distribution of ). Thus, in this case, Proposition 3.4(b) gives more general assumptions, under the requirement of to be DRHR. Note that the class DRHR contains, in particular, both log-concave and log-convex distributions. In fact, being log-concave implies that is both IFR and DRHR (recall Remark 2.4), so that, for a homogeneous Poisson process, this is a sufficient condition for the preservation of both the IFR and the DRHR property.
The next result provides preservation properties for the modified process when the random variables in the process satisfy appropriate ageing properties. This, in particular, will allow us to deal with non-homogeneous gamma deterioration processes with trend.
Assume that are DRHR for all t and . Further, assume that and are increasing and convex functions, with and Y is IFR. Then ρ is IFR.
Assume that are IFR for all t, , are increasing and concave functions and Y is DRHR. Then ρ is DRHR.
The first inequality is obtained using Lemma [, Lem. 1.B.44]) with , and , whereas the last inequality follows as the rh order is preserved by increasing transforms [, Thm. 1.B.43]). Thus, the conditions in Proposition 3.1(a) follow, since , which proves part (a).
The proof of part (b) is very similar, using Proposition 3.1(b). Note that by Lemma 3.3(b) we find that is in the class. Moreover, by [, Lem. 1.B.3]. In this case, (12) holds if we replace the rh order by the hr order, using in this case [, Lem. 1.B.3] and [, Lem. 1.B.2]. □
If and are increasing and convex functions, with and Y is IFR, then ρ is IFR.
If (no trend), is increasing and concave and Y is DRHR, then ρ is DRHR.
Proof Part (a) is an immediate application of Proposition 3.6(a). First of all note that a gamma process is in the class (recall Remark 2.8). Moreover, the random variables in are absolutely continuous, so that condition (7) is satisfied. Finally, note that in a gamma process, the are DRHR. This follows recalling Remark 2.4 as, if , has log-convex density, whereas if , has log-concave density (see [, p.99]). Then the conditions in Proposition 3.6(a) hold and the result follows.
Part (b) follows as a consequence of Proposition 3.1(b). In fact, note that (see [, p.62]), and this implies immediately that . Thus, the conditions in Proposition 3.1(b) follow as, recalling Remark 2.6, and, using Lemma 3.3(b), is in the class. □
Remark 3.8 Abdel-Hameed  proved the IFR property for non-homogeneus gamma wear process, when the mean function is convex. Observe that the previous result extends this one, by adding a convex deterministic trend.
Remark 3.9 Note that, for the gamma process, we cannot proceed in a similar way to obtain a preservation result for the DRHR property, when we have a deterministic trend. In fact, if the IFR condition for in Proposition 3.6(b) is not satisfied, we cannot ensure that . In fact, take , a gamma process with linear trend. It is readily seen by calculus that if (the interval in which the IFR property fails).
Now, we focus on the DFR property. This property will follow immediately as a consequence of part (a) in Proposition 3.10 (part (b) will be used in the preservation of log-convex densities). The method of proof (with similar ideas to that in [, Thm. 3.2]) differs substantially from the one used to obtain the preservation of the IFR property. In fact, for the DFR property we only need the stochastic ordering among the variables in the model, whereas for the IFR preservation property a stronger order (the rh one) was required in Proposition 3.1(b).
Let be a decreasing and log-convex function, with , and right-continuous at . Then is a log-convex function on .
Let be a decreasing and log-convex function, with , . Assume that and that , for all . Then is a log-convex function on .
Thus, from (13) we deduce the log-convexity of . □
As an immediate consequence of part (a) in the previous result, we have the following.
Corollary 3.11 Let be a process in the class. Consider a wear process in which is as in (8), with and being increasing and concave functions. Assume that Y, the random threshold and satisfy condition (7). Let ρ be the lifetime of the device. If Y is DFR, then ρ is DFR.
Proof The result is immediate by Proposition 3.10. First of all, our conditions ensure that is in the class, due to Lemma 3.3(b). Secondly, due to (1), we have . As Y is DFR, then satisfies assumptions on Proposition 3.10(a), from which we deduce the log-convexity of . □
Remark 3.12 Observe that this result generalizes Theorem 2.3(iii) and Theorem 2.5 in , as we are able to add a deterministic trend.
4 Preservation of log-concave and log-convex classes for subordinators
Note firstly that, for results concerning log-concavity or log-convexity we will always assume that , in order to guarantee that ρ does not have positive mass at 0, and therefore it is an absolutely continuous random variable. In fact, note that under this assumption, we have by (1), and by the fact that (as the process is a centered subordinator), .
