Physical aspects of quantile residual lifetime sequence

1 Introduction

In statistical physics, modern stochastic analysis, and traditional probability and statistics, there is a way to characterize a static or dynamic distribution using its quantile function. A direct understanding of this function offers tangible benefits that cannot be derived directly from the density function. For example, the simplest way to simulate a non-uniform random variable is to apply its quantile function to uniform deviations. Modern Monte Carlo simulation methods, techniques based on low-discrepancy sequences, and copula methods require the use of marginal distribution quantile functions. Consequently, the study of quantile functions is as important for management as many classical special functions in mathematical physics and applied analysis. In certain situations, the lifetime of an object should be measured on a discrete scale, e.g., the number of power fluctuations an electrical device endures before it fails, the number of days a patient stays in the hospital, the number of days/weeks/months/years a kidney patient survives after treatment, modeling the number of times a pendulum moves before it comes to rest, the number of times a device switches on and off, and many other applications. An appropriate model for such data is the discrete lifetime model. Many authors have defined reliability measures for discrete data and investigated their properties. Salvia and Bollinger [1] introduced the hazard rate (HR) function for a discrete life model and studied its basic properties. Takahashi [2] proposed a definition of HR of nonequispaced discrete distributions. Singer and Willett [3] have used an empirical example and mathematical arguments to show how the methods of discrete-time survival analysis provide educational statisticians with an ideal framework for studying the occurrence of events. Shaked et al. [4] established a necessary and sufficient condition for a set of functions to be discrete multivariate conditional HR functions. Gupta et al. [5] have developed techniques to determine the increasing failure rate property and the decreasing failure rate property for a broad class of discrete distributions. Sandoh et al. [6] proposed a new modified discrete preventive maintenance policy in which failures of a system can be detected only by inspection and remedied by minimal repair. Roy and Dasgupta [7] have introduced a new discretization approach to assess the reliability of complex systems for which analytical methods do not provide a closed-form solution. Xie et al. [8] have introduced a different definition of the discrete failure rate function and provide the failure rate functions according to this definition for a number of useful discrete reliability functions. A thorough overview of discrete probability distributions used in reliability theory to represent the discrete lifetime of non-repairable systems was given by Bracquemond and Gaudoin [9]. To give a methodological summary of the conservation of some classes of discrete distributions under convolution and mixture, Pavlova et al. [10] studied some commonly used classes of discrete distributions. A failure and repair model, which is a discrete-time variant of the pure birth shock model, was presented by Belzunce et al. [11]. A novel survival tree approach for discrete-time survival data with time-varying variables is presented by Bou-Hamad et al. [12]. Furthermore, Eryilmaz [13] investigated a shock model in which the shocks occur according to a binomial process and determined the probability mass function and the probability-generating function of the lifetime of the system. Schmid et al. [14] have proposed a technique based on the result that the likelihood of a discrete survival model is equivalent to the likelihood of a regression model for binary outcome data. Li et al. [15] discussed a repairable system operating in dynamic regimes under the hypothesis of discrete time. Alkaff [16] proposed modeling techniques for the exact dynamic reliability analysis of systems in which the lifetimes of all components follow independent and non-identical distributions of the discrete phase type. A discrete-time version of the nonhomogeneous Poisson process has been defined and its properties were studied by Cha and Limnios [17].

As for continuous random lifetimes, the mean residual life (MRL) of discrete models has been considered by many authors. Ebrahimi [18] proposed the decreasing and increasing MRL classes of discrete lifetime distributions, Guess and Park [19] discussed some different shapes of the MRL function, Mi [20] determined the shape of the MRL function when the HR function is bathtub (BT)-shaped or upside-down bathtub (UBT)-shaped, and Salvia [21] determined some bounds for the MRL function.

Under certain circumstances, the α -quantile residual life function ( α -QRL) may be preferred to the MRL, e.g., if we have outliers in the data or the data are skewed or heavily censored. In addition, the concept of α -QRL for continuous random lifetimes has received considerable attention. Joe and Proschan [22] have introduced two classes of life distributions defined by the α -percentile residual life function. As Jeong and Fin [23] have shown, the quantile residual life function can be strongly influenced by competing events. Franco-Pereira and de Uña-Álvarez [24] introduced a new estimator of a percentile residual life function with censored data under a monotonicity constraint.

