Estimation of monotone α-quantile of past lifetime function with application

Mohamed Kayid

doi:10.1515/phys-2024-0035

Article Open Access

Estimation of monotone α-quantile of past lifetime function with application

Mohamed Kayid

Published/Copyright: June 3, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

In many physical situations, it is reasonable to assume that the α-quantile of past life functions is increasing. In this study, we propose a new estimator for an increasing α-quantile of the past life function for right-censored data. We investigate the asymptotic properties of the new estimator and explore its weak and strong convergence. The simulation results show that the new estimator is consistent and outperforms the usual estimator from the literature. As an illustrative example, a time event from a data set of the Mayo Clinic study on primary biliary cirrhosis was analyzed.

Keywords: mathematical physics; monotone quantile of past lifetime function; right-censored data; Kaplan–Meier survival estimator

1 Introduction

In mathematical physics, modern stochastic analysis, and traditional probability and statistics, there is a way to characterize a static or dynamic distribution by 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.

The concept of past lifetime or inactivity time of a positive random variable T at time t , T t = t − T | T < t , is applied to develop new concepts in reliability theory, survival analysis, and other scientific areas, such as the reversed hazard rate (RHR) function, mean past lifetime (MPL) function, and α -quantile of past lifetime ( α -QPL) function. Let f and F be the density and distribution functions of T , respectively. The RHR function shows the instantaneous failure at ( t − δ , t ] given that it has occured at [ 0 , t ] , and can be expressed by

(1) r ( t ) = f ( t ) F ( t ) , t ≥ 0 .

The RHR function was defined by Barlow et al. [1]. Block et al. [2] studied this measure, showed that it could not be increasing for a lifetime model, and discussed its importance in studying k-out-f-n systems. Crescenzo [3] considered a proportional RHR model. Chandra and Roy [4] explored some implicative relationships with respect to the monotonic behavior of RHR. Finkelstein [5] investigated the connection between monotonicity of MPL and RHR functions. Kundu et al. [6] studied the RHR function of the order statistics and record values. Li et al. [7] focused on the RHR function of a general mixture model. Burkschat and Torrado [8] examined the RHR function of sequential order statistics, and Esna-Ashari et al. [9] applied RHR concept for comparing generalized order statistics. Also, the MPL function shows the expectation of T t absorbed many interests among researchers. For example, Ahmad et al. [10] investigated some preservation properties of the MPL order under the convolution, mixture, and shock models. Ahmad and Kayid [11] proved some results about the RHR and MPL orders. Asadi [12] studied the reliability properties of parallel systems in terms of the MPL function. Kundu and Nanda [13] proved that the second-order moment of the past lifetime characterizes the model uniquely. Eryilmaz [14] developed the MPL concept for multi-state systems. Asadi and Berred [15] obtained some results on partial ordering and characterization and estimation of the MPL function. Bhattacharyya et al. [16] proposed a nonparametric two-sample test for comparing MPL functions of two life distributions. Khan et al. [17] explored bounds, limiting behavior, and some other aspects of the MPL function.

The α -QPL function represents the α quantile of T t , 0 < α < 1 , and is simplified to

(2) q α ( t ) = t − Q ( α ̅ F ( t ) ) , t ≥ 0 ,

in which α ̅ = 1 − α and Q ( y ) = inf { x : F ( x ) ≥ y } is the inverse function of F . Mahdy [18] defined an estimator of α -QPL function for complete data using the empirical distribution function of data and discussed some attributes of it. Balmert and Jeong [19] discussed nonparametric inferences on the median past lifetime function for censored data. In addition, Balmert et al. [20] considered a log-linear quantile regression model for inactivity time for right-censored data. Recently, Kayid [21] studied the problem of estimating the α -QPL function for right-censored data and the asymptotic behavior of the defined estimator.

Finkelstein [5] proved that if the hazard rate function decreases, the RHR also decreases. Also, for a distribution with positive support, the RHR cannot increase on the entire support. Furthermore, Finkelstein [5] showed that the limiting behavior of the RHR and the density function are the same, so for supports on the entire positive real line, the RHR should be eventually decreasing. Unnikrishnan and Vineshkumar [22] showed that the RHR and the α -QPL functions are related via the following relation:

(3) q α ′ ( t ) = 1 − r ( t ) r ( t − q α ( t ) ) ,

where q α ′ ( t ) represents the first derivative of the α -QPL function in trems of t . They concluded that:

• There is no any distribution with strictly decreasing α -QPL function on the entire positive support.

• Whenever r ( t ) is decreasing, q α ′ ( t ) is increasing.

