Stochastic aspects of reversed aging intensity function of random quantiles

Abstract

This paper studies some stochastic properties of random quantiles according to the newly defined reliability measure called reversed aging intensity function. Preservation property of reversed aging intensity order under random quantile is obtained and using it, a lower bound and an upper bound for the reversed aging intensity function of a random quantile are derived. Preservation of two related monotonic reliability classes under random quantiles is also studied. We finally apply our results for reliability analysis of series systems with heterogeneous component lifetimes. Examples are included to examine and analyze the obtained results.

1 Introduction

In the context of reliability and service life science, various characteristics for a service life distribution have been introduced in the literature. For example, a well-known feature of a lifetime distribution is the hazard rate (hr) function, which measures the immediate risk of failure of a device after a certain time t, assuming the device has survived up to that point. An intuitive aspect of a lifetime distribution that is often used in reliability engineering and survival analysis is the phenomenon of aging. In practice, the problem of aging affects many systems and their components. The properties of random lifetimes are usually characterized by their respective cumulative distribution function (cdf), survival function (sf) and the hr function. The aging intensity function is a relatively new concept that can also be used in lifetime analysis. The aging property of a lifetime distribution fitted to the observations of a sample of failure times of a device provides information about the degree of deterioration or the degree of improvement of a device over time. The concept of aging is closely related to the remaining life random variable (rv). Let X be a nonnegative rv with sf $\bar{F}_{X}$. Then, the conditional rv $X_{t}=[X-t\mid X>t]$, which is well-defined for all $t\geq 0$, for which $\bar{F}_{X}(t)>0$ is called the residual lifetime rv, which has sf

$$ \bar{F}_{t}(x)=\frac{\bar{F}_{X}(t+x)}{\bar{F}_{X}(t)},~~x\geq 0. $$

The hr function of X is also closely related to the residual life rv $X_{t}$ and its distribution. This is because the hr function of an rv X with probability density function $f_{X}$ is defined as

$$ h_{X}(t):=\lim _{\delta \rightarrow 0^{+}} \frac{P(X_{t} \leq \delta )}{\delta}=\frac{f_{X}(t)}{\bar{F}_{X}(t)}. $$

The hazard rate of a system can serve as a qualitative indicator of its aging. The idea of aging intensity allows one to evaluate a system’s aging characteristic objectively and more accurately. In order to quantitatively assess the aging property of a unit, which might be a system or a living organism, the aging intensity (AI) function has been introduced in the literature. The function $L_{X}$ denotes the AI function of random life length X, and it is the ratio of the instantaneous hazard rate (hr) $h_{X}(t)$ divided to the average hazard rate $H_{X}(t)=\frac{1}{t}\int _{0}^{t} h_{X}(\tau )d\tau$, which is defined as follows:

$$\begin{aligned} L_{X}(t)&:=\frac{h_{X}(t)}{H_{X}(t)} \\ &=\frac{-tf_{X}(t)}{\bar{F}_{X}(t)\ln (\bar{F}_{X}(t))},~~t>0. \end{aligned}$$

(1.1)

In reliability engineering, the AI function has been utilized by many reliability analysts in describing lifetime events. The study of AI function by Nanda et al. [33] aimed to realize some stochastic features of different distributions and to determine the connections of the AI function with other reliability concepts. They presented two monotonic aging classes of life distributions based on the AI function and used it to establish a stochastic order known as aging intensity order. Bhattacharjee et al. [8] examined the behavior of several generalized Weibull models and specific system characteristics with respect to the AI function. Szymkowiak [49] using the AI function, reported some characterisation results on Weibull- and inverse-Weibull-related distributions.

In contrast to remaining life, the past life or reversed remaining life of a system is also a useful concept in the context of reliability (see, e.g., Nanda et al. [35]). When designing a system of components in reliability engineering, the solidarity of the components in operation is very important. To quantify the degree of solidarity in a coherent system in reliability studies, measuring the inactivity time of components at the time when the system fails is a useful tool. Some examples of abnormal inactivity times are worker strikes, extended power outages and machine failures due to poor maintenance procedures. Let X be a nonnegative rv with cdf $F_{X}$. Then the conditional rv $X_{(t)}=[t-X \mid X\leq t]$, for all $t>0$ for which $F_{X}(t)>0$ is considered as the inactivity time (or past time since failure) of a living organism at the time t. The rv $X_{t}$ has sf $\bar{F}_{(t)}$ acquired as:

$$ \bar{F}_{(t)}(x)=\frac{F_{X}(t-x)}{F_{X}(t)},~~0< x< t. $$

The reversed hazard rate (rhr) of a lifetime unit is closely related to $X_{(t)}$, as it is defined as

$$ \breve{h}_{X}(t)=\lim _{\delta \rightarrow 0^{+}} \frac{P(X_{(t)}\leq \delta )}{\delta}=\frac{f_{X}(t)}{F_{X}(t)}. $$

(1.2)

The rhr function has attracted the attention of researchers and can be thought as the instantaneous failure rate occurring just before the time point t, given that the unit has not survived longer than time t. In a certain sense, it is the dual function of the hr function and it bears some interesting features useful in reliability analysis (see also Block et al. [10] and Finkelstein [14]). The rhr function has been used to establish stochastic orders among random lifetimes (see, for examples, Rezaei et al. [42] and Hazra and Nanda [16]). This function has also been used to construct new classes of life distributions (see, e.g., Nanda et al. [35], Nanda et al. [36] and Oliveira and Torrado [38]).

Recently, the concept of reversed aging intensity ($RAI$) function has been introduced in literature (see, e.g., Szymkowiak [50] and Buono et al. [11] and the references therein). Kundu and Ghosh [23] characterized a few statistical distributions using $RAI$ function. Buono et al. [11] introduced a family of generalized $RAI$ functions and established some characterization results. Then, Buono et al. [12] provided some improvements on generalized versions of $RAI$ functions and detect the relation between two cumulative distribution functions leading to the same generalization. Goodarzi [15] consdiered the $RAI$ function in discrete case and obtained some of its properties. The $RAI$ function is defined as the ratio of the instantaneous reversed hazard rate to the baseline value of the reversed hazard rate. Let us denote by $\breve{L}_{X}$ the $RAI$ function of the rv X. The $RAI$ function is the ratio of the instantaneous reversed rhr $\breve{h}_{X}(t)$ divided to the baseline rhr function $\breve{H}_{X}(t)=\frac{1}{t}\int _{t}^{+\infty} \breve{h}_{X}(u)du$ given by:

$$\begin{aligned} \breve{L}_{X}(t)&:=\frac{\breve{h}_{X}(t)}{\breve{H}_{X}(t)} \\ &=\frac{-tf_{X}(t)}{F_{X}(t)\ln (F_{X}(t))},~~t>0. \end{aligned}$$

(1.3)

Let X and Y be two nonnegative rvs representing the lifetime of two items with $RAI$ functions $\breve{L}_{X}$ and $\breve{L}_{Y}$, respectively. We say then that X is smaller than Y in the $RAI$ order (denoted as $X\leq _{RAI}Y$) if $\breve{L}_{X}(t)\geq \breve{L}_{Y}(t)$, for all $t>0$. Through such stochastic comparison, one may be able to make an ordering of lifetime distributions based on their instantaneous risks for failure occurring just before a given time t, say. Stochastic ordering among probability distributions has been a useful aid to compare distributions from their meaningful aspects (cf. Shaked and Shanthikumar [47], Li and Li [26] and Belzunce et al. [6]). In general, there are three kinds of stochastic orders in literature, namely, the stochastic orders which consider magnitude of random variables, the stochastic orders which take the variability of random variables into account and the stochastic orders which deal with dispersion of random variables (see, Shaked and Shanthikumar [47]). Recently, some stochastic orders have been considered in literature which consider the relative aging of lifetime units (see, e.g., Misra and Francis [29]). As we illustrate in Sect. 3, the $RAI$ order is also related to the relative aging of two lifetime distributions. We also consider two reliability classes, namely increasing reversed aging intensity ($IRAI$) and decreasing reversed aging intensity ($DRAI$) classes. We shall say that X has a distribution with $IRAI$ (resp. $DRAI$) property whenever its $RAI$ function, i.e., $\breve{L}_{X}(t)$ increases (resp. decreases) in $t>0$. The problem of preservation of stochastic orders and aging classes of life distributions under various reliability models have been exhaustively studied in literature (see, e.g., Nanda et al. [34], Li and Yam [27], Navarro et al. [37], Sangüesa et al. [46], Fang and Tang [13], Kayid et al. [21], Izadkhah et al. [17] and Rao and Naqvi [40]). The aim of the present paper is to develop some preservation properties of the $RAI$ order as well as of the $IRAI$ and $DRAI$ classes under random quantiles.

The structure of the paper is as follows. In Sect. 2 we give some preliminary remarks and useful definitions that are used throughout the paper. In Sect. 3, we provide some interpretations of the $RAI$ order and the $IRAI$ class and the $DRAI$ class of life distributions in the context of reliability and information theory. In Sect. 4 we explain the motivation for applying the $RAI$ order and the $IRAI$ and $DRAI$ classes to random quantiles. In Sect. 5 we develop the $RAI$ order for random quantiles. In Sect. 6 we derive a lower and an upper bound for the $RAI$ function of random quantiles. In Sect. 7 we obtain the preservation properties of the $IRAI$ and $DRAI$ classes under random quantiles. In Sect. 8, we apply our main results to the reliability analysis of series systems with heterogeneous components under the proportional hazard rate ($PHR$) model. In Sect. 9, we conclude the paper with further illustrations and descriptions of the paper’s results and also provide insights into future studies.

2 Preliminaries

In this section, we provide several preliminaries including some useful definitions which are necessary for our investigation in the remaining part of the paper. Denote by X, the lifetime of a device or a system of components which has cdf $F_{X}$, sf $\bar{F}_{X}$ and pdf $f_{X}$ whenever it exists. The usual stochastic order is defined as below (see e.g,, Shaked and Shanthikumar [47]).