With respect to log-convexity, we have the following result.
Proposition 4.1 Let be a centered subordinator. Consider a wear process in which is defined as , and let Y be the random threshold. Let ρ be the lifetime of the device. If Y is log-convex and is differentiable, with , being non-negative, decreasing, and log-convex, then ρ is log-convex.
The conditions about guarantee that this function is concave, and therefore by Lemma 3.3(b), the process is in the class. As Y is log-convex, then is log-convex, decreasing, and strictly positive on (see [, Prop. C.11, p.117]). Thus, we can apply Proposition 3.10(b), so that the second factor in (19) is a log-convex function, and the result follows for the log-convexity of , as the product of log-convex functions is log-convex. □
Remark 4.2 The previous result guarantees, obviously, the preservation of the log-convexity for a process in the class in which , as is a centered subordinator and in this case . A non-trivial example of a function satisfying the hypotheses in the previous results is such that , , with .
For the preservation of log-concavity, stronger assumptions, concerning stochastic ordering properties of the derived process, are needed. For this reason we present specific examples in which these properties can be checked. First of all, we present a log-concavity result for the compound Poisson process.
in which is a homogeneous Poisson process, and is a sequence of independent, identically distributed non-negative random variables, having finite mean and being independent of the process. Let Y be a log-concave random threshold and let ρ be the lifetime of the device. If is log-concave, then ρ is log-concave.
The process is in the class. Moreover, as is log-concave, [, Thm. 1.C.11]. On the other hand, if is log-concave, then it is IFR, and, therefore, is log-concave. Thus, [, p. 47], and by Theorem 2.9(c), the expression in (20) is a log-concave function, so that the conclusion holds. □
Note that the previous expression shows us that is completely monotone, and henceforth log-convex [, p.123]. Thus, UT is log-convex. The log-convexity of this random variable allows us to give the following result, under the assumption of a log-concave decreasing density of Y.
Proposition 4.4 Let be a gamma wear process. Consider a wear process in which . Let Y be the random threshold, and let ρ be the lifetime of the device. If Y has a log-concave and decreasing density, is differentiable, with , being non-negative, increasing, and log-concave, then ρ is log-concave.
Our conditions guarantee that is convex, and thus is in the class, thanks to Lemma 3.3(a). Moreover, is DRHR for all . This follows as UT has a log-convex density, and therefore it is DRHR (recall Remark 2.6). Moreover, is always DRHR (as its density is either log-concave or log-convex). Thus, the DRHR property for follows, as this property is closed under convolution [, p.179]. Then it follows easily [, Lem. 1.B.44]) that . Thus, the conclusion follows by (22) and Theorem 2.9(a). □
Remark 4.5 The conditions in the previous result are quite restrictive. However, the random threshold Y will have a log-concave decreasing density if it has an exponential distribution, or a uniform distribution on the interval , for some . On the other hand, the function , or , for verifies the conditions in Proposition 4.4.
If the random variables in the derived process were ordered in the likelihood ratio order, then we could generalize the previous result, by using Theorem 2.9(c). The technical problem is the complexity of the density of (we can find integral expressions, but we are not able to find a closed-form expression). As a partial result we are able to check the preservation of log-concavity on the interval . The next lemma will be very useful to this end.
Lemma 4.6 The random variable , in which and T are exponential random variables with mean 1 and U is uniform (all of them independent), is log-concave.
thus showing the discrete log-concavity for and, therefore, the log-concavity for . □
Proposition 4.7 Let be a gamma wear process. Let Y be the random threshold, and let ρ be the lifetime of the device. If Y has a log-concave density, then ρ is log-concave on .
Consider now the stochastic process , . Note that . This follows as , in which is a gamma random variable of shape parameter t independent of . Thus, for , we have As by Lemma 4.6 is log-concave, we have (see [, p.46]), thus proving the likelihood ratio order assumption. Then, by Theorem 2.9(c) , , is a log-concave function, and so is (26), as if , is a log-concave function on , then , is a log-concave function on . Then the conclusion follows by the log-concavity of (26) and (1). □
Remark 4.8 If we could extend the fact that from to , then we could prove the preservation of log-concavity on . However, due to the technical complexity of the density function we are not able, at this point, to prove or disprove this fact.
This work has been supported by the Spanish research project MTM2012-36603-C02-02. The first and second authors acknowledge the support of DGA S11 and E64, respectively. The work of the third author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0017338). The work of the third author was also supported by Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0093827).
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