Summarizing these results, it seems that many lifetime models have decreasing RHR and increasing α -QPL functions. Even the class of decreasing RHR is a subclass of the class of models with increasing α -QPL functions. Specifically, the class of distributions with increasing α -QPL function consists of the Weibull, inverse Weibull, gamma, lognormal, Burr, reciprocal beta, reciprocal Lomax, Pareto, power, and many others (see Unnikrishnan and Vineshkumar [22]). This led us to set up a new estimator for the α -QPL function, assuming that it is increasing. Kochar et al. [23], Franco-Pereira and Una-Alvarez [24], and Shafaei Noughabi and Franco-Pereira [25] have proposed similar estimators for the mean residual life function, the α-quantile residual life function, and the bivariate α -quantile residual life function.

In the next section, some preliminary remarks are made, including the notations for censored data, the well-known Kaplan–Meier survival estimator, and the estimator of the α -QPL function discussed by Kayid [21], which will be referred to as the “usual estimator” in the following. In Section 3, a new increasing estimator of the α -QPL function is introduced and its asymptotic properties are investigated. Section 4 uses a simulation study to investigate the behavior of the proposed estimator and compares it with the usual estimator. To illustrate the applicability of the proposed estimator, a data set from the Mayo Clinic study on primary biliary cirrhosis is analyzed in Section 5. The final section concludes the study with suggestions for future research.

2 Right-censored data

2.1 α -QPL estimator

This section provides necessary notations and context for bringing up the new idea in th next section. Assume that T i , i = 1 , 2 , . . . , n be a sample of n iid instances from a distribution function F . Each T i is may be censored by a random censorship variable C following a distribution function H . If T i > C i , we say that it is censored, and otherwise, it is observed (uncensored). Thus, the censored data consist of pairs ( X i , δ i ) , i = 1 , 2 , . . . , n , where X i = min { T i , C i } is the observed value and δ i = I ( T i ≤ C i ) is the censoring indicator. Due to the fact that X i s are identical, their distribution function could be denoted by G . Let b F = inf { x : F ( x ) = 1 } represent the upper bound of the support of F , and similarly, take b H and b G for distributions H and G , respectively. Note that b G = min { b F , b H } . The survival functions related to distributions F , H , and G are represented by F ̅ , H , ̅ and G , ̅ respectively. Then, we have

G ̅ ( t ) = F ̅ ( t ) H ̅ ( t ) .

In order to state the well-known KM survival estimator, we summarize the data by

N ̅ ( t ) = ∑ i = 1 n N i ( t ) , ‍

which N i ( t ) = I ( X i ≤ t , δ i = 1 ) . Note that N ̅ ( t ) exhibits the number of items failed up to or at time t . Also, the number of items at risk at time t is

Y ̅ ( t ) = ∑ i = 1 n Y i ( t ) , ‍

where Y i ( t ) = I ( X i ≥ t ) . Applying these notations, the survival function F ̅ can be estimated by

(4) F ̅ n ( t ) = ∏ X i ≤ t ( 1 − Δ N ̅ ( X i ) Y ̅ ( X i ) ) , 0 ≤ t ≤ X ( n ) , ‍

where X ( n ) = max { X 1 , X 2 , . . . , X n } and Δ N ̅ ( X i ) = N ̅ ( X i ) − N ̅ ( X i − ) is the number of failed items at X i . Note that F ̅ n is the well-known KM survival estimator. Naturally, the distribution function F can be estimated by F n ( t ) = 1 − F ̅ n ( t ) .

The inverse of the distribution function F is defined by F − 1 ( p ) = inf { t : F ( t ) ≥ p } , 0 < p < 1 . The estimator of this function is defined as in the following:

F n − 1 ( p ) = inf { t : F n ( t ) ≥ p } ,

which shows a left continuous function with jump at F n ( X i ) , i = 1 , 2 , . . . , n .

The usual estimator of the α -QPL function at time t , q α , n ( t ) is defined by using the estimator F n in place of distribution function F in (2) (see Kayid [21]).

(5) q α , n ( t ) = t − F n − 1 ( α ̅ F n ( t ) ) , 0 < α < 1 , 0 ≤ t ≤ X ( n ) .

This function is a left continuous function and consists of line segments each with slope 1. To plot the function for a sample, we take

t i = inf { y : F n − 1 ( α ̅ F n ( y ) ) = X i } , i = 1 , 2 , . . . , n − 1 .

Clearly, t 1 < t 2 < . . . < t n − 1 . Based on the data, just some of t i s are computable. The plot of q α , n ( t ) is discontinuous at every t i at which the plot decreases by X i − X i − 1 . A schematic plot of q α , n ( t ) function is presented in Figure 1. To compute the q α , n ( t ) for a given dataset, we calculate all computable t i values, namely, t 1 , t 2 , . . . , t m , and apply the following simplified version:

(6) q α , n ( t ) = t 0 < t ≤ t 1 t − X 1 t 1 < t ≤ t 2 t − X 2 t 2 < t ≤ t 3 … t − X m − 1 t m − 1 < t ≤ t m t m − X m − 1 t > t m .