Definition 2.1

Let X and Y denote two random lifetimes such that X and Y have cdfs $F_{X}$ and $F_{Y}$, respectively. The rv X is is said to be less than (or equal to) Y in the usual stochastic order (denoted by $X\leq _{st} Y$) whenever $F_{X}(t)\geq F_{Y}(t)$, for all $t\geq 0$.

The following definition presents an aging class of lifetime distributions.

Definition 2.2

(Lai and Xie [24]). We say that X has a distribution with

• (i) increasing [resp. decreasing] failure rate property, written as $X\in IFR$ (resp. $X\in DFR$), when $h_{X}(t)$ is non-decreasing [resp. non-increasing] in $t>0$.

• (ii) increasing [resp. decreasing] failure rate in average property, written as $X\in IFRA$ (resp. $X\in DFRA$), when $\frac{1}{t}\int _{0}^{t} h_{X}(x)dx$ is non-decreasing [resp. non-increasing] in $t>0$.

The following technical definition can be found in Karlin [18].

Definition 2.3

Let $f(x,y)$ be a nonnegative function. We say that f is totally positive of order 2 (or shortly, f is $TP_{2}$) in $(x,y)\in \mathcal{X} \times \mathcal{Y}$, where $\mathcal{X}$ and $\mathcal{Y}$ are two arbitrary subsets of $\mathbb{R}=(-\infty ,+\infty )$, when

(2.1)

If the direction of the inequality given after determinant in (2.1) is reversed, then f is said to be reverse regular of order 2 (f is $RR_{2}$) in $(x,y)\in \mathcal{X} \times \mathcal{Y}$.

Righter et al. [43] proposed some classes of distributions based on aging. The following definition presents the notion of increasing [resp. decreasing] failure rate relative to average hazard rate (written as $IFR/A$ [resp. $DFR/A$]) from Righter et al. [43].

Definition 2.4

Suppose that X is a nonnegative rv with hr function $h_{X}(\cdot )$. It is said that X is $IFR/A$ [resp. $DFR/A$], whenever the AI function of X, i.e., $L_{X}(t)=\frac{h_{X}(t)}{\frac{1}{t}\int _{0}^{t}h_{X}(x)dx}$ is non-decreasing [resp. non-increasing] in $t>0$.

3 Motivation for the $RAI$ order and the $IRAI$ and $DRAI$ classes

In this section, we present some intuitive discussions of the $RAI$ order and the associated classes of life distributions, namely the $IRAI$ class and the $DRAI$ class. This is necessary to recognize the role of the $RAI$ function, as a recently proposed reliability measure, in reliability studies. We also illustrate some connections to information theory.

As one reviewer of this paper mentioned, unlike the hazard rate, which is related to the aging property of a device, the reversed hazard rate is not known to be related to aging, and unless this point is clarified, the results of the paper will have no practical use. This is because the $RAI$ function is mainly constructed using the reversed hazard rate function. The reversed hazard rate function has been widely used in reliability studies as well as survival analysis in various contexts, from applied probability and distribution theory to statistical inference. As Block and Savits [10] mentioned, the reversed hazard rate function has proven useful in analyzing data with left-censored observations and is also obvious when discussing lifetimes with reversed time scales, for example, in situations involving the study of past events. In practice, there are many situations in which a failure is observed with a delay and thus the duration between the actual time of failure, X, and the time t at which it is observed (i.e., $X_{(t)}=[t-X \mid X \leq t]$) is considered important for further analysis. Examples can be found in criminological studies, where there are many reasons for a delay in observing a crime, such as difficulties in finding a suspect or the fact that the crime was not reported immediately (see, e.g., Read and Connolly [41]). Examples can also be found in military cases where a plane crashes and observation is postponed, and in medical problems, where an illness in a person’s body only develops at a later stage when it is diagnosed by a doctor. The concept of reversed hazard rate is closely related to $X_{(t)}$, as is clear from (1.2), which shows that for a very small $\delta >0$, $P(X>t-\delta \mid X\leq t)\simeq \delta \breve{h}_{X}(t)$ reversed hazard rate function is closely related to the probability of observing a past failure, immediately after its occurrence.

To obtain the time scale, we consider the scale-free inactivity time $X^{\ast }_{(t)}=\frac{1}{t}X_{(t)}$, which is a percentage of inactivity of a previously failed device at time t (see, e.g., Righter et al. [43]). Belzunce et al. [5] introduced the proportional reversed failure rate (PRFR) [called elasticity in Belzunce et al. [4]] of X as $t\breve{h}_{X}(t)$. Note that the PRFR function is related to $X^{\ast }_{(t)}$ as follows

$$ t\breve{h}_{X}(t)=\lim _{\delta \rightarrow 0^{+}} \frac{P(X^{\ast }_{(t)}\leq \delta )}{\delta }. $$

(3.1)

The PRFR function has been used in economics to evaluate growth rates and elasticity (see e.g. Veres-Ferrer and Pavia [51]) and in the context of reliability for stochastic comparisons of systems (see Khaledi et al. [22]). Oliveira and Torrado [38] have considered the PRFR functions and the associated class of lifetime distributions in the context of reliability theory.

In the context of information theory, the surprise caused by the occurrence of the event A leads to the concept of entropy and also uncertainty (see, e.g., Ross [44]). Indeed, the quantification of surprise is an essential step in introducing the concept of entropy. From a mathematical point of view, the surprise induced by the observation of the event A is recognized as $-\ln (P(A))$, where $P(A)$ is the probability of the event A. Therefore, the function $-\ln (F_{X}(t))$ measures the extent of the surprise caused by the occurrence of the event $\{X\leq t\}$ with the probability $F_{X}(t)$. The function $-\ln (F_{X}(t))$ can be regarded as an evaluation function that takes into account the importance of the event $\{X \leq t\}$. For smaller values of t, for example, the score is high. The inactivity time of a device in the early stages of operation is also very important and needs serious attention. We define a weighted scale-free inactivity time as $X^{\ast \ast }_{(t)}=-\ln (F_{X}(t))X^{\ast }_{(t)}$. The advantage of considering the inactivity time in this form is that an information-based percentage of the inactivity time is obtained. The $RAI$ function then results as

$$ \breve{L}_{X}(t)=\lim _{\delta \rightarrow 0^{+}} \frac{P(X^{\ast \ast }_{(t)}\leq \delta )}{\delta }. $$

(3.2)

Given the limits in (1.2), (3.1) and (3.2), the reversed hazard rate function, the proportional reversed hazard rate function and the $RAI$ function appear to be quantities that measure the probability of observing a failure immediately after its occurrence in different senses. To describe the role of the $RAI$ order in our context, in the following lemma we derive an equivalent condition for $X\leq _{RAI} Y$.

Lemma 3.1

$X \leq _{RAI} Y$ if and only if $\frac{-\log (F_{X}(t))}{-\log (F_{Y}(t))}$ is non-increasing in $t>0$.

Proof

We prove that $\breve{L}_{X}(t) \geq \breve{L}_{Y}(t)$, for all $t>0$ if and only if $\frac{-\log (F_{X}(t))}{-\log (F_{Y}(t))}$ is non-increasing in $t>0$. Let us write for all $t>0$:

$$\begin{aligned} \frac{d}{dt}\left (\frac{-\log (F_{X}(t))}{-\log (F_{Y}(t))}\right )& \overset{sgn}{=} -\frac{f_{X}(t)}{F_{X}(t)}(-\log (F_{Y}(t)))+ \frac{f_{Y}(t)}{F_{Y}(t)}(-\log (F_{X}(t))) \\ &=\breve{h}_{Y}(t).(-\log (F_{X}(t)))-\breve{h}_{X}(t).(-\log (F_{Y}(t))), \end{aligned}$$

which is non-positive for all $t>0$ if and only if $-\breve{h}_{X}(t)\log (F_{Y}(t)) \geq -\breve{h}_{Y}(t)\log (F_{X}(t))$ for all $t>0$. This is also equivalent to $-t\breve{h}_{X}(t)\log (F_{Y}(t)) \geq -t\breve{h}_{Y}(t)\log (F_{X}(t))$, for all $t>0$, which means that $\breve{L}_{X}(t) \geq \breve{L}_{Y}(t)$ for all $t>0$. The proof is completed. □

Stochastic orders compare two distributions on the basis of some physical property possessed by them. Let us assume that X and Y represent the lifetimes of device A and device B, respectively. From Lemma 3.1, if $X\leq _{RAI}Y$, then the relative surprise of the event $\{Y\leq t\}$ increases with increasing t compared to that of the event $\{X\leq t\}$. In other words, the relative surprise of $\{X\leq t\}$ compared to $\{Y\leq t\}$ decreases as t increases. This means that the device A ages faster than the device B, since the failure of the device A in the time interval $(0,t]$ causes less surprise compared to the failure of the device B in the same time interval as t increases.

Before we explain the reliability concepts of the classes $IRAI$ and $DRAI$, we state the following lemma.

Lemma 3.2

Let X be a nonnegative rv with rhr function $\breve{h}_{X}$ and baseline rhr function $\breve{H}_{X}$. Then, $X\in DRAI (X\in IRAI)$ if and only if $\frac{-\ln (F_{X}(\alpha t))}{-\ln (F_{X}(t))}$ is decreasing (increasing) in $t>0$ for all $\alpha \in (0,1)$.

