Mean Residual Lifetime Frailty Models: A Weighted Perspective

)e mean residual life frailty model and a subsequent weighted multiplicative mean residual life model that requires weighted multiplicative mean residual lives are considered. )e 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. )e 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. )e 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.


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. e 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. e conclusion was that the PH model can be applied in many fields, although the motivation is medical. Suppose that T 0 is a nonnegative random variable, which has probability density function (PDF) f and cumulative distribution function (CDF) F with hazard rate h 0 (t) � (f(t)/(1 − F(t))) when F(t) < 1.
en, the random variable T with hazard rate h(t|θ) � θh 0 (t) is said to have the PH model. Nanda and Das [2] considered the model where b(t) > 0, for all t ≥ 0 and h 0 is the baseline HRF. e hazard rate may not be proportional over the entire interval of time but may be proportional differently in different smaller intervals. ey considered model (1) for different aging classes. e 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 b(t) 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 h(t|θ) � θb(t)h 0 (t).
(2) ey called this model the weighted proportional hazards model and considered mixture distributions arising from variations of θ. ey described in examples some potentials for the usability of the model in the context of reliability engineering. ey proved that the frailty random variable and population lifetime are negatively likelihood ratio dependent. ey obtained closure properties of the model with respect to some stochastic orderings and some aging classes of life distributions. ey showed that if two frailty distributions satisfying the ordering properties are chosen for the parameter θ, the resulting lifetime distributions have similar ordering properties. ey 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. e 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.
e proportional mean residual life (PMRL) model as an alternative to the Cox proportional hazards model was proposed by Oakes and Dasu [4]. ey 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 T 0 have a finite expectation with mean residual ) whenever F(t) < 1. en, the lifetime T with MRLF m(t|θ) � θm 0 (t), where θ > 0 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 T has MRL where B(t) > 0 and m 0 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 B(·) on the basis of data can be chosen. e 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. is 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. e 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].
e 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. e 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.