$Figure 1 A schematic plot of the usual α \alpha -QPL function, the black solid line, which consists of line segments with slop one and is not increasing. The blue dashed lines show the new estimator of α \alpha -QPL function, which is increasing.$

Figure 1

A schematic plot of the usual α -QPL function, the black solid line, which consists of line segments with slop one and is not increasing. The blue dashed lines show the new estimator of α -QPL function, which is increasing.

3 Increasing α -QPL estimator

The new estimator of α -QRL is defined and studied in this section. Assume that the α -QRL is increasing. In this case, we propose the following estimator:

(7) i q α , n ( t ) = sup y ≤ t q α , n ( y ) , 0 < α < 1 , 0 ≤ t ≤ X ( n ) .

Figure 1 presents an illustrative plot for this estimator and visually shows that how we can compute the estimator at a given point t .

Lemmas 1 to 4 are necessary for proving Theorem 5, which states a result about convergence of | i q α , n ( t ) − q α , n ( t ) | to zero. The whole idea for proving this statement is similar to that in Kochar et al. [23] and Franco-Pereira and Una-Alvarez [24]. To show that how q α , n ( t ) and i q α , n ( t ) are asymptotically close, a continuous piecewise linear version of q α , n ( t ) and L n q α , n ( t ) is defined. Let integer k n ↑ ∞ , Δ n = b k n and

a j n = j b k n = j Δ n , j = 0 , 1 , . . . , k n .

For any function g , define the linear interpolation L n g on [ 0 , b ] by

L n g ( a j n ) = g ( a j n ) , j = 0 , 1 , 2 , . . . , k n ,

and for a j n < x < a j + 1 n ,

L n g ( x ) = g ( a j n ) + g ( a j + 1 n ) − g ( a j n ) Δ n ( x − a j n ) , j = 0 , 1 , 2 , . . . , k n − 1 .

Lemma 1

Assume that the following conditions hold:

(C1) F is twice differetiable.

(C2) The density function f = F ′ is bounded away from zero on the interval ( 0 , Q ( α ̅ ) ) .

(C3) q α ′ ( t ) exists and there is a c 1 > 0 such that q ′ α ( t ) ≥ c 1 for all 0 ≤ t ≤ b .

Then, we have P ( liminf A n ) = 1 , where A n = { L n q α , n ( t ) is strictly increasing on [ 0 , b ] } .

Proof

Recently, Kayid [21] showed that

(8) sup 0 ≤ t ≤ b | q α , n ( t ) − q α ( t ) | = O n − 1 2 ( log log n ) 1 2 .

Thus, uniformly on 0 ≤ j ≤ k n ,

(9) | q α , n ( a j n ) − q α ( a j n ) | = O n − 1 2 ( loglog n ) 1 2 ,

and by condition (C3),

(10) q α ( a j + 1 n ) − q α ( a j n ) ≥ c 1 Δ n , j ≤ k n − 1 .

So, by combining Eqs. (9) and (10), for sufficiently large n and j ≤ k n − 1 , we have

(11) q α , n ( a j + 1 n ) − q α , n ( a j n ) ≥ c 1 Δ n + O n − 1 2 ( log log n ) 1 2 ≥ c ′ 1 Δ n , j ≤ k n − 1 ,

and the proof follows by the fact that L n q α , n ( a j n ) = q α , n ( a j n ) .

Lemma 2

Assume that the conditions C1, C2, and C3 hold. Then, we have P ( liminf B n ) = 1 , where

B n = { sup 0 ≤ t ≤ b | i q α , n ( t ) − L n q α , n ( t ) | ≤ sup 0 ≤ t ≤ b | q α , n ( t ) − L n q α , n ( t ) | } .

Proof

By Lemma 1, for sufficiently large n , we have

(12) L n q α , n ( t ) = sup y ≤ t L n q α , n ( y ) ,

then by triangle inequality of the sup-norm

(13) | i q α , n ( t ) − L n q α , n ( t ) | = | sup y ≤ t q α , n ( y ) − sup y ≤ t L n q α , n ( y ) | ≤ sup y ≤ t | q α , n ( y ) − L n q α , n ( y ) | ,

which gives the result immediately.

Lemma 3

Assume that the following condition holds: (C4) q α ″ ( t ) exists and sup 0 ≤ t ≤ b | q α ″ ( t ) | ≤ c 2 < ∞ . Then, we have

sup 0 ≤ t ≤ b | q α ( t ) − L n q α ( t ) | ≤ c 2 Δ n .

