Abstract
The mean residual life frailty model and a subsequent weighted multiplicative mean residual life model that requires weighted multiplicative mean residual lives are considered. The expression and the shape of a mean residual life for some semiparametric models and also for a multiplicative degradation model are given in separate examples. The frailty model represents the lifetime of the population in which the random parameter combines the effects of the subpopulations. We show that for some regular dependencies of the population lifetime on the random parameter, some aging properties of the subpopulations’ lifetimes are preserved for the population lifetime. We indicate that the weighted multiplicative mean residual life model generates positive dependencies of this type. The copula function associated with the model is also derived. Necessary and sufficient conditions for certain aging properties of population lifetimes in the model are determined. Preservation of stochastic orders of two random parameters for the resulting population lifetimes in the model is acquired.
1. Introduction and Preliminaries
In survival analysis, various semiparametric models have been introduced. Cox [1] introduced the proportional hazards (PH) model as one of the semiparametric models that has contributed greatly to the literature. He used censored failure times and assumed that multiple explanatory variables are available for each value. The hazard rate function (HRF) was assumed to be a function of the explanatory variables and the unknown regression coefficients were multiplied by an arbitrary and unknown function of time. He obtained a conditional likelihood leading to inferences about the unknown regression coefficients and presented some generalizations of his method. The conclusion was that the PH model can be applied in many fields, although the motivation is medical. Suppose that is a nonnegative random variable, which has probability density function (PDF) and cumulative distribution function (CDF) with hazard rate when . Then, the random variable with hazard rate is said to have the PH model. Nanda and Das [2] considered the modelwhere , for all and is the baseline HRF. The hazard rate may not be proportional over the entire interval of time but may be proportional differently in different smaller intervals. They considered model (1) for different aging classes. The closure of the model under different stochastic orders was studied and examples are presented to highlight different properties of the model. As conclusion, they showed that certain monotonicity properties of the function generate specific stochastic characteristics of the model that may contribute for inferential purposes and model selection strategies. Jarrahiferiz et al. [3] added a parameter to the family of distributions given rise to (1) to obtain a model with HRF
They called this model the weighted proportional hazards model and considered mixture distributions arising from variations of . They described in examples some potentials for the usability of the model in the context of reliability engineering. They proved that the frailty random variable and population lifetime are negatively likelihood ratio dependent. They obtained closure properties of the model with respect to some stochastic orderings and some aging classes of life distributions. They showed that if two frailty distributions satisfying the ordering properties are chosen for the parameter , the resulting lifetime distributions have similar ordering properties. They concluded that external effects on the model (1) can be included by adding a parameter which in turn provides further aspects of the model for the purpose of application.
The HRF measures the instantaneous probabilities of failure of lifetime units, while the mean residual lifetime function (MRLF) quantifies the expected entire residual life after a certain time. The proportional mean residual life (PMRL) model as an alternative to the Cox proportional hazards model was proposed by Oakes and Dasu [4]. They characterized the family of parametric lifetime distributions with linear MRLF when the PH model coincides with the PMRL model. Characterizations are of particular interest when they are used to evaluate the plausibility of certain distributional assumptions through appropriate hypothesis testing.
Let have a finite expectation with mean residual lifetime (MRL) whenever . Then, the lifetime with MRLF , where is said to have the PMRL model. Stochastic comparisons and aging properties in connection with the PMRL model have been studied by Gupta and Kirmani [5], Nanda et al. [6], and Nanda et al. [7]. Recently, a dynamic PMRL model has been introduced by Nanda et al. [8]. To establish their model, they considered a time-dependent constant of proportionality in the PMRL model so that has MRLwhere and is the baseline MRLF. In the cases where the ordinary PMRL model fails to fit data, model (3) can be alternatively applied when a proper function on the basis of data can be chosen.
The majority of lifetime systems operate in environments that change randomly over time, as the systems may operate in a collaborative environment where the performance of the systems depends on multiple dynamic sources that change over time. This contemplation makes a description of failure time using mixture distributions instead of a single distribution. In the context of survival analysis, these distributions are called frailty models and the consideration of mixture distributions in survival models is sensible.
