Analysis of the Past Lifetime in a Replacement Model through Stochastic Comparisons and Differential Entropy

: A suitable replacement model for random lifetimes is extended to the context of past lifetimes. At a ﬁxed time u an item is planned to be replaced by another one having the same age but a different lifetime distribution. We investigate the past lifetime of this system, given that at a larger time t the system is found to be failed. Subsequently, we perform some stochastic comparisons between the random lifetimes of the single items and the doubly truncated random variable that describes the system lifetime. Moreover, we consider the relative ratio of improvement evaluated at x ∈ ( u , t ) , which is ﬁnalized to measure the goodness of the replacement procedure. The characterization and the properties of the differential entropy of the system lifetime are also discussed. Finally, an example of application to the ﬁring activity of a stochastic neuronal model is provided.


Introduction
The problem of the reliability and survival analysis of a system has been widely studied in recent years. In various cases, the availability of the system is improved by means of replacements or duplications of the involved subsystems. Indeed, a typical problem in Engineering Reliability is the determination of a suitable policy for the replacement or the improvement of the system components. The literature is quite large in this area, as an example we refer to Jardine and Tsang [1] and Thomas [2].
In some cases, the reliability of a system is analyzed on the ground of partially available information that is concerning the status of the system or its components at certain fixed instants. In these instances, it is necessary to study the reliability measures of interest under the condition of truncated or doubly truncated random variables. In this contribution, we refer to a previous investigation centered on a stochastic model dealing with the replacement of items occurring at deterministic arbitrary instants (see Di Crescenzo and Di Gironimo [3]). We aim to generalize this model, by assuming that a given item is replaced by another item at a fixed instant, in such a way that their lifetimes have the same age but possess different distributions. Afterwards, we assume that, at a subsequent fixed inspection time, the system is found to be failed. Thus, we investigate the corresponding past lifetime within the considered replacement model. Hence, differently from the previous investigation, which was centered on the residual lifetime, in this case we focus on the past lifetime of the system. Due to the nature of the treated model, specific attention is given to the

The Model
Let X be an absolutely continuous nonnegative random variable with cumulative distribution function (CDF) F(x) = P(X ≤ x), probability density function (PDF) f (x), survival function F(x) = 1 − F(x). Bearing in mind possible applications to reliability theory and survival analysis, we assume that X describes the random lifetime of an item or a living organism. Let us now recall two functions of interest; as usual we denote by x ∈ R + , F(x) > 0 the hazard rate (or failure rate) of X, and by x ∈ R + , F(x) > 0 the reversed hazard rate function of X. See Barlow and Proschan [26] and Block et al. [27] for some illustrative results on these notions. Denote by Y another absolutely continuous nonnegative random variable with CDF G(x), PDF g(x), survival function G(x), hazard rate λ Y (x), and reversed hazard rate r Y (x).
We assume that X and Y are independent lifetimes of two suitable items. Both items start working at time 0. A replacement of the first item by the second one (having the same age) is planned to occur at time u > 0, provided that the first item is not failed before. Assume that at the inspection time t > u, the system is found to be failed. We denote by X (Y) u,t the random past lifetime of the system, which can be expressed as Indeed, we take into account that, at time t > u, the system is inspected and it is found failed. If the first item has failed before the replacement time u, and then the system lifetime is equal to the lifetime of first item. Otherwise, if the first item is replaced at time u, then the system lifetime is equal to the lifetime of the second unit. According to Equation (1), throughout the paper we implicitly assume that P(u < Y ≤ t) > 0, i.e., G(u) < G(t) for fixed 0 < u < t. Moreover, for any Borel set B, and for all 0 < u < t, we have the following relation: In particular, if B = (0, x] one has and the corresponding PDF reads Consequently, the survival function of X Let us now give the hazard rate of X and its reversed hazard rate We remark that the functions that are given in (5) and (6) are not necessarily continuous at x = u. Moreover, we have λ Hereafter, we provide a brief example of interest in industrial engineering. Example 1. Let X and Y be independent lifetimes of two items, having Weibull distribution, with F( According to the assumptions given above, we assume that a replacement of the first item by the second one is planned at time u > 0, and that at time t > u an inspection finds the system failed. Clearly, the replacement produces a modification in the system reliability. As example, in Figure 1 we show the system hazard rate (5) in two different cases. In both cases, at time u = 2 the hazard rate performs a jump. In the first case the jump is downward, since X is smaller than Y in the usual stocastic order (see Definition 1), which means that, at time u, the first item is replaced by a more reliable item. On the contrary, in the second case the jump is upward, since the condition on the items is reversed.

