1. Introduction
A large segment of statistical methodology deals with the modeling and analysis of data representing the time until the occurrence of an event (cf. Nanda et al. [
1]). In industrial and biomedical studies, these events represent the time to failure of a machine, an organ, an individual, or the completion of a certain task. These times are referred to as survival times in biomedical studies and as lifetimes or failure times in actuarial and engineering studies (cf. Henley and Kumamoto [
2], Fleming [
3] and Miller [
4]). Reliability measures are ordinarily considered in lifetime sciences as characteristic quantities to quantify the amount of uncertainty during the lifetime process of an item to presignify its life span. Endless numbers of models are being introduced in the literature to model failure time data. The proportional hazard rates (PHR) model, used to model left-truncated and right-censored failure times, is a reputable model that has been greatly developed in the literature (see, e.g., Cox [
5]). To concentrate on the average amounts in place of probability values Oakes and Dasu [
6] introduced the proportional mean residual life (PMRL) model for the analysis of reliability and survival data in the situations where left-truncated or right-censored observations are available. Nanda et al. [
7] studied some reliability aspects and further stochastic properties including a number of stochastic comparisons and ageing properties in the PMRL model. In contrast to the PHR and the PMRL models, there have been proposed specific models to account for right-truncated and left-censored observations. The proportional reversed hazard rates (PRHR) model has been introduced as an alternative model for the PHR model by Gupta et al. [
8] and some stochastic aspects along with further inferential procedures about it have been studied by Gupta and Gupta [
9]. The proportional mean past lifetime (PMPL) model that focuses on the mean inactivity time of items has been considered by Asadi and Berred [
10] and several stochastic properties of it have been developed by Rezaei [
11].
The advantage of the current investigation is to establish another model based on the vitality function of a distribution which is a common measure of life expectancy among demographers. The vitality function of a distribution is the conditional mean of the random variable on the right tail of distribution (see, e.g., Navarro et al. [
12], Sunoj et al. [
13] and Abdul-Sathar et al. [
14]). The proposed model can be used for modelling right-censored and left-truncated distributions.
Let
X be a non-negative random variable (RV) having absolutely continuous cumulative distribution function (CDF)
F and probability density function (PDF)
f. The vitality function of
X is
and the hazard rate (HR) function of
X is
The mean residual life (MRL) of
X after time point
t is
If the support of
X is
, then
for all
. In such a case,
,
and
are not defined for
. But it is customary, in such cases, to define
. The vitality function has a relationship with the HR and the MRL functions as follows:
where
and
For the random variable
X with a finite mean such that
the survival function (SF) of
X (
) can be retrieved from vitality function
by the inversion formula
Demographers have used the vitality function or life expectancy or expectation of life function for centuries in studies of human populations (see, e.g, Vaupel et al. [
15], Yashin et al. [
16] and Sharrow et al. [
17]). The vitality function is monotonically increasing over
. With this in mind, it is a valid conjecture that the curve of the ratio of the parametric or non-parametric estimated vitality functions based on two sets of data will indicate significant departures from a horizontal line with less probability than the case where hazard rate or mean residual life functions are utilized.
The regression models proposed for lifetime data generally consider the assumption of the PH model or the assumption of PMRL. For instance, suppose that we use the PMRL model, i.e., for all and for some unknown where and m are, respectively, the baseline and the response mean residual life functions assumed to be unknown. In the cases that two samples from the underlying distributions of m and is available, there is a principle regarding the shape of the two MRL functions. The MRL functions may exhibit quite different behaviours, such as monotonically increasing, monotonically decreasing, bathtub shaped, upside-down bathtub shaped and roller-coaster shaped depending on the fluctuations of data. The data analyst observes that a strong assumption in the PMRL model exists as the behaviour of the MRL curves must be the same under the setup of the model. For example, consider that according to data in a two sample problem the shape of the estimated mean residual lives are a lot different, thus, it is unlikely to find an appropriate value for to incorporate that possibility and this leads to an inadequacy for fitting the model to data.