Proof

We prove the non-parenthetical part of the lemma. The other part has a similar proof. Suppose that $\frac{-\ln (F_{X}(\alpha t))}{-\ln (F_{X}(t))}$ is decreasing in $t>0$ for all $\alpha \in (0,1)$. We can get

$$ \frac{\partial }{\partial t} \left ( \frac{-\ln (F_{X}(\alpha t))}{-\ln (F_{X}(t))}\right ) = \frac{1}{(-\ln (F_{X}(t)))^{2}}\left (\frac{f_{X}(t)}{F_{X}(t)} \breve{H}_{X}(\alpha t)-\frac{f_{X}(\alpha t)}{F_{X}(\alpha t)} \breve{H}_{X}(t)\right ), $$

which is nonpositive for all $t>0$ and for every $\alpha \in (0,1)$ if and only if

$$ \alpha \breve{h}_{X}(\alpha t)\breve{H}_{X}(t) \geq \breve{h}_{X}(t) \breve{H}_{X}(\alpha t),~\forall ~t>0,~~ \forall ~\alpha \in (0,1), $$

or equivalently if

$$ \alpha \frac{\breve{h}_{X}(\alpha t)}{\breve{H}_{X}(\alpha t)}\geq \frac{\breve{h}_{X}(t)}{\breve{H}_{X}(t)},~\forall ~t>0,~~ \forall ~ \alpha \in (0,1). $$

This is also equivalent to

$$ \alpha t \frac{\breve{h}_{X}(\alpha t)}{\breve{H}_{X}(\alpha t)}\geq \frac{t\breve{h}_{X}(t)}{\breve{H}_{X}(t)},~\forall ~t>0,~~ \forall ~ \alpha \in (0,1), $$

i.e., $\breve{L}_{X}(\alpha t)\geq \breve{L}_{X}(t)$, for all $t>0$ and for all $\alpha \in (0,1)$. This is also satisfied if and only if $X\in DRAI$. □

Consider t as an arbitrary point in time and let X be the lifetime of a unit. Since the event $\{X \leq \alpha t\}$ has a lower probability than $\{X\leq t\}$ for each $\alpha \in (0,1)$, it is trivial that $\{X \leq \alpha t\}$ causes more surprise than $\{X\leq t\}$ when it occurs. However, the result of Lemma 3.2 states, however, that the relative surprise of these events with respect to time as it passes through $[0,\infty )$ is available to the classes $DRAI$ and $IRAI$. For example, if X is in $IRAI$, then by Lemma 3.2, $\frac{-\ln (F_{X}(\alpha t))}{-\ln (F_{X}(t))}$ increases in $t>0$, for all $\alpha \in (0,1)$. This illustrates that for each $\alpha \in (0,1)$, the amount of surprise achieved by observing $X \leq \alpha t$ increases relative to the amount of surprise achieved by observing $X \leq t$ as t increases. This means that the reliability of the underlying lifetime unit increases with time, making the unit more reliable at older ages. Similarly, if $X \in DRAI$, the reliability of the unit decreases with time and therefore the unit is more reliable in the early stages. This can open a new perspective to identify long-lived units and short-lived units. In this context, items with a lifetime distribution belonging to the class $IRAI$ need more attention at the beginning of their use when they have a low age, while items with a lifetime distribution belonging to the class $DRAI$ need more attention when they have a high age.

4 Motivation for analysis and comparison of random quantiles

Quantile functions are equivalent alternatives to distribution functions in modeling and analysis of statistical data. However, statistical analysis is facilitated by the fascinating properties of the quantile functions of distributions rather than distribution functions. These properties do not hold for distribution function. To illustrate: The product of two positive quantile functions results in another quantile function as the distribution function that has the same property. For order statistics, the quantile function has explicit generic distribution forms. In addition, quantile functions can be used to generate data from any distribution, which provides a way to conduct a simulation analysis. The estimates obtained by the method of moments, maximum likelihood and least squares are significantly affected by outliers. For example, in the foregoing estimation methods, the sample mean as an estimate of the population mean when samples are drawn from a normal distribution; yet the population mean values vary dramatically when an outlier observation is included. Since the asymptotic variances are usually in the form of higher-order moments, which tend to be large in this case, the asymptotic efficiency of the sampling moments is quite poor for heavy-tailed distributions. A single long-term survivor can significantly change the mean lifetime in the reliability analysis, especially in the case of heavy-tailed models, which are typical of lifetime data. Quantile-based estimates tend to be more accurate and resilient to outliers in such situations. Choosing quantiles also has the advantage that one can provide useful estimates for life testing before all the test objects fail; one only has to wait until some of the test objects fail. So there is a strong case for using quantile functions as models for lifetime and supporting their analysis with derived functions. Many other facets of the quantile approach will become clearer below in the form of alternative methods, new possibilities and unique cases where there are no corresponding results when taking the distribution function approach (cf. Nair et al. [31]).

Some studies have been conducted in literature to the reliability analysis and statistical applications concerning lifetime distributions using quantile functions (see, for instance, Nair and Sankaran [30] and Bin et al. [25]). Some researchers considers stochastic orderings of lifetime distributions using quantile-based reliability measures (see, e.g., Nair et al. [32], Vineshkumar et al. [52], Arriaza et al. [1] and Sadeghi et al. [45]). Sunoj and Rasin [48] defined a quantile-based aging intensity function and studied several aging properties regarding it. They also obtained some stochastic comparison results of rvs based on the proposed quantile-based measure.

Let X be a nonnegative rv with cdf $F_{X}(x)$, which is continuous from the right. Then, the quantile function $Q_{X}(u)$ of X is defined as:

$$ Q_{X}(u)=F_{X}^{-1}(u)=\inf \{x:F_{X}(x)\geq u\},~~u\in [0,1]. $$

(4.1)

Note that for every $x\geq 0$, and also for every $u\in [0,1]$: $F_{X}(x)\geq u$, if and only if $Q_{X}(u)\leq x$. So if there is an x for which $F_{X}(x)=u$, then $F_{X}(Q_{X}(u))=u$ and $Q_{X}(u)$ is the smallest value of x that satisfies $F_{X}(x)=u$. If $F_{X}(x)$ is continuous and strictly increasing, $Q_{X}(u)$ is the only value x for which $F_{X}(x)=u$ is satisfied. By solving the equation $F_{X}(x)=u$, we can therefore find x based on u, which is the quantile function of X. In this paper, we shall consider random quantiles as transformations of rvs with support $[0,1]$. In this context, the following properties of quantile functions are required in this paper:

• (a) Let U be a uniform rv on $[0,1]$. Then, $Q_{X}(U)$ has cdf $F_{X}(x)$. This is because

  $$ P(Q_{X}(U)\leq x)=P(U\leq F_{X}(x))=F_{X}(x). $$

  The above property enables us to transform a given data set from the uniform distribution on $[0,1]$ into a data set from distribution $F_{X}$ on $(-\infty ,+\infty )$.

• (b) Let Z be an rv on $[0,1]$ with cdf $F_{Z}(z)$. Then, $Q_{X}(Z)$ has cdf $F_{Z}(F_{X}(x))$, since

  $$ P(Q_{X}(Z)\leq x)=P(Z\leq F_{X}(x))=F_{Z}(F_{X}(x)). $$

  (4.2)

  This is a property that is more general than the property given in part (a). This means that if one generates an observation on Z from distribution $F_{Z}$, which is not necessarily the uniform distribution on $[0,1]$, then $Q_{X}(Z)$ will produce an observation from an altered version of $F_{X}$ rather than $F_{X}$, and that is $F_{Z}oF_{X}$.

Błażej [9] considered weighted distributions as a family of distributions adopted from the random quantile model (4.2) whose distribution follows a composition of two distribution functions. To be more specific, let $X_{w}$ be an rv with cdf $F_{X_{w}}$, called the weighted distribution associated with lifetime distribution $F_{X}$ with weight function $w:R_{+}\rightarrow R_{+}$ for which $0< E(w(Q_{X}(U)))<+\infty$, where U is a uniform rv on $[0,1]$ given by

$$ F_{X_{w}}(t)=\int _{0}^{t}\frac{w(x)}{E(w(X))}dF_{X}(x)=\int _{0}^{F_{X}(t)} \frac{w(Q_{X}(u))}{E(w(Q_{X}(U)))} du. $$

Then, it can be seen, in spirit of (4.2), that $F_{X_{w}}(t)=F_{Z}(F_{X}(t))$, where

In this case, $f_{Z}(z)=\frac{w(Q_{X}(z))}{E(w(Q_{X}(U)))}$ is the associated density of the cdf $F_{Z}$. Note that the distribution of Z in this case depends on $F_{X}$ in its general form. Bartoszewicz and Skolimowska [3] and Bartoszewicz [2] studied several stochastic orderings concerning $Q_{Y}(Z)$ and $Q_{X}(U)$ according to likelihood ratio order, hazard (reversed) hazard rate order, usual stochastic order and dispersive order. They also derived some sufficient conditions for the aging properties of $Q_{X}(Z)$.

In the context of the current investigation, in the sequel we shall consider the random quantile model (4.2) in situations, where $F_{Z}$ is induced by weight functions of the form $w(x)=v(F_{X}(x))$, where $v:[0,1]\rightarrow R_{+}$ is independent of $F_{X}$. This weight function is a special case of the weight function considered in Theorem 1 of Bartoszewicz [2]. The weighted distributions produced in this approach encompass typical models in reliability theory and they are known as semiparametric models (cf. Marshall and Olkin [28]). In this setting, $F_{Z}$ is characterized as follows (see, e.g., Kayid and Al-Shehri [19]):

The random quantile model (4.2) produces many models in reliability and survival analysis. For example, the distribution of order statistics, record values, k record values and, in a more general setting, the distribution of generalized order statistics and the distribution of the lifetime of coherent systems with possibly dependent but identical components lifetime are at the disposal of (4.2), where $F_{Z}$ is characterized. For example, Błażej [9] showed in Example 1 of his work that if Z follows beta distribution $\boldsymbol{B}(i,n-i+1)$, then $Q_{X}(Z)$ is the distribution of the ith order statistic, $X_{i:n}$, of a random sample from $F_{X}$. Suppose that $X_{1},\ldots ,X_{n}$ denote the lifetime of n components, which are independent. Then, the order statistic $X_{i:n}$, i.e., the ith smallest among $X_{1},\ldots ,X_{n}$, is the lifetime of an $(n-i+1)$-out-of-n system in reliability theory.