The Expression of MRLF for Some Models
We consider a random MRL m(t, Θ), which measures the expected time of survival after time t at which the unit is alive in the presence of unobserved parameter Θ. e MRL is thus considered in the presence of a frailty parameter. We assume that m(0, θ) < ∞ for given θ, that is, the random variable in subpopulations has a finite mean. e random variable Θ which is called the frailty is an uncertain quantity of an individual randomly drawn from the population. e conditional survival function (SF) of the lifetime T * given Θ � θ, which may describe a specific individual, is 2 Mathematical Problems in Engineering e random variable T * given Θ � θ has PDF f(t|θ). e mean residual lifetime of T * given Θ � θ is then obtained as which is valid for all θ that F(t|θ) > 0. e hazard rate function also plays a vital role in survival analysis, and it measures the instantaneous risk of failure at the time t at which the unit is alive. e hazard rate function of T * given Θ � θ is given by which is associated with MRLF as Using inversion techniques and being aware of (the accurate form of ) h(t|θ), the conditional survival function F(t|θ) is acquired as Replacing (7) in (8) is characterized as e 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 m(t, θ) to be valid as the mean residual lifetime of a lifetime random variable X with CDF F (for which F(0 − ) � 0) for the realization θ of random frailty Θ, the following conditions need to hold: (i) m(t, θ) > 0 for all t ≥ 0 and the given θ (ii) (z/zt)m(t, θ) > − 1, for all t ≥ 0 (iii) t 0 ((1 + (z/zs)m(s, θ))/m(s, θ))ds < ∞, for all t ≥ 0 (iv) +∞ 0 ((1 + (z/zs)m(s, θ))/m(s, θ))ds � ∞ Building a variety of models based on the mean residual lifetime is based on the association of time point t and the realization θ of frailty in m(t, θ). For instance, in the presence of a multiplicative effect, the PMRL model is outlined as m(t, θ) � θm 0 (t), where m 0 (t) is the baseline MRLF (see, e.g., Oakes and Dasu [4], Gupta and Kirmani [5], and Nanda et al. [6]). e foregoing multiplicative association has been developed by Kayid et al. [30] to consider a general PMRL model in which m(t, θ) � a(θ)m 0 (t) where a(·) represents some positive functions. From another perspective, when the association no longer moves beyond the linearity, the additive MRL model is arisen in which m(t, θ) � θ + m 0 (t) with θ > 0 and m 0 being the baseline MRLFs (cf. Das and Nanda [31]).
e additive model has also been extended to a more general model by Kayid et al. [32] so that m(t, θ) � a(θ) + m 0 (t) in which a(·) is a proper (positive) function.
In a more general setting, a broader semiparametric model is m(t, θ) � ϕ(m 0 (t), θ), where ϕ is a proper bivariate function selected in the way m(t, θ) 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 1] is, for all θ in a specified domain, an increasing function in u ∈ [0, 1] with c(0, θ) � 0 and c(1, θ) � 1 (see, e.g., Kayid et al. [33] for further typical cases of such kind of semiparametric models), the MRLF is in which F − 1 is the right continuous inverse function of F. e CDF F plays the role of baseline distribution. e 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 F with the exponential distribution (cf. Shaked and Shanthikumar [34]). When F is exponential with mean μ F , then f(F − 1 (u)) � ((1 − u)/μ F ). In this case, equation (10) is translated to Mathematical Problems in Engineering By substituting c 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 5. For p ∈ (0, 1), let us consider the family of distributions . For the expression of the MRLF of finite mixtures the readers are referred to Navarro and Hernandez [39]. We assumed that F is exponential and thus we deal with a mixture of two exponential distributions with different parameters (cf. Jewell [40]) which has the MRL function where m i is the MRLF of F i for i � 1, 2. en, it can be plainly verified that m(t, p) is increasing (resp. decreasing) in p ∈ (0, 1) for all t ≥ 0, if and only if m 1 (t) ≥ (resp. ≤ )m 2 (t), for all t ≥ 0. In Example 5, a special case was considered when  Figure 4 the MRLF is decreasing in p � (1/4), (1/2), (3/4), (5/6). e next example presents a situation where the lifetime of a device is considered to be caused by degradation of the system. e 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 (D f ). Consider that the general multiplicative degradation model D(t) � Xη(t) with X follows a log-logistic distribution with parameters α and β and η(t) � ( Depending on the parameter θ � (α, β, θ 1 , θ 2 ), the lifetime distribution of the device is (see, e.g., Bae et al. [41]) To establish an identifiable model, we take e MRL function of the device is then In this step, we are ready to develop a mixture model by mixing subpopulations indexed by the amount of θ to produce a new population where Λ is the CDF of Θ and F(t|θ) is the SF as given in equation (9). We suppose that T * follows the CDF F * � 1 − F * . e output random variable T * is called the inclusive random variable and Θ denotes the random frailty parameter in the population. e associated density functions of Θ and T * will be signified by λ and f * , respectively. By a realization θ * of Θ the population with SF F * (·) is reduced to the subpopulation F(·|θ * ).
Being aware of the random parameter Θ to be equal with θ, the conditional MRLF is given by in which θ identifies an individual in the population. In equation (22), m(·|θ) stands for the MRLF of T * given Θ � θ.
In conformity with the mathematical expectation, one writes In the case when T * and Θ are independent, m(t|θ) � m * (t), for all t ≥ 0 which indicates that the MRLF does not depend on θ. e frailty component Θ may be unobservable and, as a result, the individual level model in (9) is not applied. It is sensible to contemplate the subsequent population model The mean residual lifetime m (t, p, α

Mathematical Problems in Engineering
where F(t|θ) has been inserted according to (9). By taking the expectation of the conditional PDF of T * given Θ � θ with respect to θ, the unconditional PDF of T * is obtained by in which h(t, θ) � ((1 + (z/zt)m(t, θ))/m(t, θ)) is the HRF associated with (22) (cf. equation (7)). e function F * in (24) and the function f * 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 that (26) in which λ(θ|T * > t) is the conditional PDF of Θ provided that T * > t given by is the corresponding CDF. Note that Λ(θ|T > t) converges to Λ(θ) as t ⟶ 0 + . 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 by We demonstrate below that the amount of the MRL of population at time t is the average of the MRL of individuals whose lives prolonged to time t. e CDF Λ of Θ is assumed to be absolutely continuous.
e goal of the current investigation is to consider a general mean residual lifetime frailty model and a particular weighted PMRL model. is model is raised by adding a multiplicative external parameter to the family of distributions generated by (3). is provides the possibility of external effects, separate from the variation of time-dependent coefficientB(t), to be entertained by the new model. e 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 T i be a lifetime random variable with PDF π i and CDF Π i , for i � 1, 2. e HRF of T i is defined by In other words, the HRF of T i is the ratio of the PDF of T i to its SF, and the MRLF is the ratio of the sum of the tail area of the SF of T i divided by the SF of T i . 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.
e random lifetime T 1 is less than the random lifetime T 2 in (v) Usual stochastic order (denoted by T 1 ≤ st T 2 ) whenever Equivalently, T 1 ≤ st T 2 if, and only if, for all increasing e 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 X is a nonnegative random variable with density function f X and survival function F X .