Badia et al. [9] considered the MRLF of mixtures. Gupta and Kirmani [10] made some preliminary stochastic comparisons in the multiplicative hazards frailty model. Xie et al. [11] made a comparison of multiplicative frailty models. In the context of linear regression models, Shepherd et al. [12] considered a new amount of residual based on difference of two probabilities. Kayid et al. [13] investigated some relative ordering properties in a multiplicative hazards frailty model. Da et al. [14] make stochastic comparisons between a population and subpopulations in some typical frailty models. For a detailed review of recent advances in frailty models in survival analysis, we refer to Govindarajulu and D’Agostino Sr [15]. For recent research in engineering that can be related to the context of this paper, we refer to Bocchetti et al. [16]; Amini et al. [17]; and Khalil and Lopez-Caballero [18]. We also refer to Mirzaeefard et al. [19] to address the importance of risk-based decision-making process of aging infrastructures. The PMRL model has found some applications in the context of reliability engineering to present an efficient and practical approach for designing accelerated life testing (see, e.g., Zhao and Elsayed [20]). For recent literature on MRLF, we refer to Gupta and Bradley [21]; Ali [22]; Zamanzade et al. [23]; Hall and Wellner [24]; Ma et al. [25]; and Jin and Liu [26].
The aim of the present study was to consider a general MRL frailty model and to add a parameter to the family of distributions generated by (3) in order to consider the effects of a multiplicative parameter together with the effects of the baseline population and the effects of the weight function on the life span of the population. Considering the parameter as a random variable, mixture models of the MRL frailty model and the specific weighted PMRL model are generated. A variety of distributional properties of the models are useful for making inferences about populations. In particular, dependence structures between the population life span and the frailty variable, sufficient/equivalent conditions for the population life span to follow specific aging paths, and the preservation of the stochastic orders of the frailty variable for the corresponding population life spans are determined.
The paper is organized as follows. Section 2 provides an explanation of the general MRL frailty model and some preliminary properties. Section 3 proposes the weighted multiplicative MRL model, including some justifications. Section 4 characterizes the dependency structures induced by the weighted multiplicative MRL model and derives the copula function of the model. Section 5 presents necessary and sufficient conditions for the lifetime distribution to exhibit some aging properties. In Section 6, the preservation properties of various well-known stochastic orderings are studied. Finally, Section 7 precedes a summary of the paper with some conclusions. We have moved the proofs of the results to the appendix to reduce the complexity of the paper and smooth the content.
2. The Expression of MRLF for Some Models
We consider a random MRL , which measures the expected time of survival after time at which the unit is alive in the presence of unobserved parameter . The MRL is thus considered in the presence of a frailty parameter. We assume that for given , that is, the random variable in subpopulations has a finite mean. The random variable which is called the frailty is an uncertain quantity of an individual randomly drawn from the population. The conditional survival function (SF) of the lifetime given , which may describe a specific individual, is
The random variable given has PDF . The mean residual lifetime of given is then obtained aswhich is valid for all that . The hazard rate function also plays a vital role in survival analysis, and it measures the instantaneous risk of failure at the time at which the unit is alive. The hazard rate function of given is given bywhich is associated with MRLF as
Using inversion techniques and being aware of (the accurate form of) , the conditional survival function is acquired as
Replacing (7) in (8) is characterized as
The identity in (9) shows that the mean residual lifetime uniquely characterizes the underlying lifetime distribution (cf. Proposition 2(e) in Hall and Wellner [27]). For a preliminary distributional theory of the mean residual lifetime function and its properties, see Hall and Wellner [27] and Guess and Proschan [28]. A comprehensive review of previous research on the mean residual lifetime was provided by Sun and Zhang [29].