In this paper, we finally apply our results on a series system with heterogeneous components having independent lifetimes $X_{1},\ldots ,X_{n}$ following the $PHR$ model with sfs $\bar{F}_{X}^{\lambda _{1}},\ldots ,\bar{F}_{X}^{\lambda _{n}}$, respectively. The lifetime of the series system constructed by these components is then $Q_{X}(Z)\equiv X_{1:n}$, i.e. the minimum order statistic of $X_{1},\ldots ,X_{n}$, which follows the cdf

$$ F_{Q_{X}(Z)}(t)=F_{Z}(F_{X}(t))=1-(1-F_{X}(t))^{n\bar{\lambda}}, $$

(4.3)

where $F_{Z}(z)=1-(1-z)^{n\bar{\lambda}}$, where λ̄ is the arithmetic mean of $\lambda _{1},\ldots , \lambda _{n}$. Note that Z follows a beta distribution $\boldsymbol{B}(1,n\bar{\lambda})$. Preservation properties of stochastic orderings as well as aging classes of life distributions under the formation of coherent systems are important topics in literature (see, Nanda et al. [34], Belzunce et al. [7], Pellerey and Petakos [39], Li and Yam [27], Navarro et al. [37] and Izadkhah et al. [17] among others). The aim of the investigation undertaken in this work is to derive conditions under, which

$$Q_X(U){\le }_{RAI}{Q}_Y(U)⟹Q_X(Z){\le }_{RAI}{Q}_Y(Z),$$

(4.4)

and also to find sufficient conditions to get

$$Q_X(U)\in IRAI(resp.DRAI)⟹Q_X(Z)\in IRAI(resp.DRAI).$$

(4.5)

The implications introduced in (4.4) and (4.5) are useful in many situations to obtain the preservation property of $RAI$ order for different distributions and the preservation of monotone $RAI$ classes arising from different reliability operations. In particular, in the context of coherent systems, the preservation property of any new specific stochastic order as well as any novel special reliability class of component lifetimes provides new insights into the lifetime of coherent systems built from these components. For example, if such a preservation property is validated, one can expect that of two systems, the one with components that are more reliable in one sense is also the more reliable system in that sense. In light of the discussion in Sect. 3, an advantage of preserving the $RAI$ order may look like what we describe here. Consider a series system ($S_{A}$) with independent components of type A and also consider another series system ($S_{B}$) with independent components of type B. If then the components of type A age faster than the components of type B, then $S_{A}$ ages faster than $S_{B}$ in the sense of the $RAI$ order. For the reliability classes of $IRAI$ and $DRAI$, a preservation property of these behaviors from the components of a series system into the series system shows that a certain behavior of the components is inherited by the system. Due to the discussions provided in Sect. 3, the properties of being more reliable at old age (as induced in the $IRAI$ class) and being more reliable at early age (as induced in the $DRAI$ class) can be transferred from the components of a series system to the entire system.

5 Preservation of $RAI$ ordering under random quantiles

In this section, we provide conditions under which the $RAI$ order between lifetime random variables is preserved under their random quantiles. Some examples are also given to illustrate and specifically to examine the results obtained. The following theorem presents the main result of this section.

Theorem 5.1

Let Z be a nonnegative rv on $[0,1]$ with cdf $F_{Z}$ and let U be a uniform rv on $[0,1]$. Suppose that the assertions (i) and (ii) below hold:

• (i) $-\log (Z) \in DFR/A$ (resp. $-\log (Z) \in IFR/A$).

• (ii) $X \leq _{st}Y$ (resp. $X \geq _{st}Y$).

Then,

$$Q_X(U){\le }_{RAI}{Q}_Y(U)⟹Q_X(Z){\le }_{RAI}{Q}_Y(Z).$$

Proof

We proof the non-parenthetical part of the theorem. The other part’s proof is quite similar. Note that, for all $t\geq 0$,

(5.1)

Since $X\leq _{RAI}Y$, one has

(5.2)

Now, we prove that $Q_{X}(Z)\leq _{RAI} Q_{Y}(Z)$. To this end, by using Lemma 3.1, it suffices to show that

Using Equations (5.1), we observe for all $t>0$ that:

$$\begin{aligned} \frac{d}{dt}\left ( \frac{-\log (F_{Q_{X}(Z)}(t))}{-\log (F_{Q_{Y}(Z)}(t))}\right )&= \frac{d}{dt}\frac{-\log (F_{Z}(F_{X}(t)))}{-\log (F_{Z}(F_{Y}(t)))} \\ &\overset{sgn}{=}\frac{f_{X}(t)f_{Z}(F_{X}(t))}{F_{Z}(F_{X}(t))}.\log (F_{Z}(F_{Y}(t))) \\ &\quad{} - \frac{f_{Y}(t)f_{Z}(F_{Y}(t))}{F_{Z}(F_{Y}(t))}.\log (F_{Z}(F_{X}(t))) \\ &=\frac{f_{X}(t)}{F_{X}(t)} \frac{F_{X}(t)f_{Z}(F_{X}(t))}{F_{Z}(F_{X}(t))}.\log (F_{Z}(F_{Y}(t))) \\ &\quad{} -\frac{f_{Y}(t)}{F_{Y}(t)} \frac{F_{Y}(t)f_{Z}(F_{Y}(t))}{F_{Z}(F_{Y}(t))}.\log (F_{Z}(F_{X}(t))) \\ &=\breve{h}_{X}(t)\frac{F_{X}(t)f_{Z}(F_{X}(t))}{F_{Z}(F_{X}(t))}. \log (F_{Z}(F_{Y}(t))) \\ &\quad{} -\breve{h}_{Y}(t) \frac{F_{Y}(t)f_{Z}(F_{Y}(t))}{F_{Z}(F_{Y}(t))}.\log (F_{Z}(F_{X}(t))), \end{aligned}$$

which is non-positive if

(5.3)

From (5.2), the inequality in (5.3) is satisfied if for all $t>0$:

$$ \frac{-\log (F_{X}(t))}{-\log (F_{Y}(t))}\geq \frac{F_{Y}(t)}{F_{X}(t)}. \frac{\breve{h}_{Z}(F_{Y}(t))}{\breve{h}_{Z}(F_{X}(t))}. \frac{-\log (F_{Z}(F_{X}(t)))}{-\log (F_{Z}(F_{Y}(t)))}, $$

or equivalently if

Since by assumption $X \leq _{st}Y$, thus from definition $F_{X}(t) \geq F_{Y}(t)$, for all $t>0$. Therefore, it suffices to prove that $\frac{-u \breve{h}_{Z}(u)\log (u)}{-\log (F_{Z}(u))}$ is non-decreasing in $u\in [0,1]$. Suppose that U has uniform distribution on $[0,1]$, then one can see that $\frac{\breve{L}_{Z}(u)}{\breve{L}_{U}(u)}= \frac{-u \breve{h}_{Z}(u)\log (u)}{-\log (F_{Z}(u))}$, for every $u\in [0,1]$. Hence, it is enough to demonstrate that $\frac{\breve{L}_{Z}(e^{-y})}{\breve{L}_{U}(e^{-y})}$ is non-increasing in $y\geq 0$. Note that $-\log (F_{Z}(u))=\int _{u}^{1} \breve{h}_{Z}(z)dz$. Thus,

$$\begin{aligned} -\log (F_{Z}(e^{-y}))&=\int _{e^{-y}}^{1} \breve{h}_{Z}(z)dz \\ &=\int _{0}^{y} e^{-w}\breve{h}_{Z}(w)dw \\ &=\int _{0}^{y} h_{-\log (Z)}(w)dw. \end{aligned}$$

Now, we can get

Since from assumption $-\log (Z)\in DFR/A$, thus $L_{-\log (Z)}(y)$ is non-increasing in $y\geq 0$, which completes the proof of the theorem. □

Next, we present an example where Theorem 5.1 is applied to show that the $RAI$ order of component lifetimes of two parallel system composed of components having independent, and identically distributed (i.i.d.) component lifetimes is transmitted to the lifetime of the entire parallel systems, provided that the component lifetimes of one parallel system dominates the component lifetimes of the other parallel system in the sense of the usual stochastic order. In the setting of Theorem 5.1, it is known that if U is a uniform random variable on [0,1], then using probability integral transform, $Q_{X}(U)$ is identical in distribution with X. Therefore, $X\leq _{RAI}Y$ is equivalent to $Q_{X}(U)\leq _{RAI} Q_{Y}(U)$.

Example 5.2

Suppose that $X_{1},\ldots ,X_{n}$ are n i.i.d. nonnegative rvs with a common cdf $F_{X}$ and also suppose that $Y_{1},\ldots ,Y_{n}$ are n i.i.d. nonnegative rvs with a common cdf $F_{Y}$. In the spirit of Eq. (4.2), one can easily see that if Z has a beta distribution, $Z\sim \boldsymbol{B}(n,1)$, then $Q_{X}(Z)$ and $Q_{Y}(Z)$ are identical in distributions with $X_{n:n}$ and $Y_{n:n}$, respectively. Note that $X_{n:n}$ and $Y_{n:n}$ represent the lifetimes of two parallel systems composed of components with lifetimes $X_{1},\ldots ,X_{n}$ and $Y_{1},\ldots ,Y_{n}$, respectively. It can be plainly seen that $T=-\log (Z)$ follows exponential distribution with mean $\frac{1}{n}$. Therefore, $L_{T}(t)=1$, for all $t\geq 0$, showing that $-\log (Z)\in DFR/A$ and also $-\log (Z)\in IFR/A$. By using Theorem 5.1, if $X \leq _{RAI} Y$ and either $X\leq _{st}Y$ or $X\geq _{st} Y$, then $X_{n:n}\leq _{RAI} Y_{n:n}$.