Definition 2.
e random lifetime X with the HRF h X (t) � (f X (t)/F X (t)) and the MRLF m It is straightforward that DMRL ⊆ NBUE and that IMRL ⊆ NWUE. Furthermore, IFR ⊆ DMRL and also DFR ⊆ IMRL. e 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. en, It has been established in literature that e result of eorem 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 T * as well as distribution of (Θ|T * > t) is not known. e following result presents sufficient conditions for NBUE (NWUE) and DMRL (IMRL) properties of T * that follows the general mixture model (24). property such that T * and Θ are PQD (NQD), then T * has also the NBUE (NWUE) property (ii) If (T * |Θ � θ), for all θ, has the DMRL (IMRL) property such that Θ is RTD (RTI) in T * , then T * has also the DMRL (IMRL) property Sufficient conditions for the stochastic relation To this end, we will get some observations showing that the shape of m(t|θ) or more generally the shape of F(t|θ) with respect to (t, θ) is useful criteria. From the Proof of eorem 1, and specifically, it holds if and only if, for all θ We present a useful definition and a technical lemma. en, in Proposition 1, convenient conditions will be obtained under which the aforementioned sufficient conditions are satisfied.
To present a shortcut intelligible method for inspecting the TP 2 (RR 2 ) property, we can see g is TP 2 (RR 2 ) in (x, y) ∈ S, if (g(x 2 , y)/g(x 1 , y)) is nondecreasing (nondecreasing) in y for all (x i , y) ∈ S for which x 1 ≤ x 2 and g(x 1 , y) > 0 and further if for values of y for which g(x i , y) � 0, i � 1, 2 the inequality in (37) holds true. e latter requirement is readily examined.

Proposition 1. Let m(t|θ) be nondecreasing (nonincreasing) in θ, for all
e following example shows that the sufficient condition RTD(Θ|T * ) in eorem 2 (a) is not necessary to establish the DMRL property of T * .