The survival models have developed rapidly in the literature to model time to event data (cf. Hosmer Jr and Lemeshow [
18] and Liu [
19]). When a semi-parametric model is proposed, there may exist some challenging issues regarding that model. For instance, one may question the suitability of the model to model data in many practical situations and that how well does it fit the data. The main goal of this paper is to introduce a semi-parametric model based on the concept of vitality function, called proportional vitalities (PVIT) model. In the context of the PVIT, the involved age-specific vitality function is inherently monotonically increasing. The credibility of modelling lifetime data using the PVIT model may be worthy.
The entire structure of this paper is organized as follows. In
Section 2, the model is proposed. In
Section 3, making inferences about the parameter(s) of the model is discussed. In
Section 4, we lay the foundations of several stochastic orderings and aging properties. In
Section 5, a brief conclusion of the paper is emphasized and some generalizations for future work are outlined.
2. The PVIT Model
First of all, in what follows in the paper, we refer the readers for the definitions of the utilized stochastic orders to Shaked and Shanthikumar [
20] and for the concepts of aging notions that will be used in the sequel to Barlow and Proschan [
21]. The property of Totally positive of order 2 (TP
) for bivariate functions is also adopted from Karlin [
22]. To introduce the model, some motivations shall be provided. An examples of parametric distributions is brought to clarify the structure of the model. The following definition is stated.
Definition 1. X and Y with respective vitality functions and are said to satisfy the PVIT model, if for all where ξ is a positive constant that we call it vitality growth parameter. We can infer that
is the expected life-length of an individual randomly drawn from the individuals grouped at the age
t. The PVIT model induces that the ratio
is
t-free which means that the vitality functions of
X and
Y are relatively free of their underlying aging processes. From a mathematical point of view,
purporting that relative mean life-lengths of survivor individuals in both populations remain unchanged when the age of survival is variable. In particular, for almost surely positive random variables,
In view of (
3), if
X and
Y satisfy the PVIT model as described in Definition 1, then
The identity (
7) may not give a closed form for the SF of the random variable
Y in terms of SF of
X, whereas in the PMRL model the inversion formula provides an explicit expression for the SF of the response variable in terms of a combination of the SF of the underlying distribution function with that of its equilibrium distributions (see, e.g., Nanda et al. [
23]). On that account, we are capable of proving that (
7) presents a proper SF if the following conditions hold:
- (i)
For all
- (ii)
for all
- (iii)
In what follows, the PVIT model is characterized in terms of the relationship between the ratios of survival (or hazard rate) functions of the length-biased distributions and the ratio of those of the underlying distributions. Recall that
and
with PDF’s
and
(provided that the expectations exist and are finite) are said to follow length-biased distributions associated with
f and
respectively (cf. Gupta and Keating [
24]). Length-biased sampling arises when a component already in use is sampled at a fixed time and then allowed to fail (cf. Scheaffer [
25]). The concept of length-biased sampling and the arising hazard rates, as well as cumulative distribution functions and the relations with other reliability measures, are appreciated in the literature (cf. Gupta and Keating [
24]).
Proposition 1. The non-negative random variables X and Y satisfy the PVIT model, if and only if one of the following assertions holds;
- (i)
, for all
- (ii)
for all
Proof of Proposition 1. By Proposition 1 in Izadkhah et al. [
26], for all
we have
and
Hence,
, for all
, if, and only if,
, for all
which is equivalent to (i). By a further application of Proposition 1 of Izadkhah et al. [
26], one has, for all
and
It is perceptible that under the setup of the PVIT model, is free of Thus, one can deduce that for all This, together with the recent identities above, establishes the equivalence of (ii) and the PVIT model. □
For studies in reliability, biometry and survival analysis, the length-biased distributions have been frequently appropriate for quite a number of sampling plans. Suppose that two random samples on X and Y cannot be reached due to an ungovernable biased sampling procedure. In spite of that, imagine that the available data come from the associated length-biased distributions. Proposition 1 states that in the context of the PVIT model, the ratio of survival functions (or the hazard rates) of the original distributions could be estimated under length-biased sampling as effectively as acting under random (unbiased) sampling. The following example indicates that by adding the vitality parameter to the exponential distribution, the well-known Hall–Wellner family of distributions is generated.