The following technical lemma will be used in the sequel.

Lemma 5.3

For every $a< b \in R_{+}$, $\frac{\log (1+\frac{1}{bt})}{\log (1+\frac{1}{at})}$ is non-increasing in $t>0$.

Proof

For all $t>0$, we have:

$$\begin{aligned} \frac{\log (1+\frac{1}{bt})}{\log (1+\frac{1}{at})}&= \frac{\log (1+bt)-\log (bt)}{\log (1+at)-\log (at)} \\ &= \frac{\int _{t}^{+\infty} \frac{1}{x(1+bx)}~dx}{\int _{t}^{+\infty} \frac{1}{x(1+ax)}~dx} \\ &=\frac{\psi (2,t)}{\psi (1,t)}, \end{aligned}$$

where

$$ \psi (j,t):=\int _{0}^{\infty} \phi (j,x).I[x > t]~dx,~~j=1,2;~ t>0, $$

(5.4)

in which $I[x>t]$ is the indicator function of $[x>t]$, and further, $\phi (j,x)=\frac{1}{x(1+ax)}$ for $j=1$ and $\phi (j,x)=\frac{1}{x(1+bx)}$ for $j=2$. It is plain to verify that $\phi (j,x)$ is $RR_{2}$ in $(j,x)\in \{1,2\} \times R_{+}$. In addition, it is easy to see that $I[x>t]$ is $TP_{2}$ in $(x,t) \in R_{+} \times R_{+}$. By using the general composition theorem of Karlin [18] in Eq. (5.4), we deduce that $\psi (j,t)$ is $RR_{2}$ in $(j,t)\in \{1,2\} \times R_{+}$. Equivalently, $\frac{\psi (2,t)}{\psi (1,t)}$ is non-increasing in $t>0$ and this completes the proof. □

The next example illustrates that the condition $-\log (Z)\in IFR/A$ in Theorem 5.1 is not a necessary condition. Recall that a nonnegative rv has Lomax distribution with shape parameter $\alpha >0$ and the scale parameter $\lambda >0$, whenever X has sf $\bar{F}_{X}(x)=\left (1+\frac{x}{\lambda}\right )^{-\alpha}$, for $x \geq 0$, and we shall write $X\sim Lo(\alpha , \lambda )$.

Example 5.4

Consider $X\sim Lo(1,b^{-1})$ and $Y \sim Lo(1,a^{-1})$, where $0< a< b$. We can see that $F_{X}(t)=\frac{bt}{1+bt}$ and $F_{Y}(t)=\frac{at}{1+at}$, for all $t\geq 0$. Then, it is trivial to see that $F_{X}(t) \leq F_{Y}(t)$, for all $t\geq 0$, i.e., $X\geq _{st}Y$. On the other hand $\frac{-\log (F_{X}(t))}{-\log (F_{Y}(t))}= \frac{\log (1+\frac{1}{bt})}{\log (1+\frac{1}{at})}$, which is non-increasing from Lemma 5.3. Hence, by using Lemma 3.1, we deduce that $X\leq _{RAI} Y$, and equivalently, $Q_{X}(U)\leq _{RAI}Q_{Y}(U)$, where $U\sim U(0,1)$. Now, contemplate the rv Z as an rv with cdf $F_{Z}(z)=z.\exp (1-z)$, for $z\in [0,1]$. Set $T=-\log (Z)$. By some routine calculation, one gets $L_{T}(t)=\frac{t(1-e^{-t})}{t+e^{-t}-1}$, for all $t>0$, which is a non-increasing function. To prove this, let us see that

Thus, $L_{T}(t)$ is non-increasing in $t>0$, if and only if,

which holds true if and only if $1-e^{-t} \geq te^{-\frac{t}{2}}$, for all $t\geq 0$. This inequality is also satisfied since $e^{-\frac{t}{2}} \geq 1-\frac{t}{2}$, for all $t>0$. One can also see Fig. 1. This proves that $-\log (Z) \in DFR/A$. It is then seen that, for all $t>0$

We can see that $Q_{X}(Z)\leq _{RAI} Q_{Y}(Z)$ as the plot depicted in Fig. 2 shows that $L_{Q_{X}(Z)}(t) \geq L_{Q_{Y}(Z)}(t)$, for all $t\geq 0$, whenever $a=2$ and $b=5$.

Figure 1

Plot of the AI function of $T=-\log (Z)$ in Example 5.4

Figure 2

Plot of the $RAI$ functions of $Q_{X}(Z)$ (dashed line) and $Q_{Y}(Z)$ (solid line) in Example 5.4

6 Bounds for the $RAI$ function of random quantiles

In this section, we obtain a lower bound and an upper bound for the $RAI$ function of random quantile of X under the condition that the distribution of X possesses some reliability properties. One of the advantages of stochastic orders is to find inequalities and bounds for statistical measures. In the context of reliability, stochastic orders among lifetime distributions could be used to establish bounds for reliability measures and aging characteristics. It is a common approach in the theory of reliability analysis to find a bound or to set up some inequalities for a certain reliability measures of a probability distribution based on existing partial information from this probability distribution (see, e.g., Proposition B.7 and Proposition B.8 in Marshall and Olkin [28]). In this regard, we make use of Theorem 5.1 to derive some bounds for the $RAI$ function of transformation of a random lifetime, i.e., the random quantile of it.

We need to recall the Fréchet distribution. The rv T is said to have Fréchet distribution with the shape parameter $\alpha >0$, the scale parameter $s>0$ and the location parameter $m \in (-\infty .+\infty )$ (written as $T\sim Fr(\alpha ,s,m)$) when it has cdf

$$ F_{T}(t)=\exp (-\left (\frac{t-m}{s}\right )^{-\alpha}),~~t>m. $$

The Fréchet distribution is also called inverse Weibull distribution. Next, we obtain a lower bound for $\breve{L}_{Q_{X}(Z)}(t)$.

Theorem 6.1

Let X be a nonnegative rv such that $F^{t}_{X}(t)$ is non-decreasing in $t>0$. Further, let Z be an rv on $[0,1]$ with cdf $F_{Z}$. Suppose at least one of the following assertions hold:

• (i) $-\log (Z)$ is $DFR/A$ and $s=\lim _{t\rightarrow 0^{+}} (-t\log (F_{X}(t)))<\infty$.

• (ii) $-\log (Z)$ is $IFR/A$ and $s=\lim _{t\rightarrow \infty} (-t\log (F_{X}(t)))>0$.

  Then, $\breve{L}_{Q_{X}(Z)}(t) \geq \breve{L}_{-\frac{s}{\log (Z)}}(t)$, for all $t>0$.

Proof

We prove the theorem under assertion (i). The proof for assertion (ii) is similar. Suppose $Y\sim Fr(1,s,0)$, where $s=\lim _{t\rightarrow 0^{+}} (-t\log (F_{X}(t)))$. Then, it is seen that for all $t>0$, $-\log (F_{Y}(t))=\frac{s}{t}$, and that $\breve{h}_{Y}(t)=\frac{s}{t^{2}}$, which proves that $t\breve{h}_{Y}(t)=-\log (F_{Y}(t))$ for all $t>0$. Hence, it follows that $\breve{L}_{Y}(t)=1$ for all $t>0$. We now show that if $F^{t}_{X}(t)$ is non-decreasing in $t>0$, then $X \leq _{RAI} Y$ and $X\leq _{st}Y$. It is plain to see that if $F^{t}_{X}(t)$ is non-decreasing in $t>0$, then $-t\log (F_{X}(t))$ is non-increasing in $t>0$. Thus $\frac{d}{dt}(-t\log (F_{X}(t)))=-\log (F_{X}(t))-t\breve{h}_{X}(t)$, which is non-positive for all $t>0$ if and only if $\breve{L}_{X}(t)\geq 1$, for all $t\geq 0$. Since $\breve{L}_{Y}(t)=1$, for all $t>0$, thus we proved that $\breve{L}_{X}(t)\geq \breve{L}_{Y}(t)$, for all $t\geq 0$, i.e., $X\leq _{RAI} Y$. On the other hand, $X\leq _{st} Y$ if and only if $F_{X}(t) \geq \exp (-\frac{s}{t})$, for all $t>0$, or equivalently, $(-t\log (F_{X}(t))) \leq s$, for all $t>0$, and this means that $(-t\log (F_{X}(t))) \leq \lim _{t\rightarrow 0^{+}} (-t\log (F_{X}(t)))$ for all $t>0$. This is also satisfied because $-t\log (F_{X}(t))$ is a non-increasing in $t>0$. Therefore, using the non-parenthetical part of Theorem 5.1, if $-\log (Z)$ is $DFR/A$, as given in assertion (i), we then deduce that $\breve{L}_{Q_{X}(Z)}(t) \geq \breve{L}_{Q_{Y}(Z)}(t)$, for all $t>0$. Now, since $Y\sim Fr(1,s,0)$, thus $F_{Q_{Y}(Z)}(t)=F_{Z}(\exp (-\frac{s}{t}))$, $t>0$. In addition,

$$ \breve{h}_{Q_{Y}(Z)}(t)=\frac{d}{dt}\log (F_{Q_{Y}(Z)}(t))= \frac{s e^{-\frac{s}{t}}f_{Z}(e^{-\frac{s}{t}})}{t^{2} F_{Z}(e^{-\frac{s}{t}})}. $$

Therefore,

The proof of theorem is complete. □

Note that the assertion (ii) of Theorem 6.1 is obtained by using the parenthetical part of Theorem 5.1. In the proof of Theorem 6.1, if one considers X, in place of Y, as an rv with Fréchet distribution so that $X\sim Fr(1,s,0)$ then an upper bound for the $\breve{h}_{Q_{Y}(Z)}(t)$ is obtained. The following theorem specifies the result.