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, h(t|Z) � e β T (t)Z h 0 (t) in which h 0 (t) is the baseline HRF, β(t) is a p-varaite vector of external parameters and Z is the p-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 Z as Z � (Z 1 , Z 2 ) and accordingly break β(t) as β(t) � (β 1 (t), β 2 (t)) so that β 1 (t) ≡ β 1 does not depend on t but β 2 (t) does. In the PMRL model which is given by By making the choices B(t) � e b 2 (t)Z 2 and θ � e β T 1 Z 1 the definition of a weighted PMRL model is convinced (cf. Section 6 in Chen et al. [50]).
Suppose that B(·) is a nonnegative continuous function such that Model (40) is called weighted multiplicative mean residual life (WMMRL) model in which the influence t and θ on the MRLF is multiplicative and B fulfills the possibility of variations with time in the PMRL model where Mathematical Problems in Engineering m(t|θ) � θm 0 (t). e 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 T * given Θ � θ is in which V(t) � t 0 (dx/B(x)m 0 (x)) and V ′ signifies the derivative of V. e corresponding PDF is where V ″ denotes the second derivative of V. Note that One has in which the last inequality holds since m To present conditions under which (40), for a given amount of θ > 0, is a well-founded MRLF so that (41) is a valid SF we obtain: e following example indicates the existence of the WMMRL model where a theoretical situation for the applicability of the model is provided.
To carry out a regression analysis based on the WMMRL model, assume that the baseline distribution is the Pareto distribution having SF F 0 (t) � (1/(1 + t) 2 ) which has MRLF m 0 given by m 0 (t) � t + 1. We take B(t) � (1 + t) η where η > 0. To generate a sample of size n � 100 from the baseline distribution, we utilize the function run if in R so that we firstly generate uniform random variables U 1 , . . . , U 100 from (0, 1). en, by the inverse transform technique, we obtain where F 0 is the baseline CDF which has Pareto distribution as given earlier.
e values of T 1 , . . . , T 100 constitute a random sample from F 0 . e value of the underlying MRLF at the time T i is m 0 (T i ) � T i + 1, i � 1, . . . , 100. Suppose that T * is a random variable with MRLF m(·|θ, η) given at the time of T i as m(T i |θ, η) � θ(1 + T i ) η m 0 (T i ), i � 1, . . . , 100, in which θ � 2 and η � 1. Suppose that R i � ln(m 0 (T i )) and W i � ln(m(T i |θ, η)) for which W i � ln(θ) + ηln(1 + T i ) + R i + E i in which E 1 , . . . , E 100 is a random sample of generated errors from normal distribution with mean 0 and variance 0.001. To move forward, we treat erefore, Appealing to the simulated data, we get η � 0.9995931 and θ � 2.000597.
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]). e frailty factor Θ 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 Θ, the following extended mixture WMMRL model is taken into consideration: where V(t) � t 0 (dx/B(x)m 0 (x)). e random variable T * has SF (48) measures time-to-failure length or time-to-death length for a randomly drawn individual. e PDF of T * is then given by In equation (48) and also in equation (49) if Λ is an absolutely continuous CDF, then dΛ(θ) is replaced by λ(θ)dθ. In the continuing part of the paper, various stochastic properties considering the mixture model (48) are studied to provide observations on the model and to detect further characteristics of the model potentially useful for the applicability of the mode on data.

Identification of Several
Partial Dependencies e extended WMMRL model proposed in (48) provides a mechanism to introduce dependence into the model. e recognition of dependency concept(s) the model induces may be useful to obtain aging properties of the model (see, e.g., eorem 2) and vice versa.
Although the partial dependencies in Definition 3 may not hold in some situations, in the context of some typical reliability models they are satisfied after imposing some mild condition. In such situations, the dependency is perfectly explainable using prevalent conditional probabilities. Conditional stochastic orders are also useful to illustrate as they are used to characterize the dependence structures introduced (see for instance, Lemma 1 in Izadkhah et al. [55]). (49) and Θ are PLRD.
One may notice that the sufficient condition of eorem 3 in the context of WMMRL model is equivalent to saying that T * given Θ � θ is DMRL for all θ > 0. is condition which states that the model inspires a positive aging property is a mild condition when the model applies to data for lifetime units that are deteriorated with time. However, the following result indicates no condition is needed to conclude weaker dependencies as shown because T * and Θ enjoy the RCSI dependence property and, moreover, T * is increasing in Θ.
Theorem 4. T * with the SF (49) and Θ are RCSI and also T * is SI in Θ.