A method for calculating the quantile residual life function that relaxes the condition of independent censoring and takes covariates into account was proposed by Noughabi et al. [25]. Noughabi and Kayid [26] proposed and investigated the bivariate α-quantile residual life measure. Noughabi and Franco-Pereira [27] have shown that a mixture model is bounded by its components over the quantile residual life, and they investigated how mixture models are ordered with respect to the quantile residual life function when their components are ordered. However, in the reliability literature on discrete life models, the α-QRL has not received the attention it deserves. This motivated us to define and investigate the α -QRL sequence as a reliability measure.

The remainder of this study is organized as in the following. In Section 2, we set notations and present some preliminaries. In Section 3, the α -QRL sequence is defined and studied. The relation between the α -QRL sequence and the HR sequence is discussed, and in particular, the form of it when the HR is increasing, decreasing, BT, or UBT is explored. In Section 4, a new stochastic order for discrete lifetime variables based on the α -QRL concept is defined and its connection with the HR order is investigated. Section 5 includes the conclusions and future topics.

2 Notations and preparatory contents

Let T represent a discrete random lifetime with the support t 0 , t 1 , t 2 , . . , t i ∈ W and W be the set of all whole numbers, i.e., W = { 0 , 1 , 2 , 3 , . . } . The probability mass function and the reliability function are corresponding to sequences p 0 , p 1 , p 2 , … and R 0 , R 1 , R 2 , … where p i = P ( T = t i ) and R i = P ( T ≥ t i ) respectively. The support of T may be bounded from above by t m , i.e., p i > 0 for i = 0 , 1 , 2 , … , m and p i = 0 for i ≥ m + 1 . Usually, in the reliability theory and survival analysis literature, the support of a lifetime variable is considered to be W .

The residual life given survival up to time t i is denoted by the sequence of conditional random variables T ( i ) = T − t i | T ≥ t i , i = 0 , 1 , 2 , … , which guide us to the conditional reliability function

The following HR function was traditionally defined by Salvia and Bollinger [1]:

The HR function h i characterizes the distribution function. Shaked et al. [4] investigated that a sequence h i should satisfy one of the following necessary and sufficient conditions, based on the support of the underline model, to be an HR function:

• There exists one m ∈ W such that for every i < m , h i < 1 and h m = 1 .

• For every i ∈ W , h i < 1 and ∑ i = 0 ∞ ‍ h i = ∞ .

  Xie et al. [8] explored and discussed several drawbacks of the traditional definition of the HR function and redefined this concept in the following:

  (3) r i = log R i R i + 1 , i ∈ W .

  It could be checked that the sequences r i and h i are in a one-to-one relation:

  (4) r i = − log ( 1 − h i ) , i ∈ W ,

  and implies that both have the same monotonicity attribute. Also, this relation indicates that the sequence r i characterizes the model as h i does.

  The MRL function at t i , which shows the mean of the remaining time to failure given survival up to t i , is defined precisely by

  (5) m i = E ( T − t i | T ≥ t i ) = 1 R i ∑ k = i ∞ ‍ ( t k + 1 − t k ) R k + 1 , i ∈ W .

  In the discrete case, HR, MRL, and the α -QRL and other similar functions are in fact sequences, so they are referred to as sequences too. It can be checked easily that the MRL and the traditional HR sequences are related by the following relation. Refer to Salvia [21] for a similar form:

  (6) m i + 1 = m i 1 − h i − ( t i + 1 − t i ) , i ∈ W .

  The fact that an object or process has an increasing risk of failure due to aging or fatigue indicates that the true model follows an increasing HR sequence. In this case, the MRL sequence has a decreasing shape. However, in some situations, an object may be exposed to excessive hazards in its early-life phase, which decreases over time. Therefore, the lifetime model should naturally assume an early-life phase with decreasing and eventually increasing HR. A characteristic of such objects is that instances that pass through an early-life phase (i.e., the burn-in phase) inevitably become more reliable. In such cases, the MRL sequence has an increasing and then a decreasing shape. It is obvious that the larger the MRL of the object is, the more reliable the condition it has for continuation. So, we can propose the point maximizing the MRL function as the optimal burn-in time, i.e., the MRL burn-in time b * satisfies

  (7) b * = arg max i m i , i ∈ W .

  An interesting topic of reliability theory and survival analysis is the study of the shape of m i , particularly in the context of the HR sequence. Guess and Park [19] proposed a necessary and sufficient condition for which the MRL function first increases and then decreases (decreases and then increases). Mi [20] considered a discrete lifetime model and proved that if the HR sequence has a BT form, the MRL sequence is decreasing or UBT. Tang et al. [28] complemented this result for the models with UBT HR sequences and showed that in this case, the MRL sequence is increasing or BT. Nair et al. [29] presented discrete lifetime distributions with BT-HR functions.