Proof

Similar to the proof of Proposition 4.3 in Kochar et al. [23], let n and 0 ≤ j ≤ k n − 1 be fixed and define

(14) e ( t ) = q α ( t + a j n ) − L n q α ( t + a j n ) , 0 ≤ t ≤ Δ n .

Since L n is a linear interpolation, e ′ ′ ( t ) = q ′ ′ α ( t + a j n ) , t ∈ ( 0 , Δ n ) , and by the definition of this linear interpolation, e ( 0 ) = e ( Δ n ) = 0 . Using the Taylor expansion, we have

(15) e ( t ) = e ′ ( 0 + ) t + 1 2 e ′ ′ ( η t ) t 2 , 0 ≤ t ≤ Δ n ,

for some η t ∈ ( 0 , t ) . So, at point t = Δ n ,

(16) 0 = e ( Δ n ) = e ′ ( 0 + ) Δ n + 1 2 e ′ ′ ( η Δ n ) Δ n 2 ,

which implies

(17) e ′ ( 0 + ) = − 1 2 e ′ ′ ( η Δ n ) Δ n .

Applying the condition C4, we can write

(18) | e ( t ) | = | − 1 2 e ′ ′ ( η Δ n ) Δ n t + 1 2 e ′ ′ ( η t ) t 2 | ≤ c 2 Δ n 2 .

This proves the assertion.

Lemma 4

Under C1, C2, C3, and C4, and if n 1 4 = o ( k n ) , we have

n sup 0 ≤ t ≤ b | q α , n ( t ) − L n q α , n ( t ) | → 0 , inprobability .

Proof

Kayid [21] showed that the process

(19) r n α ( t ) = n f ( Q ( α ̅ F ( t ) ) ) ( q α , n ( t ) − q α ( t ) )

weakly converges to a zero mean Gaussian process. Take a piecewise shift transformation V n of q α ( t ) by

(20) V n q α ( t ) = q α ( t ) + ( q α , n ( a j n ) − q α ( a j n ) ) , a j n ≤ t ≤ a j + 1 n , j ≤ k n − 1 ,

and V n q α ( b ) = q α , n ( b ) . Completely similar with proof of Proposition 4.4 of Kochar et al. [23], and by the tightness conditions of the Skorohod topology J 1 , it implies that

(21) n sup 0 ≤ t ≤ b | q α , n ( t ) − V n q α ( t ) | → 0 , inprobability .

Note that V n q α ( a j n ) = q α , n ( a j n ) , and on the other hand, L n q α , n ( a j n ) = q α , n ( a j n ) , j ≤ k n . So, q α , n ( a j n ) = V n q α ( a j n ) for all j ≤ k n . By the fact that L n is piecewise linear, it results that L n q α , n ( t ) = L n V n q α ( t ) for all 0 ≤ t ≤ b . Now, we can write

(22) q α , n ( t ) − L n q α , n ( t ) = ( q α , n ( t ) − V n q α ( t ) ) + ( V n q α ( t ) − L n V n q α ( t ) ) + ( L n V n q α ( t ) − L n q α , n ( t ) ) = ( q α , n ( t ) − V n q α ( t ) ) + ( V n q α ( t ) − L n V n q α ( t ) ) = ( q α , n ( t ) − V n q α ( t ) ) + ( q α ( t ) − L n q α ( t ) ) .

The last equality follows from the fact that V n is a piecewise-shifted transformation. Since n 1 4 = o ( k n ) , it implies from Lemma 3 that

(23) sup 0 ≤ t ≤ b | q α ( t ) − L n q α ( t ) | → 0 ,

so the proof is completed by (21).

The following result follows from Lemmas 2 and 4 immediately.

Theorem 5

Under C1, C2, C3, and C4, we have

(24) n | i q α , n ( t ) − q α , n ( t ) | → 0 , inprobability .

Similar to the process adopted for the α -QRL function by Csorgo and Csorgo [26], Franco-Pereira and Una Alvarez [24], and the process for the α -QPL function defined by Kayid [21], the scaled α -QPL process r n * α ( t ) can be defined by

(25) r n * α ( t ) = n f ( Q ( α ̅ F ( t ) ) ) ( i q α , n ( t ) − q α ( t ) ) , 0 < t < X ( n ) .