Remark 2.
From the chain of implications among dependence concepts, brought after Definition 3, one can further develop by eorem 4 that T * with the SF (49) is RTI in random frailty Θ and also that T * and Θ are PQD. e studied properties of dependencies in the WMMRL model can also be evaluated using the copula function, which plays a prominent role to model the dependence T * and Θ have. e function C as a copula associated to the random pair (T * , Θ) is the joint distribution function of (U, V) such that U � F * U(T * ) and V � Λ(Θ). e copula function captures the dependencies in a pair of random variables without influences of marginal distributions of (T * , Θ). Based on the Sklar's theorem, mathematical derivation of copula function is possible in terms of a composition of the joint and marginal distribution functions as where F * − 1 and Λ − 1 are the right-continuous inverses of F * and Λ, respectively. e above identity is equivalent to For more details and properties of copulas, the readers are referred to Nelsen [47].
We use (50) to obtain the copula function of T * and Θ. First, the joint CDF of T * and Θ is grasped from (41) as From (50), the formation of copula function C is independent of the choices of marginal distribution. us, by choosing an arbitrary distribution Λ for Θ and then deriving F * � 1 − F * from (48) and substituting their inverses in (52) the copula function is characterized by (50). In order to provide a tool to reach mathematical tractability for solving the integral in (52), as is well-known and broadly applied in the context of Bayesian statistics, conjugate priors can be assigned. To this end, let us suppose that Θ has Inverse Weibull distribution with PDF λ(ω) � ω − 2 exp(− ω − 1 ), ω > 0 and thus CDF Λ(θ) � exp(− θ − 1 ), θ > 0 and Λ(θ) � 0 for θ ≤ 0. By applying (48) the CDF of T * is obtained as . (53) e inverse function F * − 1 has no closed form whereas the inverse function Λ − 1 has a closed form as we have ∈ (0, 1). From (52) and in the spirit of (53) one gets By (54) and equation (50), we obtain where k − 1 is the inverse function of k given by 1 + y)). It can be checked that since k is an strictly increasing function thus k − 1 (1) � ∞ and k − 1 (0) � 0. erefore, C(0, v) � 0, for all v ∈ [0, 1] and also C(u, 0) � 0 for all u ∈ [0, 1]. Furthermore, C (1, v) � v, for all v ∈ [0, 1] and C(u, 1) � u, for all u ∈ [0, 1]. ese conditions are requirements for a valid copula function. e regular positive dependencies between T * and Θ in the mixture WMMRL model inspire such that a stochastic variation of Θ will lead to a stochastic alteration of T * in the accordant direction.
is motivates us to study the preservation properties of some stochastic orders that develop a theory applicable to investigate the conclusion of the positive dependencies the model instigate. e next result is a readily obtainable conclusion of Lemma 1 in Izadkhah et al. [55] which is a translation of the result of eorem 3 and the result of eorem 4 into stochastic ordering relations between some conditional random variables. e proof is not given as the result is plain to prove.

Proposition 2.
Let T * have the SF (49). en, and thus and consequently, e values θ 1 and θ 2 in Proposition 2 may be two potential numbers, since as far as the random frailty Θ is unobservable in practice, the information about the accurate values of θ is not attainable. erefore, the study of distribution of Θ among individuals with lifetimes in specific intervals can be carried out using observations on T * . e possibility of the observing T * , and using the data to fit mixture WMMRL model has been considered here as it might be seen in the results of eorems 1-4 in Gupta and Gupta [56] in the context of a general frailty model and also in the result of eorem 3 in Rezaei and Gholizadeh [57] in the context of a mixture PMRL model. eorem 1 utilized the distribution of frailty Θ among survivors of age t to establish the MRLF of the general MRL frailty model. Proposition 1 acquired a stochastic property involving the distribution of frailty among survivors in the general MRL frailty model.

Sufficient Conditions for Some Aging Properties
A considerable percentage of reliability theory is given over the study of aging concepts, their properties, entanglements, and applications. Detecting aging paths of lifetime distributions is useful for recognizing the performance of a model and for model selection purposes as life distributions are classified on the basis of their aging behaviour and thus knowledge of this aging behaviour could be used effectively (cf. Nair et al. [58]). In reliability and survival analysis, there is often the need to recognize the aging behaviour for a lifetime unit to take into account the process of deterioration over time to determine how expectations on the failure are raised. In Section 2, we gave sufficient conditions for standing the DMRL (IMRL) and the NBUE (NWUE) properties of lifetime distribution in the setup of general MRL frailty model (22). In this section, our concentration is on a similar investigation in the context of several aging classes of lifetime distributions for the WMMRL mixture model to find conditions under which T * , following the lifetime distribution (with the SF (48)) possesses several aging properties. In the WMMRL model, a specific individual has MRL m(t|θ) � θB(t)m 0 (t) which is increasing in θ > 0 for all t ≥ 0.
is makes use of eorem 2 (a). For the population, the random lifetime T * which follows the mixture model (49) is the time-to-failure of a random individual.