To meet the requirements for to be valid as the mean residual lifetime of a lifetime random variable with CDF (for which ) for the realization of random frailty , the following conditions need to hold:(i) for all and the given (ii), for all (iii), for all (iv)
Building a variety of models based on the mean residual lifetime is based on the association of time point and the realization of frailty in . For instance, in the presence of a multiplicative effect, the PMRL model is outlined as , where is the baseline MRLF (see, e.g., Oakes and Dasu [4], Gupta and Kirmani [5], and Nanda et al. [6]). The foregoing multiplicative association has been developed by Kayid et al. [30] to consider a general PMRL model in which where represents some positive functions. From another perspective, when the association no longer moves beyond the linearity, the additive MRL model is arisen in which with and being the baseline MRLFs (cf. Das and Nanda [31]). The additive model has also been extended to a more general model by Kayid et al. [32] so that in which is a proper (positive) function.
In a more general setting, a broader semiparametric model is , where is a proper bivariate function selected in the way is valid as a mean residual lifetime function. To fulfill a mean residual lifetime frailty model and to guarantee the validation of the model, a study is needed for checking out the conditions (i)–(iv) enumerated earlier to detect whether a formation presents a valid MRL.
In semiparametric models of the form , where is, for all in a specified domain, an increasing function in with and (see, e.g., Kayid et al. [33] for further typical cases of such kind of semiparametric models), the MRLF isin which is the right continuous inverse function of . The CDF plays the role of baseline distribution. The examples below present the MRLF in several applications. We have considered the exponential distribution as the baseline distribution which has the no aging property to quantify departures from this stable situation within the framework of the model. In reliability theory, different aging classes of lifetime distributions are produced by stochastic comparisons of a distribution with the exponential distribution (cf. Shaked and Shanthikumar [34]). When is exponential with mean , then . In this case, equation (10) is translated to
By substituting γ into equation (11), the parameter θ is added to the family of the exponential distribution, and equation (11) generates a new distribution. In the context of reliability engineering and survival analysis, there are many specific semiparametric models, thus obtaining the expression of the MRLF and also plotting the graph of MRLF which can be useful to recognize new aspects of the models including aging behaviours of lifetimes. To obtain the MRLF and examine its behaviour for some reputable typical semiparametric models, we present some examples where the exponential distribution is utilized as a baseline distribution so that the aging properties of the resulting lifetime distribution are structured.
Example 1. In the proportional odds model, where (see, e.g., Kumar and Sankaran [35]), we have . By considering the baseline exponential distribution with mean and thus using equation (11), the expression of the MRL when is In the case when , we have for which the MRL function is constant, i.e., . Figure 1 presents the graphic of for some values of the tilt parameter .
Example 2. In the proportional hazard rates (PHR) model (see, for instance, Kochar and Xu [36]). The implied SF, with baseline exponential distribution with mean , is , which corresponds to the SF of the exponential distribution with mean . Therefore, for all . In the case of a general baseline distribution in the PHR model, Gupta [37] represented the expression of the MRLF.
Example 3. Considering the PMRL model, we getUpon taking as the baseline distribution, we obtain . The MRL function is for all .
Example 4. Consider for , the conditional random variable which is the tail of distribution function on the right area with probability (cf. Nair and Sankaran [38]). The random variable has CDFwhere and . In view of (14) using (11), we obtainin which . Figure 2 plots the graph of for different percentages of distribution tail identified by values of .
The following example deals with a mixture of two semiparametric distributions.
Example 5. For , let us consider the family of distributionsin which and as a result, . For the expression of the MRLF of finite mixtures the readers are referred to Navarro and Hernandez [39]. We assumed that is exponential and thus we deal with a mixture of two exponential distributions with different parameters (cf. Jewell [40]) which has the MRL functionFigures 3 and 4 plot the curves of for two selected values of and when and , respectively, with different values of and a fixed amount of in the vector .
Remark 1. Let and be CDFs of two lifetime random variables. Let be the relevant two-component mixture model with MRLF given bywhere is the MRLF of for . Then, it can be plainly verified that is increasing (resp. decreasing) in for all , if and only if , for all . In Example 5, a special case was considered when and shared the common baseline . In that instance, and therefore, , for all if and only if, . Figures 3 and 4 clarify and confirm this issue where when in Figure 3 the MRLF is increasing in and further when in Figure 4 the MRLF is decreasing in .
The next example presents a situation where the lifetime of a device is considered to be caused by degradation of the system. The MRL of a system under degradation may be proper criteria for predicting the remaining lifetime of the system after a time point.