Theorem 6.2

Let Y be a nonnegative rv such that $F^{t}_{Y}(t)$ is non-increasing in $t>0$. Let at least one of the following conditions hold:

• (i) $-\log (Z)$ is $DFR/A$ and $s^{\star}=\lim _{t\rightarrow 0^{+}} (-t\log (F_{Y}(t)))>0$.

• (ii) $-\log (Z)$ is $IFR/A$ and $s^{\ast}=\lim _{t\rightarrow \infty} (-t\log (F_{Y}(t)))<\infty$.

  Then, $\breve{L}_{Q_{Y}(Z)}(t) \leq \breve{L}_{-\frac{s^{\star}}{\log (Z)}}(t)$ for all $t>0$.

It is remarkable that if $Q_{Fr(1,s,0)}(z)$ and $Q_{Fr(1,s^{\star},0)}(z)$ denote the quantile functions of of two rvs with Fréchet distributions with parameters $(1,s,0)$ and $(1,s^{\star},0)$, then one can see that $Q_{Fr(1,s,0)}(z)=-\frac{s}{\log (Z)}$ and also $Q_{Fr(1,s^{\star},0)}(z)=-\frac{s^{\star}}{\log (Z)}$. On that account, in Theorem 6.1, $\breve{L}_{-\frac{s}{\log (Z)}}(t)=\breve{L}_{Q_{Fr(1,s,0)}(Z)}(t)$ and in Theorem 6.2, $\breve{L}_{-\frac{s^{\star}}{\log (Z)}}(t)=\breve{L}_{Q_{Fr(1,s^{ \star},0)}(Z)}(t)$.

The following example illustrates a situation for the applicability of Theorem 6.2(i).

Example 6.3

Suppose that $Y\sim Lo(1,1)$ which has cdf $F_{Y}(t)=\frac{t}{t+1}$, $t\geq 0$. We show that $F^{t}_{Y}(t)$ is non-increasing in $t>0$. To this end, we prove that $-t\log (F_{Y}(t))$ is non-decreasing in $t>0$. Note that $\frac{d}{dt}(-t\log (F_{Y}(t)))=-\log \left ( \frac{t}{t+1}\right )-1+\frac{t}{t+1}$, which is nonnegative for all $t\geq 0$, since for all $u\in (0,1)$, we have $-\log (u) -1+u\geq 0$. Now, let us see that

$$ s^{\ast}=\lim _{t\rightarrow \infty} (-t\log (F_{Y}(t)))=\lim _{t \rightarrow \infty} -t\log \left (\frac{t}{t+1}\right )=1. $$

We consider Z as an rv with cdf $F_{Z}(z)=z\exp (-(-\log (z))^{2})$. We can see that $L_{-\log (Z)}(t)=\frac{1+2t}{1+t}$, for all $t\geq 0$. Hence, $-\log (Z)\in IFR/A$. Therefore, the sufficient conditions given in Theorem 6.2(i) are fulfilled. Thus, according to this theorem, $\breve{L}_{Q_{Y}(Z)}(t) \leq \breve{L}_{-\frac{s^{\star}}{\log (Z)}}(t)$ for all $t>0$. We can get

In Fig. 3, we plot the graph of $\breve{L}_{Q_{Y}(Z)}(t)$ and $\breve{L}_{-\frac{s^{\star}}{\log (Z)}}(t)$ in terms of t by which it is verified graphically that $\breve{L}_{Q_{Y}(Z)}(t) \leq \breve{L}_{-\frac{s^{\star}}{\log (Z)}}(t)$ for $0< t<20$.

Figure 3

Plot of the $RAI$ function of $Q_{Y}(Z)$ (solid line) and the $RAI$ function of $\frac{s^{\star}}{-\log (Z)}$ (dashed line) in Example 6.3

Remark 6.4

In the proof of Theorem 6.1, as well as in what could be developed for proving Theorem 6.2, the monotonicity of $F^{t}_{W}(t)$ in $t>0$, for a nonnegative rv W is equivalent to $\breve{L}_{W}(t)$ being non-strictly less or greater than 1, for all $t>0$. However, a sufficient condition based on the rhr function of W may also be appreciated. Let $\breve{h}_{W}$ be the rhr function of W. It is known that $F^{t}_{W}(t)$ is non-decreasing (resp. non-increasing) in $t>0$ if and only if $-t\log (F_{W}(t))$ is non-increasing (resp. non-decreasing). We can see that

(6.1)

where

$$ \psi ^{\star}(j,t):=\int _{0}^{\infty} \phi ^{\star}(j,w)I[w>t]~dw, $$

(6.2)

in which $\phi ^{\star}(j,w)=w^{-2}$ for $j=1$, and $\phi ^{\star}(j,w)=\breve{h}_{W}(w)$ for $j=2$. It is shown that if $w^{2}\breve{h}_{W}(w)$ is non-increasing (resp. non-decreasing), then $\phi ^{\star}(j,w)$ is $RR_{2}$ (resp. $TP_{2}$) in $(j,w)\in \{1,2\} \times R_{+}$. On the other hand, it is obvious that $I[w>t]$ is $TP_{2}$ in $(w,t) \in R_{+} \times R_{+}$. Then, by applying the general composition theorem of Karlin [18] in Eq. (6.2), $\psi ^{\star}(j,t)$ is $RR_{2}$ (resp. $TP_{2}$) in $(j,t)\in \{1,2\} \times R_{+}$. This proves that $\frac{\psi ^{\star}(2,t)}{\psi ^{\star}(1,t)}$ is non-increasing (resp. non-decreasing) in $t>0$. In view of Eq. (6.1), this is equivalent to saying that $-t\log (F_{W}(t))$ is non-increasing (resp. non-decreasing) in $t>0$. Therefore, if $w^{2}\breve{h}_{W}(w)$ is non-increasing (resp. non-decreasing) in $w>0$, then $F^{t}_{W}(t)$ is non-decreasing (resp. non-increasing) in $t>0$.

It should be noted at this point that the bounds derived in this section are not distribution-free, as they depend on the underlying life distribution due to some boundary value restrictions. Therefore, the results in this section may be useful to dominate certain life distributions by their $RAI$ function of random quantiles.

7 Preservation of some monotonic reliability classes

In this section, we study the preservation properties of two reliability classes constructed based on the $RAI$ function under quantile transformation. The following definition is basic to our development.

Definition 7.1

The rv X with $RAI$ function $\breve{L}_{X}$ is said to have increasing (resp. decreasing) reversed aging intensity function, denoted by $X\in IRAI$ (resp. $X \in DRAI$), whenever $\breve{L}_{X}(t)$ is non-decreasing (resp. non-increasing) in $t>0$.

The following theorem is the main result of this section.

Theorem 7.2

Let X be a nonnegative rv with quantile function $Q_{X}$ and let Z be a nonnegative rv on $[0,1]$. Let $U\sim U(0,1)$.

• (i) If $-\log (Z)\in DFR/A$. Then, $Q_{X}(U) \in IRAI$ implies that $Q_{X}(Z)\in IRAI$.

• (ii) If $-\log (Z)\in IFR/A$. Then, $Q_{X}(U) \in DRAI$ implies that $Q_{X}(Z)\in DRAI$.

Proof

We only prove part (i). The proof of part (ii) analogously follows. According to Definition 7.1, we need to show that $\breve{L}_{Q_{X}(Z)}(t)$ is non-decreasing in $t>0$. From (5.1), we have

Now, in spirit of the proof of Theorem 5.1, we can get

$$\begin{aligned} \breve{L}_{Q_{X}(Z)}(t)&= \frac{t\breve{h}_{Q_{X}(Z)}(t)}{-\log (F_{Z}(F_{X}(t)))} \\ &=\frac{t\breve{h}_{Z}(F_{X}(t))F_{X}(t)}{-\log (F_{Z}(F_{X}(t)))}. \breve{h}_{X}(t) \\ &= \frac{-F_{X}(t)\breve{h}_{Z}(F_{X}(t))\log (F_{X}(t))}{-\log (F_{Z}(F_{X}(t)))}. \frac{t\breve{h}_{X}(t)}{-\log (F_{X}(t))} \\ &=\frac{\breve{L}_{Z}(F_{X}(t))}{\breve{L}_{U} (F_{X}(t))}.\breve{L}_{Q_{X}(U)}(t). \end{aligned}$$

(7.1)

From assumption, since $Q_{X}(U)\in IRAI$, thus $\breve{L}_{Q_{X}(U)}(t)$ is non-decreasing in $t>0$. On the other hand, since from assumption $-\log (Z)\in DFR/A$, thus from the proof of Theorem 5.1 it is deduce that $\frac{\breve{L}_{Z}(u)}{\breve{L}_{U} (u)}$ is non-decreasing in $u\in [0,1]$. Hence, $\frac{\breve{L}_{Z}(F_{X}(t))}{\breve{L}_{U} (F_{X}(t))}$ is non-decreasing in $t>0$. Therefore, in view of (7.1), $\breve{L}_{Q_{X}(Z)}(t)$ is the product of two nonnegative and non-decreasing functions of $t>0$, which is non-decreasing in $t>0$. The proof is completed. □

The following example illustrates an application of Theorem 7.2.