Remark 4.
In the setting of WMMRL model, by Proposition 2 (ii), it holds that (T * |Θ > θ 1 ) ≤ hr (T * |Θ > θ 2 ), for all nonnegative θ 1 < θ 2 . In particular, for values θ 1 � 0 and θ 2 � θ > 0 one gets (T * |Θ > 0) ≤ hr (T * |Θ > θ) for all θ > 0. By applying Definition 1 (ii) in a conditional setting and denoting T i ≡ (T * |Θ > θ i ) when θ 1 � 0 and θ 2 � θ, since T 1 ≤ hr T 2 and P(T i > t) � P(T * > t|Θ � θ i ), i � 1, 2, we thus conclude By Bayes' rule one reverses the conditioning as which by (61) is nondecreasing in t ≥ 0, for all θ > 0. is is equivalent to saying that Θ is RTI in T * . From Remark 2, T * and Θ follow the PQD structure either. us, application of eorem 2 (a) in the setup of negative aging properties of the NWUE and the IMRL is possible. e conclusion is that, if m(t|θ) ≥ m(0|θ) for all t ≥ 0 and for all θ > 0, then T * is NWUE and if m(t|θ) increases in t ≥ 0 for all θ > 0 or since However, in view of Remark 4, the result of eorem 2 does not provide any conclusion on the positive aging classes of the DMRL and the NBUE. From eorem 1, by substituting m(t|θ) � θB(t)m 0 (t) as the characteristic of the WMMRL model contributes to (A.2), it is deduced that T * has the MRLF e HRF of T * following the WMMRL model is also given by (cf. Nanda and Das [43]) e following result establishes necessary and sufficient conditions for the positive aging classes of the IFR, the NBUE, and the DMRL.
In the general frailty model (22), the amount of the parameter θ is random and it is a realization of the random frailty Θ. e study of the influence of the heterogeneity caused solely by the frailty on the population lifetime distribution in the WMMRL model can be conducted using stochastic orders.
is can be done in the way that the effect of variation of frailty variable when Θ 1 as the frailty of first group is replaced by Θ 2 as the frailty of second group on the alteration of the resultant lifetime distribution under the setup of the model. Suppose that Θ i has an absolutely continuous CDF Λ i with corresponding PDF λ i , for i � 1, 2. e response variable when frailty is Θ i is denoted by T * i . By an application of (48), the SF of T * i is obtained as in which V(t) � t 0 (dx/B(x)m 0 (x)). e characterized typical dependencies between T * and Θ as found in eorems 3, 4 and Remark 2 make stochastic ordering relation of one of these random variables correspond to the stochastic ordering relation of the other one.
To investigate how a stochastic variation of Θ makes stochastic alteration of T * , the property of transmission of stochastic orders of Θ 1 and Θ 2 to stochastic orders between T * 1 and T * 2 is considered. e existence of positive dependence between Θ and T * implies that stochastic variation of Θ towards one direction inspires T to be varied in the similar direction. e choice of frailty distribution for Θ, for which there are no observations available, is usually made by referring to its mathematical tractability. e knowledge of preservation of stochastic orders from frailty distributions to stochastic orders of the implied lifetime distributions may be beneficial for clarification of the effect of miss-specification of the frailty distribution. e miss-specification is sometimes regular so that it is explained perfectly by a stochastic order. For example, the influence of a wrong choice of the measurement scale of Θ on the lifetime variable T * is detected when a stochastic order of Θ and σΘ where σ is a scale parameter with a value in a certain interval leads to a stochastic order of their corresponding lifetime variables. In an analogous situation, the implication of an incorrect selection of the location for measurements of Θ on the lifetime variable T * is realized when a stochastic order is between Θ and μ + Θ where μ is a location parameter implies a stochastic order of the associated lifetime variables. e main result for the translation of stochastic orders of random frailties in the WMMRL model to stochastic orders of the resulting lifetime distributions is given next. It is assumed, we suppose that interchanging derivatives and integrals is allowed and further that Θ i and T * i for i � 1, 2 have absolutely continuous distribution functions.
A potential application of the result of eorem 6 may be as follows. In view of eorem 6 (ii), if a miss-specification in the choice of frailty distribution happens, for example, for location parameter in the sense that when the true parameter is μ 1 the parameter μ 2 > ( < )μ 1 is used. en since Θ + μ 1 ≤ st ( ≥ st )Θ + μ 2 , then distribution function is overestimated (underestimated). In the spirit of eorem 6 (iii), if a missspecification in the choice of an IFR (DFR) frailty distribution happens for location parameter in the sense that when the true parameter is μ 1 the parameter μ 2 > μ 1 is used then since, from the IFR (DFR) property, Θ + μ 1 ≤ hr ( ≥ hr )Θ + μ 2 then the hazard rate function is underestimated (overestimated). By eorem 6 (iv) if a miss-specification happens for location parameter in the choice of a DMRL (IMRL) frailty distribution so that F(t|θ) is convex in θ for all t ≥ 0 such that when the true parameter is μ 1 the parameter μ 2 > μ 1 is mistakenly used then since, from the DMRL (IMRL) property, Θ + μ 1 ≤ mrl (≥ mrl )Θ + μ 2 then the MRLF is overestimated (underestimated). An analogue discussion can be conducted if, for example, a missspecification in the choice of frailty distribution happens for the scale parameter.