Stochastic Comparisons
In the following, we recall some useful definitions and comparison properties of X, Y with X (Y) u,t . We refer to [28] or [29] for more details. Note that the terms increasing and decreasing are used in a non-strict sense. Definition 1. Let X be an absolutely continuous random variable with support (l X , u X ), CDF F, and PDF f . Similarly, let Y be an absolutely continuous random variable with support (l Y , u Y ), CDF G, and PDF g. We say that X is smaller than Y in the (a) usual stocastic order ( where λ X (t) = f (t)/F(t) and λ Y (t) = g(t)/G(t) are, respectively, the hazard rates of X and Y, or equivalently, g(t)/ f (t) increases in t over the union of supports of X and Y; (d) reversed hazard rate order (X ≤ rh Y) if G(t)/F(t) increases in t ∈ (min(l X , l Y ), +∞).
We recall the following relations among the above defined stochastic orders: Now, we study the effect of replacement when the lifetime of the first item is stochastically smaller than the second in the sense of Definition 1. Theorem 1. Let X and Y be absolutely continuous nonnegative random variables. Subsequently, we easily have that λ this giving the proof of (i).
Concerning point (ii), from (4) the case 0 ≤ x ≤ u is immediate. In the second case, we have Hence, noting that G(u) − F(u) ≥ 0 by assumption, and that G(x) ≥ G(t) for x ≤ t, we have or, equivalently, This shows that Through the upcoming theorem, we show that some implications between stochastic comparisons involving the random variables X, Y and X Proof. In general, we can prove that the ratio is not decreasing in x. We recall that, from (3), one has Clearly, since the right-hand-side tends to ∞ as u → t. Point (i) is thus proved. The proof of the other points of the theorem is given through Example 2.
Example 2. Let X and Y be exponentially distributed with parameters 2 and 1, respectively, so that f (x) = 2e −2x , x > 0, and g(x) = e −x , x > 0. Thus, the conditions on X and Y given in Theorem 2 are fulfilled. Note that, from (3), we have and, thus, for This confirms the validity of the Statement (iii) of Theorem 2.
Let us now compare stochastically the random variables X and X Hence, (iv) of Theorem 2 holds true. Due to (6), we have In general, such reversed hazard rates are not ordered. This can be seen in Figure 3, for instance. Hence, (v) of Theorem 2 is true. u,t (x) and r Y (x) (from bottom to top near the origin) for the random variables considered in Example 2, for u = 0.6 and t = 1. Figure 4), so that (vi) of Theorem 2 is fulfilled.  Let us now give another result that is based on stochastic orderings.
From Equation (9), we get In the limit for x ↓ u, we have λ X (u) ≥ λ Y (u) for all u > 0 and, thus, X ≤ hr Y.

Further Comparison Results
Aiming to perform suitable comparisons, hereafter we introduce the random lifetime of the system when the replacement mechanism involves random variables having the same distribution. Specifically, we consider the case when that the random lifetimes of the two items are identically distributed, i.e., X d = Y. In this case, for simplicity we denote by X u,t , instead of X (X) u,t , the random past lifetime of the system. Hence, from Equation (1) and assumption G(x) = F(x), ∀ x ∈ R, we have Clearly, in this case we implicitly assume that P(u < X ≤ t) > 0, i.e., F(u) < F(t) for fixed u < t.
The relevant functions concerning X u,t , such as the distribution function F u,t (x), the probability density f u,t (x), the survival function F u,t (x), the hazard rate λ u,t (x), and the reversed hazard rate r u,t (x) can be easily obtained from (2)-(6) by replacing G(·) with F(·). For instance, due to (4), the survival function of X u,t is given by Aiming to analyze the effect of the replacement with an item having a different distribution, hereafter we stochastically compare the random lifetimes X u,t and X  Proof. Recalling the survival functions (4) and (11), we have that Hence, the thesis follows from Theorem 1.C.5 of Shaked and Shanthikumar [29].
With the objective of comparing the means of X (Y) u,t and X u,t , we now recall that the mean inactivity time of X is expressed as for all u > 0 such that F(u) > 0. Furthermore, the doubly truncated mean of X is for all 0 ≤ u < t, such that F(t) − F(u) > 0, and similarly for Y.
We are now able to compare the means of X u,t and X u,t .
Theorem 5. Let X, Y be absolutely continuous nonnegative random variables. Subsequently, for 0 ≤ u < t, we have Proof. Making use of Equations (3), (12) and (13), we have When X and Y are identically distributed, we clearly have The result (14) thus immediately follows.
Aiming to compare the means considered in Theorem 5, we recall the following definition given in Navarro et al. [30].