Example 7.3

Suppose that $X\sim Lo(1,1)$ with cdf $F_{X}(t)=\frac{t}{t+1}$, $t\geq 0$. One can get $\breve{L}_{Q_{X}(U)}(t)= \frac{1}{(t+1)\log \left (1+\frac{1}{t}\right )}$, for $t>0$. Since for all $x>0$, $\log (x+1)\leq x$, thus $\frac{d}{dt}(t+1)\log \left (1+\frac{1}{t}\right )=\log \left (1+ \frac{1}{t}\right )-\frac{1}{t}\leq 0$, for all $t>0$. That is $\breve{L}_{Q_{X}(U)}(t)$ is non-decreasing in $t>0$. Therefore, $X\in IRAI$. Let us take Z as an rv with cdf $F_{Z}(z)=z\exp (1-z)$. Then, from Example 5.4, we have $-\log (Z) \in DFR/A$. Therefore, by Theorem 7.2, it follows that $Q_{X}(Z)\in IRAI$. Since $\breve{h}_{Z}(z)=\frac{1-z}{z}$ and $\breve{h}_{X}(t)=\frac{1}{t(t+1)}$, thus from the proof of Theorem 7.2, we can get

$$ \breve{L}_{Q_{X}(Z)}(t)= \frac{t\breve{h}_{Z}(\frac{t}{t+1})\left (\frac{t}{t+1}\right )}{-\log (F_{Z}\left (\frac{t}{t+1}\right ))}. \breve{h}_{X}(t) = \frac{1}{(t+1)^{2}\log \left (1+\frac{1}{t}\right )-(t+1)}. $$

In Fig. 4, the graph of $\breve{L}_{Q_{X}(Z)}(t)$ in terms of t is depicted for $0< t<30$, which acknowledges that $\breve{L}_{Q_{X}(Z)}(t)$ increases in t.

Figure 4

Plot of the $RAI$ function of $Q_{X}(Z)$ in Example 7.3

8 Reliability analysis of series systems with heterogeneous component lifetimes

In this section, we utilize the main results achieved in this paper to systems with series structures. We assume that the system has heterogenous component lifetimes following the $PHR$ model with different parameters. To be more specific, let $X_{1},X_{2},\ldots ,X_{n}$ represent the random lifetimes of n components of a series system, which are independent such that $X_{i}$ follows sf $\bar{F}^{\lambda _{i}}_{X},i=1,2,\ldots n$ where $\bar{F}_{X}$ is the sf of a nonnegative rv X. Then, as given in (4.3), $X_{1:n}$ has cdf

$$ F_{X_{1:n}}(t)=F_{Z}(F_{X}(t))=1-(1-F_{X}(t))^{n\bar{\lambda}}, $$

(8.1)

where Z is an rv with beta distribution with parameters 1 and nλ̄. Thus, Z has cdf

$$ F_{Z}(z)=1-(1-z)^{n\bar{\lambda}},~~z\in [0,1]. $$

Suppose that X has pdf $f_{X}$. In view of Eq. (8.1), the rhr function of $X_{1:n}$ is then obtained as:

$$ \breve{h}_{X_{1:n}}(t)=\frac{d}{dt}\log (F_{X_{1:n})}(t))= \frac{n\bar{\lambda}f_{X}(t)\bar{F}^{n\bar{\lambda}-1}_{X}(t)}{1-\bar{F}^{n\bar{\lambda}}_{X}(t)}. $$

Therefore, the $RAI$ function of $X_{1:n}$ is given by

$$ \breve{L}_{X_{1:n}}(t)= \frac{n\bar{\lambda}tf_{X}(t)\bar{F}^{n\bar{\lambda}-1}_{X}(t)}{-(1-\bar{F}^{n\bar{\lambda}}_{X}(t))\log (1-\bar{F}^{n\bar{\lambda}}_{X}(t))}. $$

(8.2)

The following technical lemma is useful in the sequel. The proof can be found in Kayid et al. [20].

Lemma 8.1

Let $\psi _{\theta}(u)=\frac{\theta u(1-u)^{\theta -1}}{1-(1-u)^{\theta}}$, for $u\in (0,1)$ and $\theta >0$. Then:

• (i) If $0<\theta \leq 1$, then $\frac{\psi _{\theta}(u^{\alpha})}{\psi _{\theta}(u)}$ is non-decreasing in $u\in (0,1)$, for all $\alpha \in [0,1]$.

• (ii) If $\theta \geq 1$, then $\frac{\psi _{\theta}(u^{\alpha})}{\psi _{\theta}(u)}$ is non-increasing in $u\in (0,1)$, for all $\alpha \in [0,1]$.

Let us define $R(u,\theta , \alpha ):= \frac{\psi _{\theta}(u^{\alpha})}{\psi _{\theta}(u)}$ for $u\in (0,1)$, $\theta >0$ and $\alpha \in [0,1]$. In Fig. 5, we plot the graph of $R(u,\theta , \alpha )$, for $u\in (0,1)$ for a value of θ in $(0,1)$ and also five values of $\alpha \in [0,1]$. In Fig. 6, we plot the graph of $R(u,\theta , \alpha )$, for $u\in (0,1)$ for a value of θ in $(1,+\infty )$ and five values of α in $[0,1]$. The graphs exhibit a monotone behavior, which confirms the result of Lemma 8.1. Let us contemplate a series system with heterogenous independent component lifetimes $X_{1},\ldots ,X_{n}$ $PHR$ model with sfs $\bar{F}_{X}^{\lambda _{1}},\ldots ,\bar{F}_{X}^{\lambda _{n}}$, respectively. The lifetime of the series system constructed by these components is then $X_{1:n}$, which in view of (4.3) has cdf $F_{Z}(F_{X}(t))$, where $F_{Z}(z)=1-(1-z)^{n\bar{\lambda}}$ is the cdf of Z, in which $\bar{\lambda}=\frac{\lambda _{1}+\cdots +\lambda _{n}}{n}$. In Theorem 5.1, Theorem 6.1, Theorem 6.2 and Theorem 7.2, one commonly used sufficient condition is that $-\log (Z) \in DFR/A~(IFR/A)$. Here, we investigate that this condition holds under some mild constraints on $(\lambda _{1},\ldots ,\lambda _{n})$. If we prove that $\frac{h_{-\log (Z)}(\alpha t)}{h_{-\log (Z)}(t)}$ is decreasing [resp. increasing] in $t\geq 0$, for every $\alpha \in (0,1)$, then, we conclude that $-\log (Z)\in IFR/A$ [resp. $-\log (Z)\in DFR/A$]. To this end, we derive the hr function of $-\log (Z)$. We have

$$ \bar{F}_{-\log (Z)}(t)=F_{Z}(e^{-t})=1-(1-e^{-t})^{n\bar{\lambda}}. $$

Therefore,

$$ h_{-\log (Z)}(t)=-\frac{\partial}{\partial t}\log (\bar{F}_{-\log (Z)}(t))= \frac{n\bar{\lambda}e^{-t}(1-e^{-t})^{n\bar{\lambda}-1}}{1-(1-e^{-t})^{n\bar{\lambda}}}. $$

By considering the notations introduced in Lemma 8.1, we can reformulate the hr function of $-\log (Z)$ as follows:

Now, let us set

Now we can see that $k(t;\lambda _{1},\ldots ,\lambda _{n},\alpha )=R(e^{-t},n \bar{\lambda}, \alpha )$, for $t>0$ and $\alpha \in [0,1]$. Therefore, by using Lemma 8.1(i), if $\theta =n\bar{\lambda} \in (0,1)$, then $k(t;\lambda _{1},\ldots ,\lambda _{n},\alpha )$ is non-increasing in $t>0$ and by applying Lemma 8.1(ii), if $\theta =n\bar{\lambda} \geq 1$, then $k(t;\lambda _{1},\ldots ,\lambda _{n},\alpha )$ is non-decreasing in $t>0$. Hence, it was verified that if $n\bar{\lambda} \in (0,1)$ (resp. $n\bar{\lambda}\geq 1$), then $\frac{h_{-\log (Z)}(\alpha t)}{h_{-\log (Z)}(t)}$ is decreasing [resp. increasing] in $t\geq 0$, for every $\alpha \in (0,1)$. As a result, it is deduced that

and, moreover,

Figure 5

Plot of $R(u,\theta , \alpha )$ for $\theta =0.5$ and $\alpha =0.2,0.25,0.3,0.35,0.45$

Figure 6

Plot of $R(u,\theta , \alpha )$ for $\theta =2$ and $\alpha =0.2,0.25,0.3,0.35,0.45$

Now, we derive the following proposition using Theorem 5.1.

Proposition 8.2

Let X and Y be two nonnegative rvs with sfs $\bar{F}_{X}$ and $\bar{F}_{Y}$, respectively. Consider a series systems with independent component lifetimes $X_{1},X_{2},\ldots ,X_{n}$ and another series systems with independent component lifetimes $Y_{1},Y_{2},\ldots ,Y_{n}$ such that the component lifetimes follow the $PHR$ model, so that $X_{i}$ has sf $\bar{F}^{\lambda _{i}}_{X}$ and $Y_{i}$ has sf $\bar{F}^{\lambda _{i}}_{Y}$, where $\lambda _{i}$ is a positive parameter for $i=1,2,\ldots ,n$. Let

• (i) $\bar{\lambda}> \frac{1}{n}$ (resp. $\bar{\lambda}\leq \frac{1}{n}$).

• (ii) $X \leq _{st}Y$ (resp. $X \geq _{st}Y$).

Then, $X\leq _{RAI} Y$ implies $X_{1:n}\leq _{RAI} Y_{1:n}$.

Being important in applications, the case where the component lifetimes of the series systems are i.i.d. could also be discussed as a result of Proposition 8.2. If there exists an $\lambda >0$ such that $\lambda _{i}=\lambda$, for all $i=1,\ldots ,n$, then, according to Proposition 8.2, if $\lambda > \frac{1}{n}$ (resp. $\lambda <\frac{1}{n}$) and, further, if $X \leq _{st}Y$ (resp. $X \geq _{st}Y$), then $X\leq _{RAI} Y$ implies $X_{1:n}\leq _{RAI} Y_{1:n}$. For example, if $X_{i}$ follows sf $\bar{F}_{X}$ and $Y_{i}$ follows sf $\bar{F}_{Y}$, then $\lambda _{i}=\lambda =1$, for all $i=1,\ldots ,n$. Therefore, by using the non-parenthetical part of Proposition 8.2, it is deduced that if $X \leq _{st}Y$ and also if $X\leq _{RAI} Y$, then $X_{1:n}\leq _{RAI} Y_{1:n}$.