Summary and Conclusion
In this paper, we first considered a unified MRL frailty model. From a variety of known survival models, the MRLF was determined. Plots were constructed for the derived MRLFs to visualize the monotonicity and nonmonotonicity of the MRLF with respect to the variation of the parameters. As an extension of the model, the mixture model was also considered, for which expressions of the HRF and the MRLF were given. In Figure 8, we give a flowchart to explain the proposed methodology. e MRLF has been used to detect sufficient conditions for the lifetime distribution to possess the DMRL (the IMRL) and the NBUE (the NWUE) aging properties. e special frailty model with a weighted MRL so that the MRL of an individual is the product of the parameter θ, the weight function B(·) and the baseline MRL were also scrutinized. e PDF and SF of the resulting lifetime distribution in individual level (fixed effect of parameter) and population level (random effect of parameter) were acquired. It was argued and acknowledged that the lifetime variable and the random parameter (frailty) satisfy the partial dependencies PQD, SI, RCSI, and PLRD. Necessary and sufficient conditions for the lifetime distribution to have DMRL, IFRA, and NBUE aging properties were obtained. e hazard rate, the likelihood ratio, the MRL, and the increasing convex stochastic orderings of lifetime distributions with different frailties have been built when these stochastic orderings hold for the underlying frailties.
To clarify the scope of the study, it has to be noted that the MRL frailty model in the general form of (22) encompasses many specific parametric models. Stochastic behaviours and the sensibility of these models to the change of the input components including frailty distribution by the methods developed in this paper can be investigated. e weighted multiplicative MRL model may be applicable in situations where the ordinary proportional MRL model does not hold. In situations where data are available, there is no reason to find consistent inferential procedures to gather information about the population specially when the model is complicated. In such circumstances, to infer about the population appealing to stochastic properties such as ordering properties (for the sake of comparison) and aging properties (to predict failure times) can be consentient. It is almost acknowledged in practical works that the distribution of frailty is not specified and in parallel even if there is no data on frailty variable because its value is affected by hidden features of distributions which are not observable occasionally. However, ordering conditions between frailty distributions may be imaginable. In such cases, ordering properties in the population can be guaranteed. e study, however, has the following limitations: (1) e CDF of the general frailty model (22) and the specific model (40) does not have a closed form and thus obtaining straightforward likelihood functions on the basis of data is not possible (2) e choice of B(·) in model (40) is restricted since the function m(·|θ) has to be valid as an MRLF. e problem becomes more controversial when the baseline MRLF (m 0 (·)) is unknown In the future of this study, in the context of statistics, the following problems close to our achievements may be sought: (i) Estimation of the parameter θ in different situations including censored data and also providing techniques to recognize the adequacy of the WMMRL model in practical situations can be considered. (ii) For making statistical inferences about the parameter θ and related problems, novel nonparametric and semiparametric strategies can be adopted. e problem may lie under different sampling schemes. (iii) In the context of applied probability for making inferences on the model without data, one may strive to find weaker conditions on the frailty distributions rather than imposing stochastic orderings between them and also detecting certain assumptions on the baseline distribution and the weight function B to obtain further stochastic aspects of lifetime distribution.  where F(·|w) is the conditional SF of T * given Θ � w.

(A.18)
It is seen that E Θ|T * > t � D(− V(t)) and thus d dt By some routine calculation, one has (A.20) By above identities and since , for all t ≥ 0.
(iii) e proof obtains if we show that F * i (t) is TP 2 in (i, t) when i � 1, 2 and t ≥ 0. e partial derivatives of F(t|θ) with respect to t and also θ exist and they are continuous. We have which is nonnegative from eorem 4. By integration by parts is completes the proof of the theorem.

Data Availability
ere is no source of data applicable in this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.