  On the other hand, stochastic orderings have found a wide field of application in probability, statistics, and statistical decision theory. Two discrete random lifetimes T 1 and T 2 , with their corresponding reliability sequences R 1 , i , and R 2 , i , i ∈ W , could be compared in the sense of their different characteristics.

• The simplest comparison is based on the reliability function, which states that T 1 is smaller than T 2 in the usual stochastic order, T 1 ≤ st T 2 , if R 1 , i ≤ R 2 , i for all i .

• T 1 is smaller than T 2 in likelihood ratio order, T 1 ≤ lr T 2 , if p 2 , i p 1 , i is an increasing sequence.

• T 1 is smaller than T 2 in HR order, T 1 ≤ hr T 2 , if R 2 , i R 1 , i is an increasing sequence. Equivalently, T 1 ≤ hr T 2 , if h 1 , i ≥ h 2 , i for every i .

• T 1 is smaller than T 2 in MRL order, T 1 ≤ mlr T 2 , if ∑ k = i ∞ ‍ ( t k + 1 − t k ) R 2 , k + 1 ∑ k = i ∞ ‍ ( t k + 1 − t k ) R 1 , k + 1 is an increasing sequence. Equivalently, T 1 ≤ mrl T 2 , if m 1 , i ≤ m 2 , i for every i .

It is known that T 1 ≤ lr T 2 implies T 1 ≤ hr T 2 . Also, T 1 ≤ hr T 2 implies T 1 ≤ st T 2 and T 1 ≤ mrl T 2 . Refer to Dewan and Sudheesh [30].

3 α -QRL sequence

For the random lifetime T , the i th element of α -QRL sequence is the α -quantile of the remaining life T − t i given that T ≥ t i and can be expressed precisely by

where α ̅ = 1 − α . Note that R ( i ) ( x ) is right continuous and has step form with jumps at integer values of x and in turn q α , i receives just integer values. We can simplify the α -QRL sequence (8) as in the following:

where R − 1 ( p ) = inf { y : y = t j , R j ≤ p } is the inverse of the reliability function. If the support of T is bounded from above by t m , i.e., R i > 0 for i = 0 , 1 , 2 , . . , m and R i = 0 for i ≥ m + 1 , then

To better illustrate the α- α -QRL and its applications, let T plot the days between the treatment of a tumor and its first recurrence in breast cancer patients. For treated patients who have not experienced tumor recurrence after i days, let q α , i be the number of days from i in which tumor recurrence will occur in 100 α % of these patients. Such information could be useful for hospital management when planning future patient visits. It is worth noting that Proschan [31] considered the time interval between consecutive air conditioning failures on a Boeing 720. After i days since the last maintenance, q α , i provides the number of days from i in which the air conditioning will fail in 100 α % of these aircraft. Assume that we have a random lifetime T with the support t 0 , t 1 , t 2 , . . , t i ∈ W . For computing q α , i , we apply the following steps:

• Compute α ̅ R i = α ̅ P ( T ≥ t i ) .

• To find R − 1 ( α ̅ R i ) , find the smallest t j , for that P ( T ≥ t j ) is equal to or less than the calculated α ̅ R i . Then, q α , i = R − 1 ( α ̅ R i ) − t i .

It can be checked easily from (6) that m i + 1 − m i ≥ t i − t i + 1 . The following lines show that a similar property holds for α -QRL:

The reliability function could be expressed in terms of HR by

where h ̅ k = 1 − h k . Then, q α , i can be expressed as in the following:

Note that j ≥ i . Now, by (4), we have

Eqs. (11) and (12) reveal the close relationship between the HR and the α -QRL sequences and give the main clue in proving the following results:

When lifetime data exhibit a BT HR model, we can define the burn-in time b * = t j maximizing the α -QRL, specially the median residual life. Theorem 2 shows that the burn-in time b * is not greater than the first change point of the HR sequence.