Definition 2.
Let X and Y have supports S X and S Y , respectively, with S X ∩ S Y = ∅. Afterwards, X is said to be smaller than Y in the mean doubly truncated order ( We are now able to state a comparison result for the means of the system lifetime. Proof. The proof follows from (14) and Definition 2.
The following proposition is immediate from the above result and Proposition 3.3 and Theorem 3.6 of [30].

Relative Ratio of Improvement
In this section, we refer to a system having random lifetime X, which is replaced by Y at time u. Clearly, if X is smaller than Y according to some stochastic order, then it is reasonable that the reliability of the system improves at time x, for u < x < t, under the assumptions specified in Section 2. Aiming to measure the usefulness of the replacement, let us now introduce the relative ratio of improvement evaluated at u < x < t, defined in terms of (4) as Clearly, from (15), one has lim The measure of differential with the new variable Y is denoted with In this case, (16) gives Theorem 6. Let X and Y be absolutely continuous nonnegative random variables. It results that (i) if X ≥ hr Y, then the function R u,t (x) is increasing in x, for u < x < t; (ii) the function R * u,t (x) is increasing in x, for u < x < t.
Proof. From (15) and from straightforward calculations, we get and, thus, we obtain the result (i). Moreover, Equation (16) gives so that the result (ii) follows.
The following example investigates R u,t (x) and R * u,t (x) when X and Y are exponentially distributed.
Example 3. Let X and Y be exponentially distributed with parameters 1 and λ, respectively. Figure 5 shows R u,t (x) and R * u,t (x) for some choices of λ.

Results Based on Entropies
In this section, we study some informational properties of the random past lifetime X (Y) u,t . It is well known that the differential entropy of a nonnegative and absolutely continuous random variable X, with PDF f , can be expressed as Hence, the differential entropy of [X|X ≤ t], t > 0, also named past entropy, is defined as (see [21,31]) The latter identity implies that We recall that the partition entropy of X evaluated at time t is given by (see, for instance, Bowden [32,33]) The interval entropy of X in the interval [t 1 , t 2 ] is (see, for instance, Sunoj et al. [19], and Misagh and Yari [34]) The interval entropy of Y can be defined similarly. Moreover, from (20), it is not hard to see that when where the latter term is also known as residual entropy of X, i.e., the differential entropy of [X − t|X > t] (see [31,35,36]).
We are now able to determine an expression of the differential entropy of the random past lifetime defined in (1).
Hence, by taking into account the interval entropy of Y in the interval [u, t] (see, e.g., (20)), we get Making use of (18) and (19), it is not hard to see that Equation (22) holds. The proof is thus completed.
Clearly, the partition entropy of X immediately follows from (19). Hence, by combining the above quantities, from Equation (22) we can determine the differential entropy of X (Y) u,t . We omit the expression for brevity. Nevertheless, in Figure 6, we provide some plots of H X (Y) u,t , obtained by means of Proposition 3 and resorting to numerical evaluations. In all cases, we take u = 1, and the differential entropy of X (Y) u,t is increasing in t > 1 and approaches a finite limit as t → ∞. Hence, the information amount about the replacement at time u is increasing in the inspection time t, if at time t the considered system is found to be failed. From the given plots, we note that the entropy under investigation is not always monotone in the given parameters for fixed values of t.  Figure 6. Plots of the differential entropy of X (Y) u,t for the case considered in Example 4, for u = 1, and (a) α = 2, β = 2, µ = 1, with λ = 1, 2, 3 (from top to bottom near t = 5); (b) λ = 1, β = 2, µ = 1, with α = 2, 3, 4, 5, 6 (from top to bottom); (c) λ = 1, α = 2, β = 2, with µ = 1, 2, 3, 4, 5, 6 (from bottom to top for large t); (d) λ = 1, α = 2, µ = 1, with β = 2, 3, 4, 5, 6, 7 (from top to bottom).
We point out that the distributions considered in Example 4 satisfy the proportional reversed hazard rate property. Indeed, distributions satisfying this property are often used in stochastic models that involve the past lifetime and the past entropy (see, for instance, Nanda and Das [37]), because the proportionality condition of the reversed hazard rates leads to more manageable results.