Example 1. Suppose that X is distributed with mean in accordance with exponential distribution. In an easy way, we get Let Y be a random variable with life vitality for where Then, Y has mean residual life function which is the characteristic of the Hall–Wellner family of distributions. To be clear, recall from Hall and Wellner [27], their family of survival functions which is valid for and . When and , it gives respectively a Pareto, an exponential (at limit) and a rescaled beta distribution. Here, by choosing and , it is clarified that the distribution of Y is in the Hall–Wellner family. Now, we discuss a characterization result within the class of distributions satisfying the PVIT model.
Theorem 1. Let X and Y be two non-negative random variables with finite means and hazard rate functions and respectively, such that . Then, X and Y satisfy the PVIT model if and only if they are equal in distribution.
Proof of Theorem 1. By the construction of the model,
for all
which is free of
Therefore,
Remark that since
thus, when
and
. As a result, using l’Hopital’s rule,
In a similar manner,
It thus concludes that
i.e.,
for all
which means that
X and
Y have equal distributions. □
4. Some Closure and Preservation Properties
In this section, the closure (preservation) property of some stochastic orders (aging classes) under the formation of the model is studied. The problem has attracted the attention of many researchers in the recent past decades (cf. Amini-Seresht and Zhang [
28], Amini-Seresht and Khaledi [
29], Barmalzan and Najafabadi [
30]). The following result indicates that the model by restricting the domain of the parameter
induces the hazard rate (mean residual life) ordering property between the underlying random variables.
Theorem 2. , if and only if,
Proof of Theorem 2. From the relationships among the HR and the MRL orders, it suffices to prove that
implies
and that
implicates
Under the set up of the model, by (
1)
which is non-negative for
as
for all
because
is increasing in
(see, e.g., Proposition 2.4 of Nanda and Jain [
31]) and also
is decreasing in
for any
This indicates that when
Now, we prove that
implies that
From (
2), for random variables
X and
Y that satisfy the PVIT model, we can get
for all
and thus
, which is greater than 1 by definition. □
However, the following example is an indication that the hazard rate order in Theorem 2 cannot be replaced by the likelihood ratio order.
Example 2. Let X and Y have, respectively, exponential distribution and Lomax distribution with survival functions and It can be seen that for all , i.e., vitality of Y has been evolved with respect to vitality of X according to the PVIT model with a frailty growth identified by By some routine calculations, it is perceived that .
In the recent past decade, Righter et al. [
32] propounded some fresh ageing notions on the basis of scaled conditional lifetime. The scaled conditional life of a random variable
X is the total life relative to the current age, conditioned on the current age, and is given by
where
for all
t with
is called the residual life of an item with age
in which
means equality in distribution. For further studies of the family of distributions of
, see Righter et al. [
32], Belzunce et al. [
33] and the references therein. A new class of life distributions could be as the following. Recall that a function
is anti-star-shaped on
if
is non-increasing in
Definition 2. The non-negative random variable X is said to have decreeing mean scaled conditional life (denoted as ) if is non-increasing in or equivalently, is an anti-star-shaped function in
It has been investigated by the authors that some ageing properties of X are inherited by the random variable Y in the context of the model. Before stating our findings about that problem, we need to prove the following definition and key lemma.
Definition 3. The non-negative random variable X is said to have
- (i)
increasing failure rate (IFR) whenever the hazard rate function of X is non-decreasing.
- (ii)
increasing mean residual life (IMRL) whenever the mean residual life function of X is non-decreasing.
- (iii)
increasing failure rate in average (IFRA) whenever is non-decreasing in .
- (iv)
new worse than used (NWU) property whenever for all and for all .
- (v)
new worse than used in expectation (NWUE) whenever for all
- (vi)
increasing generalized failure rate (IGFR) whenever is non-decreasing in .
The decreasing failure rate (DFR) and the decreasing failure rate in average (DFRA) aging paths are defined by reversing the required monotonicity behaviour in Definition 3(i) and Definition 3(ii), respectively. The new better than used in expectation (NBUE) property is also defined by revering the side of the inequality in Definition 3(v).