The following example illustrates a situation where Proposition 8.2 works.

Example 8.3

Suppose that $X\sim Lo(1,b^{-1})$ and $Y\sim Lo(1,a^{-1})$ where $0< a< b$. In Example 5.4 we showed that $X\geq _{st}Y$ and that $X\leq _{RAI}Y$. Let us consider two series systems with component lifetimes $X_{1},X_{2},\ldots ,X_{n}$ and $Y_{1},Y_{2},\ldots ,Y_{n}$ so that $X_{i}$ has sf $\bar{F}^{\lambda _{i}}_{X}$ and $Y_{i}$ has sf $\bar{F}^{\lambda _{i}}_{Y}$, where $\lambda _{i}$’s (for $i=1,2,\ldots ,n$) are positive values for which $\bar{\lambda}\leq \frac{1}{n}$. Then, according to the parenthetical part of Proposition 8.2, $X_{1:n} \leq _{RAI} Y_{1:n}$. From (8.2), the $RAI$ functions of $X_{1:n}$ and $Y_{1:n}$ are given by

$$ \breve{L}_{X_{1:n}}(t)= \frac{nb\bar{\lambda} t}{((1+bt)-(1+bt)^{n\bar{\lambda}+1})\log \left (1-\left (\frac{1}{1+bt}\right )^{n \bar{\lambda}}\right )}, $$

and

$$ \breve{L}_{Y_{1:n}}(t)= \frac{na\bar{\lambda} t}{((1+at)-(1+at)^{n\bar{\lambda}+1})\log \left (1-\left (\frac{1}{1+at}\right )^{n \bar{\lambda}}\right )}. $$

Let us now choose $a=2$, $b=3$, $n=4$ and $\lambda _{1}=0.1$, $\lambda _{2}=0.6$, $\lambda _{3}=0.1$ and $\lambda _{4}=0.1$ for which $\bar{\lambda}=0.225$, which is less than $\frac{1}{n}=0.25$. In Fig. 7, the graphs of $\breve{L}_{X_{1:4}}(t)$ and $\breve{L}_{Y_{1:4}}(t)$ versus $t \in (0,30)$ are plotted.

Figure 7

Plot of $RAI$ functions of $X_{1:4}$ and $Y_{1:4}$

Now, we present an application of Theorem 6.1 and Theorem 6.2 in the context of series systems with heterogeneous components. The next result makes an application of Theorem 6.1, which presents a lower bound for the $RAI$ function of a lifetime of a series system with $PHR$ component lifetimes.

Proposition 8.4

Let X be a nonnegative rv with cdf $F_{X}$ such that $F^{t}_{X}(t)$ is non-decreasing in $t>0$. Consider a series system with component lifetimes $X_{1},X_{2},\ldots ,X_{n}$ following $PHR$ model, where $X_{i}$ has the sf $\bar{F}^{\lambda _{i}}_{X}$, where $\lambda _{i}$ is a positive parameter for $i=1,2,\ldots ,n$. If at least one of the following conditions are satisfied:

• (i) $\bar{\lambda}>\frac{1}{n}$ and $s=\lim _{t\rightarrow 0^{+}} (-t\log (F_{X}(t)))<+\infty$

• (ii) $\bar{\lambda}\leq \frac{1}{n}$ and $s=\lim _{t\rightarrow \infty} (-t\log (F_{X}(t)))>0$, then $\breve{L}_{X_{1:n}}(t) \geq \breve{L}_{-\frac{s}{\log (Z)}}(t)$, for all $t>0$.

The following proposition is a direct conclusion of Theorem 6.2, which gives an upper bound for the $RAI$ function of a lifetime of a series system with $PHR$ component lifetimes.

Proposition 8.5

Let Y be a nonnegative rv with cdf $F_{Y}$ such that $F^{t}_{Y}(t)$ is non-increasing in $t>0$. Consider a series system with component lifetimes $Y_{1},Y_{2},\ldots ,Y_{n}$, which follow $PHR$ model, where $Y_{i}$ has the sf $\bar{F}^{\lambda _{i}}_{Y}$, where $\lambda _{i}$ is a positive parameter for $i=1,2,\ldots ,n$. If at least one of the following conditions are satisfied:

• (i) $\bar{\lambda}>\frac{1}{n}$ and $s^{\star}=\lim _{t\rightarrow 0^{+}} (-t\log (F_{Y}(t)))>0$

• (ii) $\bar{\lambda}\leq \frac{1}{n}$ and $s^{\ast}=\lim _{t\rightarrow \infty} (-t\log (F_{Y}(t)))<\infty$,

  then $\breve{L}_{Y_{1:n}}(t) \leq \breve{L}_{-\frac{s^{\star}}{\log (Z)}}(t)$, for all $t>0$.

The following example presents a situation where Proposition 8.4 is applicable.

Example 8.6

Consider an rv X with cdf $F_{X}(t)=e^{-\frac{1}{t(t+1)}}$, $t>0$. It can be seen that $F^{t}_{X}(t)=e^{-\frac{1}{t+1}}$, which is non-decreasing in $t>0$. Further, $s=\lim _{t\rightarrow 0^{+}} (-t\log (F_{X}(t)))=1$. Let $\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}$ be such that $\bar{\lambda}>\frac{1}{n}$. Then, according to Proportion 8.4, we can get a lower bound for the $RAI$ of $X_{1:n}$. It can be seen after some routine calculation that

$$ \breve{L}_{\frac{1}{-\log (Z)}}(t)= \frac{n\bar{\lambda} (1-e^{-\frac{1}{t}})^{n\bar{\lambda}-1}e^{-\frac{1}{t}}}{-(1-(1-e^{-\frac{1}{t}})^{n\bar{\lambda}})\log (1-(1-e^{-\frac{1}{t}})^{n\bar{\lambda}})}. $$

Therefore, for all $t>0$,

$$ \breve{L}_{X_{1:n}}(t)\geq \frac{n\bar{\lambda} (1-e^{-\frac{1}{t}})^{n\bar{\lambda}-1}e^{-\frac{1}{t}}}{-(1-(1-e^{-\frac{1}{t}})^{n\bar{\lambda}})\log (1-(1-e^{-\frac{1}{t}})^{n\bar{\lambda}})}. $$

From (8.2), one can get

$$ \breve{L}_{X_{1:n}}(t)= \frac{n\bar{\lambda}(1+2t)e^{-\frac{1}{t(t+1)}}(1-e^{-\frac{1}{t(t+1)}})^{n\bar{\lambda}-1}}{-t(t+1)^{2}(1-(1-e^{-\frac{1}{t(t+1)}})^{n\bar{\lambda}})\log (1-(1-e^{-\frac{1}{t(t+1)}})^{n\bar{\lambda}})}. $$

To examine the result of this example numerically, let us choose $n=5$, $\lambda _{i}=\frac{i}{30}$, for $i=1,\ldots ,5$. We can observe that $n\bar{\lambda}=\frac{1}{2}$, which is greater than $\frac{1}{n}$. The graphs of $\breve{L}_{X_{1:5}}(t)$ and $\breve{L}_{\frac{1}{-\log (Z)}}(t)$ versus t are plotted in Fig. 8.

Figure 8

Plot of the $RAI$ function of $X_{1:5}$ and its lower bound in Example 8.6

9 Conclusion

In this paper, we have investigated the nature of reversed aging intensity function for random quantile models. We have derived a preservation property of reversed aging intensity order under quantile transformations of probability distributions. This preservation property develops the order property of the underlying distributions in terms of reversed aging intensity function to the order relation of the modified distributions that stand for modeling random quantiles based on the reversed aging intensity function. For example, a system of components with identical lifetimes has a lifetime modeled by a random quantile. If such a system enjoys the preservation property under random quantiles, then it can be said that if one promotes all components of this system to reduce the reversed aging intensity function of the lifetime of the components at all points in time, then the reversed aging intensity function of the entire system will also be reduced over all time. Under some mild conditions for the lifetime distribution of the underlying distributions, a lower and an upper bound for the reversed aging intensity of one of their random quantiles are derived. To control the extreme magnitudes of reversed aging intensity function of a system, the obtained bounds can be useful. Preservation properties of the increasing reversed aging intensity function and the decreasing reversed aging intensity function under random quantiles were also obtained. This may be useful to preserve the knowledge that decreasing (increasing) behavior of the reversed aging intensity function of the underlying distributions over time translates into the same behavior for the random quantile distributions. Finally, the results of the work were applied in the context of a series system with independent and heterogeneous component lifetimes under the condition that the ith component lifetime follows a proportional hazard rate model with parameter $\lambda _{i},i=1,2,\ldots ,n$. The component lifetimes follow sfs $\bar{F}^{\lambda _{1}}_{X},\ldots ,\bar{F}^{\lambda _{n}}_{X}$, which have hr functions $\lambda _{1}h_{X}(x), \ldots , \lambda _{n} h_{X}(x)$, respectively. We can conclude that if in a described series system the reliability of the components increases (or decreases) according to the usual stochastic order, then if the average hazard rate of the component lifetime of the system (i.e., $\frac{h_{X}(x)}{n}\sum _{i=1}^{n} \lambda _{i}$) is is greater (or smaller) than the average hazard rate of the component lifetimes of a system with component lifetimes that follow a common sf $\bar{F}^{\frac{1}{n}}_{X}$ (i.e., $\frac{h_{X}(x)}{n}$), then an overall improvement of the components (in the entire time domain) in terms of the reversed aging intensity function leads to an improvement of the described series system in terms of the reversed aging intensity function.

In future work, the problem of preserving the reversed aging intensity order under random quantiles $Q_{X}(Z_{1})$ and $Q_{Y}(Z_{2})$ (instead of $Q_{X}(Z)$ and $Q_{Y}(Z)$) will be investigated, allowing one to develop the reversed aging intensity order under broader classes of lifetime distributions. Closure properties of parametric families of distributions can also be investigated.