The subject of the following theorem is that it weakly converges to a zero mean Gaussian process. To state the result, we need to define

d ( t ) = ∫ 0 t G ̅ − 2 ( x ) d F ∼ ( x ) , ‍

in which F ∼ ( t ) shows the probability that one event takes place before or at t and be uncensored, i.e.,

(26) F ∼ ( t ) = P ( X i ≤ t , δ i = 1 ) = ∫ 0 t H ̅ ( x ) d F ( x ) . ‍

Theorem 6

Suppose that 0 < α < 1 , 0 ≤ t < b G , Q ( α ̅ F ( t ) ) < b G and the density-quantile function f ( Q ( α ̅ F ( t ) ) ) is continuous at point α ̅ F ( t ) . Then, we can write

(27) r n * α ( t ) → N ( 0 , σ α , t 2 ) , indistribution ,

where σ α , t 2 = ( 1 − α ̅ F ( t ) ) 2 d ( Q ( α ̅ F ( t ) ) ) + α ̅ 2 F ̅ 2 ( t ) d ( t ) − 2 α ̅ ( 1 − α ̅ F ( t ) ) F ̅ ( t ) d ( Q ( α ̅ F ( t ) ) ) .

Proof

The proof follows from Theorems 5 and 1 of Kayid [21] and Theorem 2.7 (iv) from Vaart van der [27].

Now, a law of iterated logarithm for the proposed monotone estimator of the α -QPL function is presented.

Theorem 7

Under conditions C1 and C2, we have with probability 1,

(28) sup 0 ≤ t ≤ b | i q α , n ( t ) − q α ( t ) | = O n − 1 2 ( log log n ) 1 2 .

Proof

By triangle inequality of the sup-norm, we can write

(29) | i q α , n ( t ) − q α ( t ) | = | sup y ≤ t q α , n ( y ) − sup y ≤ t q α ( y ) | ≤ sup y ≤ t | q α , n ( y ) − q α ( y ) | .

On the other hand, Corollary 1 from Kayid [21] states that with probability 1:

(30) sup 0 ≤ t ≤ b | q α , n ( t ) − q α ( t ) | = O ( n − 1 2 ( loglog n ) 1 2 ) ,

which gives the desired result.

4 Simulation

In a simulation study, attributes of the usual estimator, q α , n ( t ) and the proposed monotone estimator i q α , n ( t ) are explored and compared. The Weibull and Gamma distributions, respectively, with the following distribution functions are applied for this purpose:

(31) F ( t ) = 1 − exp − t α β , α > 0 , β > 0 , t ≥ 0 ,

and

(32) F ( t ) = γ ( α , β t ) Γ ( α ) , α > 0 , β > 0 , t ≥ 0 ,

where γ ( α , x ) = ∫ 0 x y α − 1 e − y ‍ d y and Γ ( α ) = ∫ 0 ∞ y α − 1 e − y d y ‍ are the lower incomplete gamma function and the complete gamma function, respectively. All computations are made by R statistical programming language. In each run, for selected values of the parameters, r = 5 , 000 replicates of samples of sizes n = 25 and 50 are simulated. For simulation one sample of Weibull or gamma, we apply the built-in functions rweibull or rgamma of R. Then, each sample is imposed to right censorship by a random censorship variable C from a uniform distribution U ( 0 , M ) . For each simulated sample of Weibull or gamma model, we generate a random sample of uniform U ( 0 , M ) by built-in function runif in R. The value of M is computed based on the censorship rate p , which is set to be 0.05 or 0.25 . In this way, we compute M by solving the following equation in terms of M :

pM = ∫ 0 M ‍ F ̅ ( t ) d t ,

where F ̅ ( t ) = 1 − F ( t ) is the reliability function of the Weibull or gamma distribution. This equation is solved numerically by uniroot function in R. For each censored sample, the usual estimator of 0.5 -QPL and the proposed increasing estimator of it are computed at four deciles q 0.2 , q 0.4 , q 0.5 , and q 0.7 . Then, the bias (B) and the mean-squared error (MSE) of both estimators are computed. Tables 1 and 2 report the simulation results for the Weibull and gamma models, respectively. The main points that can be summarized from these results are as follows:

Table 1

Simulation results for the Weibull distribution

			p (censorship)
			0.05				0.25
			Usual		Increasing		Usual		Increasing
n	Parameters	q	B	MSE	B	MSE	B	MSE	B	MSE
25	α = 1.2 , β = 5	0.2	0.1209	0.1028	0.2961	0.1286	0.0858	0.0913	0.2535	0.1034
		0.4	0.0718	0.1698	0.2724	0.1380	0.1120	0.1844	0.2668	0.1360
		0.5	0.0836	0.2175	0.2318	0.1966	0.1093	0.2654	0.2567	0.2171
		0.7	0.0641	0.4239	0.1981	0.3210	−0.3406	2.1740	0.1462	0.3611
	α = 0.7 , β = 1	0.2	0.0062	0.00059	0.0134	0.00044	0.0048	0.00058	0.0126	0.00043
		0.4	0.0028	0.00304	0.0142	0.00104	0.0069	0.00302	0.0131	0.00102
		0.5	0.0026	0.00569	0.0164	0.00447	0.0055	0.00999	0.0180	0.00447
		0.7	−0.0053	0.0369	0.0164	0.0134	−0.2953	0.6195	−0.0158	0.02158
100	α = 1.2 , β = 5	0.2	0.0448	0.0441	0.1363	0.04317	0.0425	0.0454	0.1433	0.0474
		0.4	0.0676	0.0885	0.1467	0.06311	0.0651	0.0910	0.1491	0.0656
		0.5	0.0651	0.1151	0.1364	0.10952	0.0645	0.1246	0.1452	0.1129
		0.7	0.0558	0.1952	0.1163	0.1799	0.0462	0.2421	0.1042	0.1898
	α = 0.7 , β = 1	0.2	0.0021	0.00031	0.0064	0.00024	0.0018	0.00032	0.0055	0.00024
		0.4	0.0070	0.00164	0.0097	0.00069	0.0056	0.00162	0.0079	0.00066
		0.5	0.0064	0.00292	0.0138	0.00262	0.0068	0.00295	0.0116	0.00269
		0.7	0.0056	0.00811	0.0134	0.00766	−0.0166	0.04700	0.0060	0.00817

The usual and increasing columns show the results for the usual estimator and the new increasing estimator. The columns B and MSE provide the bias and the mean-squared error of the corresponding estimator.

Table 2

Simulation results for the gamma distribution

			p (censorship)
			0.05				0.25
			Usual		Increasing		Usual		Increasing
n	Parameters	q	B	MSE	B	MSE	B	MSE	B	MSE
25	α = 1.2 , β = 2	0.2	0.0135	0.00130	0.0328	0.00161	0.0109	0.00120	0.0284	0.00133
		0.4	0.0081	0.00236	0.0258	0.00197	0.0126	0.00258	0.0282	0.00224
		0.5	0.0097	0.00317	0.0245	0.00263	0.0101	0.00661	0.0284	0.00299
		0.7	0.0018	0.01630	0.0221	0.00490	−0.1434	0.22683	0.0150	0.00581
	α = 0.8 , β = 0.5	0.2	0.0179	0.00329	0.0387	0.00288	0.0136	0.00309	0.0337	0.00250
		0.4	0.0124	0.01095	0.0380	0.00840	0.0175	0.01148	0.0441	0.00903
		0.5	0.0120	0.01777	0.0387	0.01368	0.0180	0.02329	0.0485	0.01473
		0.7	0.0048	0.07057	0.0373	0.03267	−0.2865	0.72372	0.0067	0.04465
50	α = 1.2 , β = 2	0.2	0.0045	0.00057	0.0152	0.00056	0.0055	0.00061	0.0154	0.00057
		0.4	0.0072	0.00130	0.0156	0.00115	0.0073	0.00136	0.0163	0.00119
		0.5	0.0073	0.00176	0.0148	0.00155	0.0071	0.00179	0.0153	0.00158
		0.7	0.0076	0.00299	0.0130	0.00286	−0.0009	0.01155	0.0121	0.00281
	α = 0.8 , β = 0.5	0.2	0.0067	0.00159	0.0171	0.00134	0.0071	0.00166	0.0182	0.00135
		0.4	0.0142	0.00585	0.0280	0.00511	0.0109	0.00629	0.0267	0.00537
		0.5	0.0149	0.00917	0.0266	0.00822	0.0105	0.00972	0.0274	0.00848
		0.7	0.0138	0.01977	0.0259	0.01872	−0.0079	0.05097	0.0204	0.02002

• When n increases, the MSE decreases significantly, which indicates that both estimators are consistent.

• As expected, the results indicate that the proposed increasing estimator has smaller MSE than the usual estimator specially for larger deciles, i.e., the increasing estimator outperforms the usual one, since the MSE of the increasing MSE is smaller than the MSE related to usual estimator. Compare the usual and increasing columns of Tables 1 and 2.

5 Application

In this section, a dataset of Mayo Clinic trial in primary biliary cirrhosis (PBC) of the liver is considered. The dataset is reported by Fleming and Harrington [28] and is available in the “survival” library of the R statistical programming language. The dataset consists of a time variable, which shows the number of days between registration and the earlier of death, liver transplantation, or study analysis time in July 1986.

The KM survival function is plotted in Figure 2. Approximately, 60 % of the elements are censored so the survival function can be estimated just up to approximately t = 4 , 000 , which is 0.3534 . Every plus symbol on this plot shows a censored element.