Lemma 1. Let such that (). Then the function δ given byis non-decreasing (non-increasing) in for all ξ in the specified intervals. Proof of Lemma 1. Take
as differentiable over
. We prove that
for all
and for any
(
). One has
where
ab means that
a and
b have the same sign. The proof is completed. □
In the situations where
X and
Y with common support
satisfy the IFR (or even the ultimately IFR) property, the assumption of Theorem 1 is fulfilled. Furthermore, when
X and
Y possess the (ultimately) DMRL ageing property, then trivially both
and
tend to zero when
t is closing to infinity and
X and
Y are equal in distribution in this case either. It can be shown that when
X and
Y have IFRA property then for all
and the requirement
together with (
16) makes the assumption of Theorem 1 satisfied. Therefore, in all of these cases the model is valid if and only if distributions of
X and
Y coincide each other and as a result preservation property for the foregoing positive aging classes is an ineffective study. In spite of that, we can prove the following results. For the IGFR class of distributions, we refer the readers to Lariviere and Porteus [
34].
Theorem 3. Suppose that X and Y are as in the model. Then,
- (i)
For any IMRL implies that IMRL.
- (ii)
For any NWUE implies that NWUE.
- (iii)
For any NBUE implies that NBUE.
- (iv)
For any IGFR implies that IGFR.
Proof of Theorem 3. To prove (i), from the relationship between
and
in Theorem 2, we have
which is non-negative for all
by assumption. For proving (ii) [(iii)], note that
NWUE [NBUE] yields
for all
which concludes by assumption that
for all
considering that
for all
since
is a non-decreasing function. It thus follows that
for all
which is equivalent to
for all
i.e,
NWUE [NBUE]. To prove the assertion (iv), denote first by
and
the PHRs of
X and
Y, respectively. From the construction of the model it follows that
where
is as defined in Lemma 1. From assumption, since
IGFR thus
is non-decreasing, and according to Theorem 20 in Kayid et al. [
35],
DMSCL. On using Lemma 1, when
is non-decreasing in
which results
IGFR.
The next result establishes preservation of some other ageing properties.
Theorem 4. Let X and Y be related as in the model such that DMSCL. Then,
- (i)
For any DFR implies that DFR.
- (ii)
For any DFRA implies that DFRA.
- (iii)
For any NWU implies that NWU.
Proof of Theorem 4. To prove (i), observe that
for all
By applying Lemma 1, the result follows immediately. For the sake of proving (ii), notice that
DFRA, if and only if,
for all
Since
and
DMSCL thus Lemma 1 concludes that
is non-negative and non-increasing in
Therefore,
which clarifies that
DFRA. To demonstrate (iii), be mindful of that
NWU if and only if,
for all
and for all
Now, set
where
is as given in Lemma 1 and
is the indicator function of the set
It is clear from assumption by using Lemma 1 that
is non-increasing in
for all
and for all
with
On that account, Lemma 7.1(b) in Barlow and Proschan [
21] implies that
where
In contrast, since
is non-negative and non-increasing by assumption, thus
which is non-negative from (
17) and the result follows. □
5. Conclusions
The PVIT model has some complementary role with respect to the so called PHR model. The random varaibles
X and
Y are said to have PHR when
for all
and for some constant
(see, e.g., Nanda and Das [
36]). As in the PHR model, the data from lifetime of units that deteriorate with age could be used but in the PVIT model those units that recuperate with age can be modelled. In a regression analysis problem, the vitality growth parameter
may be expressed in terms of some covaraites, i.e.,
where
is a
vector of regression parameters, and
is a
vector of covariates. On the
ith individual is available a certain vector of observed covariates
As a result, the factor
in the study can be used to quantify some positive/negative events. For example, in the PHR model
, as a covariate, has two levels for the
ith individual,
describes a non-smoker and
a smoker. In the PVIT model,
if the
ith individual does not get enough exercises weekly, and
if he/she does have bodily activity weekly enough. It is well-known that if somebody is a smoker, the hazard rate of his/her lifetime increases, but if the person is an athlete, his/her lifetime is improved in the sense of vitality function, relatively.
Specifically, in the current investigation, some simulation studies were carried out to detect the accuracy of the PVIT model based on some data sets. The numerical iteration method of Newton–Raphson has been adopted to estimate the parameters of the model. Several reliability properties in the PVIT model, including some stochastic orders and some aging properties, are discussed. In the future of this study, time dependent proportional vitalities model will be considered.