Example 6. Suppose that the failure of a device occurs when the test items’ degradation level reaches a predetermined threshold value . Consider that the general multiplicative degradation model with follows a log-logistic distribution with parameters and and in which for . The survival function of is . Depending on the parameter , the lifetime distribution of the device is (see, e.g., Bae et al. [41])To establish an identifiable model, we take and which givesThe MRL function of the device is thenFigure 5 produces the graphs of for values . For a comparison of the exhibition, the HRF curve and the MRLF curve of the implied lifetime distribution, Figures 5 and 6 in Bae et al. [41], can be collated.
In this step, we are ready to develop a mixture model by mixing subpopulations indexed by the amount of to produce a new population with SF , where is the CDF of and is the SF as given in equation (9). We suppose that follows the CDF . The output random variable is called the inclusive random variable and denotes the random frailty parameter in the population. The associated density functions of and will be signified by and , respectively. By a realization of the population with SF is reduced to the subpopulation .
Being aware of the random parameter to be equal with , the conditional MRLF is given byin which identifies an individual in the population. In equation (22), stands for the MRLF of given . In conformity with the mathematical expectation, one writesIn the case when and are independent, , for all which indicates that the MRLF does not depend on . The frailty component may be unobservable and, as a result, the individual level model in (9) is not applied. It is sensible to contemplate the subsequent population modelwhere has been inserted according to (9). By taking the expectation of the conditional PDF of given with respect to , the unconditional PDF of is obtained byin which is the HRF associated with (22) (cf. equation (7)). The function in (24) and the function in (25) present the population-level SF and the population-level PDF of the MRL frailty model given in (22), respectively.
Marshall and Olkin [42] represented the expression of the HRF of the general mixture model . In accordance with Nanda and Das [43], one can further develop thatin which is the conditional PDF of provided that given byis the corresponding CDF. Note that converges to as . From Nanda and Das [43], where the hazard rate of mixtures was considered, we conclude that .
We get an expression for the MRLF of population, which is given byWe demonstrate below that the amount of the MRL of population at time is the average of the MRL of individuals whose lives prolonged to time . The CDF of is assumed to be absolutely continuous.
Theorem 1. If , then the population-level MRLF is the expectation of with regard to the conditional density of given , i.e., .
The goal of the current investigation is to consider a general mean residual lifetime frailty model and a particular weighted PMRL model. This model is raised by adding a multiplicative external parameter to the family of distributions generated by (3). This provides the possibility of external effects, separate from the variation of time-dependent coefficient, to be entertained by the new model. The study of aging behaviours, dependency structures, and preservation of some stochastic orders in the weighted multiplicative MRL model and the associated frailty model is also conducted.
To be familiar with several notions in applied probability useful to describe the MRL frailty models, we give some necessary definitions here. Let be a lifetime random variable with PDF and CDF , for . The HRF of is defined by for for which . The MRLF of the random variable with finite mean, is for for which . In other words, the HRF of is the ratio of the PDF of to its SF, and the MRLF is the ratio of the sum of the tail area of the SF of divided by the SF of . For the definition of these stochastic orders, we refer the readers to Shaked and Shanthikumar [34] and also Belzunce et al. [44]. Asadi and Shanbhag [45] developed the MRL and the HR orders to entertain more general distributions rather than absolutely continuous distributions and purely discrete distributions. However, the lifetimes are considered to have absolutely continuous distributions in the following definition.
Definition 1. The random lifetime is less than the random lifetime in(i)Likelihood ratio order (denoted by ), if is nondecreasing in for which .(ii)Hazard order (denoted by ) if , for all or equivalently if is nondecreasing in for which .(iii)Mean residual lifetime order (denoted by ) if , for all or equivalently if(iv)Increasing convex order (denoted by ) if(v)Usual stochastic order (denoted by ) wheneverEquivalently, if, and only if, for all increasing functions for which , we have .
The relationships among stochastic orders in Definition 1 are as below:Aging classes of lifetime distributions in terms of monotonicity of MRLF and the property that the MRLF have an initial highest peak at zero or an absolute minimum at zero, have been defined in the literature (see, e.g., Barlow and Proschan [46]) as follows. We assume is a nonnegative random variable with density function and survival function .