Figure 2

KM survival function of the data.

Figure 3, left side, shows the plots of the usual 0.25 -QPL, 0.5 -QPL, and 0.75 -QPL functions. The right side of this figure presents the increasing estimators i q 0.25 , n ( t ) , i q 0.5 , n ( t ) , and i q 0.75 , n ( t ) . Specially, the computed values of i q 0.5 , n ( t ) at two selected times 1,000 and 3,000 are i q 0.5 , n ( 1,000 ) = 483 and i q 0.5 , n ( 3,000 ) = 1 , 803 , i.e., we expect half of the patients expriencing the event before 1,000 days, experienced it before (after) 1,000 − 483 = 517 days. Similarly, half of the patients how experienced the event before 3,000 days, and experience it before (after) 1,197 days.

$Figure 3 Left: Usual estimators q 0.25 , n ( t ) {q}_{0.25,n}\left(t) , q 0.50 , n ( t ) , {q}_{0.50,n}(t), and q 0.75 , n ( t ) {q}_{0.75,n}\left(t) . Right: the proposed increasing estimators i q 0.25 , n ( t ) i{q}_{0.25,n}\left(t) , iq 0.50 , n ( t ) , {{iq}}_{0.50,n}(t), and i q 0.75 , n ( t ) i{q}_{0.75,n}\left(t) .$

Figure 3

Left: Usual estimators q 0.25 , n ( t ) , q 0.50 , n ( t ) , and q 0.75 , n ( t ) . Right: the proposed increasing estimators i q 0.25 , n ( t ) , iq 0.50 , n ( t ) , and i q 0.75 , n ( t ) .

6 Conclusion

The α -QPL function is an important measure based on the conditional prior. The special case of the 0.5 -QPL function is called the median prior function and proves to be better than the mean prior function, for example, when the underlying distribution is skewed or with a strong tail, the data are heavily censored, or its moments are not finite. Balmert and Jeong [19] and Kayid [21] have proposed an estimator for the α -QPL function based on the KM survival estimator for the right-censored data. The fact that the α -QPL function is increasing for many models and real data sets motivates us to propose a new estimator that takes this into account. It is proved that a suitable standardized process for the new estimator converges weakly to a zero-mean Gaussian process. It also converges strongly against the α -QPL function. The simulation results show that the new estimator is consistent and outperforms the usual estimator in terms of the MSE. A dataset of Mayo Clinic trial in PBC is considered. The usual estimators q 0.25 , n ( t ) , q 0.50 , n ( t ) , and q 0.75 , n ( t ) and the new increasing estimators i q 0.25 , n ( t ) , iq 0.50 , n ( t ) , and iq 0.75 , n ( t ) are computed and plotted for comparison. Future research may involve defining and studying the estimator of the α -QPL function for two ordered lifetime random variables, extending the concept of the α -QPL function and its estimation to a multivariate context, which is still an open problem.

Acknowledgments

The author would like to thank the two anonymous reviewers who kindly reviewed an earlier version of this manuscript and provided valuable suggestions and comments. This research was funded by Researchers Supporting Project Number (RSP2024R392), King Saud University, Riyadh, Saudi Arabia.

Funding information: The study was supported by Researchers Supporting Project Number (RSP2024R392), King Saud University, Riyadh, Saudi Arabia.
Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The author states no conflict of interest.
Data availability statement: The data set considered in this paper is available in the “survival” library of the R statistical programming language version 4.3.0 or later.

References

[1] Barlow RE, Marshall AW, Proschan F. Properties of probability distributions with monotone hazard rate. Ann Math Stat. 1963;34(2):375–89.10.1214/aoms/1177704147Search in Google Scholar

[2] Block HW, Savits TH, Singh H. The reversed hazard rate function. Probab Eng InformSci. 1998;12(1):69–90.10.1017/S0269964800005064Search in Google Scholar

[3] Crescenzo AD. Some results on the proportional reversed hazards model. Statist. Probab Lett. 2000;50(4):313–21.10.1016/S0167-7152(00)00127-9Search in Google Scholar

[4] Chandra NK, Roy D. Some results on reversed hazard rate. Probab Eng Inform Sci. 2001;15(1):95–102.10.1017/S0269964801151077Search in Google Scholar

[5] Finkelstein MS. On the reversed hazard rate. Reliab Eng Syst Saf. 2002;78(1):71–5.10.1016/S0951-8320(02)00113-8Search in Google Scholar

[6] Kundu C, Nanda AK, Hu T. A note on reversed hazard rate of order statistics and record values. J Stat Plann Inference. 2009;139(4):1257–65.10.1016/j.jspi.2008.08.002Search in Google Scholar