Example 1. Lai and Wang [32] studied the attributes of a discrete BT HR model with the following density function:

The reliability and HR sequences are

and

Also, the α -QRL sequence is

Lai and Wang [32] proved that the HR is increasing for a ≥ 0 , and BT for a < 0 . Thus, by Theorem 1, when a ≥ 0 , the α -QRL is decreasing. When a < 0 , the α -QRL is eventually decreases, and it has at least one maximum. Figure 1 shows the density and HR sequences of the MP model for some different parameters. For a = − 1 , − 2 , − 3 , the HR sequences take their minimums at i = 8 , 10 , 12 , respectively. Also, the median residual life sequences are maximized at i = 5 , 6 , 6 , respectively, for a = − 1 , − 2 , − 3 .

Figure 1

Density and HR sequences of the MP model for some values of parameters.

Example 2. A very flexible model which is suitable in situations we deal with monotone, BT, and UBT HR functions is the competing risk model with the following reliability function:

Table 1 shows the number of cycles to failure for 60 electrical appliances in a life test. This dataset was reported by Lawless [33] and analyzed by many authors, e.g., Bebbington et al. [34]. Shafaei et al. [35] showed that the model (21) could be a good candidate for this dataset. The maximum-likelihood estimate of the parameters is reported to be a ˆ = 0.000311 and b ˆ = 0.5954 . Figure 2 shows the density and FR function of the estimated competing risk at the data points. These plots exhibit a BT form for the HR function and a unimodal form for the median residual life. The HR is minimized at i = 776 and the median residual life is maximized at i = 234 , and this point could be considered as a burn-in time for the appliances.

Figure 2

Density and FR function of the estimated competing risk model (21).

Assume a discrete lifetime T with reliability sequence R i , i ∈ W . Now, if this object is exposed to stress, shock, or environmental factors, its corresponding reliability sequence may be expressed by

and based on the fact that the affecting factor causes smaller or bigger lifetime, θ > 1 or θ < 1 . The corresponding HR, defined by Xie et al. [8], of this model is

and due to this relation, the model (24) is called the proportional HR model. Let T be a discrete random lifetime and T * correspond to one proportional HR model of it. By the fact that the forms of h i and r i are the same, it follows that a discrete random lifetime T is IHR if and only if T * is. Dewan and Sudheesh [30] showed that T is decreasing mean residual life (increasing mean residual life) if and only if T * is. Assume that q α , i * is the α -QRL of T * . Then,

where β = 1 − α ̅ θ . The following result follows directly from (26).

4 α -QRL order

Let T 1 and T 2 represent two discrete random lifetimes with common support t 0 , t 1 , t 2 , . . and the α -QRL sequences of T 1 and T 2 be denoted by q 1 , α , i and q 2 , α , i , respectively.

From (10), it is clear that if T 1 ≤ α − QRL T 2 and the support of T 2 is bounded from above by t m , then the support of T 1 is bounded from above by t ′ m ≤ t m .

Since T 1 ≤ α i − QRL T 2 , we have

Then, comparing (27) and (28) shows that h ̅ 2 , i ≥ h ̅ 1 , i or equivalently h 1 , i ≥ h 2 , i , which completes the proof.

Assume that T 1 * and T 2 * correspond to the proportional HR models of T 1 and T 2 , respectively. The following theorem shows that the α − QRL order implies an order for the proportional HR models. The proof follows from (26) directly.

5 Conclusion

The α -QRL sequence for discrete data was defined, and its relationship to the HR sequence was investigated. An increasing (decreasing) HR sequence implies a decreasing (increasing) α -QRL. If the HR sequence has a BT (UBT) form, the α -QRL usually has a UBT (BT) form. If the HR has a BT form, the point that maximizes the α -QRL can be used as a suitable burn-in time point. The α -QRL order in the discrete context is defined, and it is proved that this order is weaker than the HR order. In addition, the basic properties of α -QRL in the proportional HR model are discussed. Some interesting topics that can be considered in future research are listed in the following:

• To the best of our knowledge, the α -QRL for a value of α does not clearly characterize the base model. Therefore, the question of how the model can be characterized by α -QRL sequences remains an open problem.

• Problems in estimating the proposed α -QRL sequence can be extended to estimate monotonic α -QRL sequences.

• For discrete models, the α-quantile past sequence can be defined and analyzed in a similar way.

• The concept of α -QRL can be extended to multivariate relationships when two or more dependent lifetime random variables are involved. In this context, the problem of estimating the α -QRL function in a multivariate context is interesting.

• Although all observations are effectively discrete, they can have a very different observational resolution/accuracy if the observations are actually related to a continuous variable. For example, an underlying time variable can be observed with the unit one of a week or with the unit one of an hour. The relationship between these two resolutions based on the quintile residual life sequence is an interesting and remains an open problem.