Definition 2. The random lifetime with the HRF and the MRLF is said to be(i)New better (worse) than used in expectation [NBUE (NWUE)] whenever , for all (ii)Decreasing (increasing) mean residual lifetime [DMRL (IMRL)] whenever is nonincreasing (nondecreasing) in (iii)Increasing (decreasing) failure rate [IFR (DFR)] whenever is nonincreasing (nondecreasing) in It is straightforward that DMRL NBUE and that IMRL NWUE. Furthermore, IFR DMRL and also DFR IMRL.
The partial dependencies given below have been adopted from Nelsen [47]. Recently, Nair and Vineshkumar [48] applied these dependencies on some bivariate lifetime variables. In case the reversed conditioning is required, the definition can be stated similarly.
Definition 3. Consider the random couple with joint PDF and joint SF . Then,(i) and are said to have positive (negative) likelihood ratio dependence (PLRD [NRLD]) structure if is in . Then, it is said that(ii) is stochastically increasing (decreasing) in , denoted by [], if is nondecreasing (nonincreasing) in , for all .(iii) and are right corner sets increasing (decreasing) if is in .(iv) and are said to have positive (negative) quadrant dependence structure, denoted by PQD (NQD), whenever , for all and .(v) is said to be right tail increasing (decreasing) in , denoted by RTI (RTD ), whenever is nondecreasing (nonincreasing) in for all .It has been established in literature thatThe result of Theorem 1 may be useful to detect the monotonicity behaviour (or some other behaviour of interest) in the shape of the MRLF of mixtures in the general frailty model (22) when distribution of as well as distribution of is not known. The following result presents sufficient conditions for NBUE (NWUE) and DMRL (IMRL) properties of that follows the general mixture model (24).
Theorem 2. (a) Let be nondecreasing in for all and for all . Then,(i)If given , for all , has the NBUE (NWUE) property such that and are NQD (PQD), then has the NBUE (NWUE) property(ii)If given , for all , has the DMRL (IMRL) property such that is RTD (RTI) in , then has the DMRL (IMRL) propertyThe following result compliments the result of Theorem 2 (a). The proof, being similar to the case (a), is omitted.
(b) Let be nonincreasing in for all and for all . Then,(i)If , for all , has the NBUE (NWUE) property such that and are PQD (NQD), then has also the NBUE (NWUE) property(ii)If , for all , has the DMRL (IMRL) property such that is RTD (RTI) in , then has also the DMRL (IMRL) propertySufficient conditions for the stochastic relation when can be sought. To this end, we will get some observations showing that the shape of or more generally the shape of with respect to is useful criteria. From the Proof of Theorem 1, with density function which is determined by the identity by which one hasTherefore, by Definition 2 (vi), if and only if,and specifically, it holds if and only if, for all
We present a useful definition and a technical lemma. Then, in Proposition 1, convenient conditions will be obtained under which the aforementioned sufficient conditions are satisfied.
Definition 4. A bivariate nonnegative function is said to be totally positive (reverse regular) of order 2, denoted by , in wheneversuch that (see, e.g., Karlin [49]) for definition and properties of and functions).
To present a shortcut intelligible method for inspecting the property, we can see is in , if is nondecreasing (nondecreasing) in for all for which and and further if for values of for which the inequality in (37) holds true. The latter requirement is readily examined.
Lemma 1 (basic composition formula, Karlin [49]). Let be a function in and let be a function in . Then, for a - finite measure the function given by is in .
It is also remarkable that if is in then is in and further, if and are two nonnegative functions, then is in .
Proposition 1. Let be nondecreasing (nonincreasing) in , for all such that is in . Then , for all .
The following example shows that the sufficient condition in Theorem 2 (a) is not necessary to establish the DMRL property of .