[7] Li X, Da D, Zhao P. On reversed hazard rate in general mixture models. Stat Probab Lett. 2010;80(7):654–61.10.1016/j.spl.2009.12.023Search in Google Scholar

[8] Burkschat M, Torrado N. On the reversed hazard rate of sequential order statistics. Stat Probab Lett. 2014;85(2):106–13.10.1016/j.spl.2013.11.015Search in Google Scholar

[9] Esna-Ashari M, Balakrishnan N, Alimohammadi M. HR and RHR orderings of generalized order statistics. Metrika. 2023;86:131–48.10.1007/s00184-022-00865-2Search in Google Scholar

[10] Ahmad IA, Kayid M, Pellerey F. Further results involving the MIT order and the IMIT class. Probab Eng Inform Sci. 2005;19(3):377–95.10.1017/S0269964805050229Search in Google Scholar

[11] Ahmad IA, Kayid M. Characterizations of the RHR and MIT orderings and the DRHR and IMIT classes of life distributions. Probab Eng Inform Sci. 2005;19(4):447–61.10.1017/S026996480505028XSearch in Google Scholar

[12] Asadi M. On the mean past lifetime of the components of a parallel system. J Stat Plann Inference. 2006;136(4):1197–206.10.1016/j.jspi.2004.08.021Search in Google Scholar

[13] Kundu C, Nanda AK. Some reliability properties of the inactivity time. Comm Stat Theory Methods. 2010;39(5):899–911.10.1080/03610920902807895Search in Google Scholar

[14] Eryilmaz S. Mean residual and mean past lifetime of multi-state systems with identical components. IEEE Trans Rel. 2010;59(4):644–9.10.1109/TR.2010.2054173Search in Google Scholar

[15] Asadi M, Berred A. Properties and estimation of the mean past lifetime. Statistics. 2012;46(3):405–17.10.1080/02331888.2010.540666Search in Google Scholar

[16] Bhattacharyya D, Khan RA, Mitra M. A nonparametric test for comparison of mean past lives. Stat Probab Lett. 2020;161(6):108722.10.1016/j.spl.2020.108722Search in Google Scholar

[17] Khan RA, Bhattacharyya D, Mitra M. On some properties of the mean inactivity time function. Stat Probab Lett. 2021;170:108993.10.1016/j.spl.2020.108993Search in Google Scholar

[18] Mahdy M. Further results involving percentile inactivity time order and its inference. Metron. 2014;72:269–82.10.1007/s40300-013-0017-9Search in Google Scholar

[19] Balmert L, Jeong JH. Nonparametric inference on quantile lost lifespan. Biometrics. 2017;73(1):252–59.10.1111/biom.12555Search in Google Scholar PubMed PubMed Central

[20] Balmert LC, Li R, Peng L, Jeong JH. Quantile regression on inactivity time. Stat Methods Med Res. 2021;30(5):1332–46.10.1177/0962280221995977Search in Google Scholar PubMed PubMed Central

[21] Kayid M. Statistical inference of an α-quantile past lifetime function with applications. AIMS Mathematics. 2024;9(6):15346–60. 10.3934/math.2024745.Search in Google Scholar

[22] Unnikrishnan N, Vineshkumar B. Reversed percentile residual life and related concepts. J Korean Stat Soc. 2011;40(1):85–92.10.1016/j.jkss.2010.06.001Search in Google Scholar

[23] Kochar SC, Mukerjee H, Samaniego FJ. Estimation of a monotone mean residual life. Ann Stat. 2000;28:905–21.10.1214/aos/1015952004Search in Google Scholar

[24] Franco-Pereira AM, de Una Alvarez J. Estimation of a monotone percentile residual life function under random censorship. Biom J. 2013;55(1):52–67.10.1002/bimj.201100199Search in Google Scholar PubMed

[25] Shafaei Noughabi M, Franco-Pereira AM. Estimation of monotone bivariate quantile residual life. Statistics. 2018;52(4):919–33.10.1080/02331888.2018.1470180Search in Google Scholar

[26] Csorgo M, Csorgo S. Estimation of percentile residual life. Oper Res. 1987;35:598–606.10.1287/opre.35.4.598Search in Google Scholar

[27] Van der Vaart AW. Asymptotic Statistics. New York: Cambridge University Press; 1998.10.1017/CBO9780511802256Search in Google Scholar

[28] Fleming TR, Harrington DP. Counting Processes and Survival Analysis. New York: Wiley; 1991.Search in Google Scholar

Received: 2024-03-03

Revised: 2024-04-26

Accepted: 2024-05-05

Published Online: 2024-06-03

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0035

Keywords for this article

mathematical physics; monotone quantile of past lifetime function; right-censored data; Kaplan–Meier survival estimator

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