Application to a Stochastic Neuronal Model
The stochastic model considered in this paper can also be used in other applied contexts, in which the replacement occurring at time u can be viewed as a relevant changing point, i.e., an event that produces a variation in the dynamics of the system under investigation. A typical case of interest in theoretical neurobiology concerns the activity of a single neuron, which is slowed during a refractory period. Specific assumptions are used to suitably describe such a refractory period within neuronal models. For instance, the modification of the upper (time-varying) firing threshold is often employed. In this context, the replacement model that is presented in Section 2 is useful to describe a modification in the neuronal dynamics occurring at time u, which can be viewed as the final instant of the refractory period. Now, we take as a reference a stochastic model for the firing activity of a neuronal unit that has been investigated by Di Crescenzo and Martinucci [38] and D'Onofrio et al. [39]. It includes the decay effect of the membrane potential in absence of stimuli, and the occurrence of Poisson-paced excitatory inputs modeled by jumps of random amplitudes. Specifically, we assume that the neuronal membrane potential at time t is denoted by where v 0 is the level of the membrane potential at initial time, just after a spike occurrence; τ > 0 is a parameter that describes how fast the membrane potential exponentially decays to the resting level in absence of stimuli; -N(t), t ≥ 0, is a Poisson process with intensity λ > 0 that describes the number of excitatory pulses received by the neuron in (0, t]; and, -(Z k ) k∈N is a sequence of i.i.d. exponentially distributed random variables with mean α −1 > 0, which are independent on N(t). For fixed value of the membrane potential before the occurrence of the n-th excitatory stimulus, the mean amplitude of the n-th jump is inversely proportional to α, so that large values of α reduce the effect of the neuronal excitatory activity.
For such a model, the probability density of the firing time is obtained in closed form for β > v 0 and it is given by (cf. Theorem 3.1 of [38]) is modified Bessel function of the first kind. We note that With reference to the replacement model considered above, we assume that the random variables X and Y possess PDF's given by with 0 < λ 1 < λ 2 and 0 < α 2 < α 1 , and where h corresponding to the firing PDF given in (25). The assumptions given on the parameters λ i , α i , i = 1, 2, imply that after the replacement time u, the neuronal unit undergoes different dynamics producing more frequent excitatory pulses, having greater and greater strength. Hence, it is expected that the performed modification produces an increment of the relevant rates of the stochastic system under investigation. For a suitable illustration, we consider the following choices of the involved parameters suggested by various investigations dealing with reasonably physiologic values (see [38] and references therein): τ = 0.2 ms −1 , β = 20 mV, v 0 = 10 mV, λ 1 = 2 ms −1 , λ 2 = 2.2 ms −1 , α 1 = 6 mV −1 , and α 2 = 5 mV −1 . Figure 7 shows the effect of the replacement on the hazard rate (5) and on the reversed hazard rate (6), confirming that the replacement produces an increment of such rates. u,t as functions of x, with u = 1 ms, with t = 30 ms (a) and t = 5 ms (b), for X and Y having PDF's given in (26). See the text for the other parameters.

Conclusions
The determination of a policy for the replacement or the improvement of the components of a system is a relevant problem in Engineering Reliability. In this framework, several criteria have been proposed and analyzed in the literature in the recent years. The availability of the system is improved often by means of replacements or duplications of the involved subsystems. Large attention has been devoted to the reliability analysis based on information on the past history of the system. Less attention has been devoted to the cases when the uncertainty is related to the previous status of the system. Within such framework, in this paper we continued the study of a replacement model considered in [3]. In the previous paper, the replacement is planned in advance, the replaced item possessing a different failure distribution and having the same age of the replaced item. Here, the new results are obtained assuming that the replacement of the first item by the second one (having the same age) is planned to occur at time u, provided that the first item has not failed before. Moreover, we assume that the system is interrupted at the inspection time t > u.
The investigation has been centered on the stochastic comparison of the resulting random lifetimes. We also performed suitable stochastic comparisons between the system past lifetime and the lifetimes of the single items. Furthermore, the goodness of the replacement criteria has been studied by means of the relative ratio of improvement. The dynamic information concerning by the system lifetime has been analyzed using the differential entropy. An application to a stochastic neuronal model has also been provided.
Possible future developments on this model can be finalized to the extension to multidimensional instances in which several replacement are planned at subsequent times in order to improve the reliability of the system.