Example 7. Suppose that given has MRL where . This is the MRLF of Hall–Wellner family of distributions (cf. Hall and Wellner [27]). It is seen for the values of in , as determined, that . By (9), we get for . We assume that follows truncated inverse Weibull distribution with CDF and is a positive parameter. For fixed , taking the average of with respect to the randomly drawn values of as in (24), yieldsfrom which the MRLF obtains bySince is nondecreasing in for all and further it is in , thus Proposition 1 implies that is RTI in . Now, because is decreasing in for all , thus Theorem 2 (a) cannot be applied. However, we can observe that is decreasing and hence has DMRL property. Figure 6 clarifies this issue with exhibition of plot of MRLF .
3. Weighted Multiplicative Mean Residual Life Model
In survival analysis, multiplicative effect of covariates on performance of lifetime distributions is considered. For instance, in the Cox PH model, in which is the baseline HRF, is a -varaite vector of external parameters and is the -dimensional vector of covariates. To develop the PH model to entertain mutual effect of time and covariates, Jarrahiferiz et al. [3] proposed the weighted PH model as in (2). In a similar manner, a weighted version of the PMRL model can be defined. Let us split the vector as and accordingly break as so that does not depend on but does. In the PMRL model which is given by with baseline MRLF , assume we are in a position to test against , where is a known given vector. The lifetime distribution under commonly plays an important role, as needed for constructing a test statistic for statistical inference about . The PMRL model under reduces to . By making the choices and the definition of a weighted PMRL model is convinced (cf. Section 6 in Chen et al. [50]).
Suppose that is a nonnegative continuous function such that
Model (40) is called weighted multiplicative mean residual life (WMMRL) model in which the influence and on the MRLF is multiplicative and fulfills the possibility of variations with time in the PMRL model where . The PMRL model has been studied in the literature (see, e.g., Oakes and Dasu [4], Zahedi [51], and Nanda et al. [6]). Although in the PMRL model the CDF has a closed form, from (9) when the WMMRL model stands, the conditional SF of given isin which and signifies the derivative of . The corresponding PDF iswhere denotes the second derivative of . Note that
One hasin which the last inequality holds since for all . Consequently,which makes (42) a valid statement for the density function of given . Model (40) does not lie in the class of semiparametric models of the form , for some proper function . The function may not depend on the baseline distribution, and records external variations with time. The influence of time and frailty on the MRLF in (40), as one important aspect of such model, is multiplicative.
To present conditions under which (40), for a given amount of , is a well-founded MRLF so that (41) is a valid SF we obtain:(i), for all (ii), for all (iii), for all (iv)
The following example indicates the existence of the WMMRL model where a theoretical situation for the applicability of the model is provided.
Example 8. Let where , which is the MRLF of the exponential distribution, and let by which one gets . Thus, , so we have . Presume is the realization of contributes to model (22). It is plain to see that (i) for all , (ii) , for all , (iii) , for all , and also (iv) . The conditions (i)–(iv) thus hold and is a valid MRLF. The resulting CDF is thus obtained from (41) as . Figure 7 plots the graph of for values , and 2.
To carry out a regression analysis based on the WMMRL model, assume that the baseline distribution is the Pareto distribution having SF which has MRLF given by . We take where . To generate a sample of size from the baseline distribution, we utilize the function run if in R so that we firstly generate uniform random variables from . Then, by the inverse transform technique, we obtain where is the baseline CDF which has Pareto distribution as given earlier. The values of constitute a random sample from . The value of the underlying MRLF at the time is . Suppose that is a random variable with MRLF given at the time of as , in which and . Suppose that and for which in which is a random sample of generated errors from normal distribution with mean 0 and variance 0.001. To move forward, we treat , as a simple linear regression model with unknown parameters and and apply the method of least squares to estimate the parameters. Based on estimations of and are obtained from equations belowTherefore,Appealing to the simulated data, we get and .
Mixture distributions usually provide a way to add more flexibility to (semiparametric) parametric distributions in modelling data when a prior distribution for the parameter is assigned (see e.g., Bučar et al. [52]; Elmahdy and Aboutahoun [53] and Dang and Xu [54]). The frailty factor in the mixture of our suggested model is unobservable in population and thus the conditional model (41) is unobservable at the individual level. To entertain heterogeneity among individuals in the population, which is a source of variation of , the following extended mixture WMMRL model is taken into consideration: