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Reversed percentile residual life and related concepts
Journal of the Korean Statistical Society 40 (2011) 85–92
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Reversed percentile residual life and related concepts N. Unnikrishnan Nair, B. Vineshkumar ∗ Department of Statistics, Cochin University of Science and Technology, Cochin, India
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Article history: Received 6 October 2009 Accepted 3 June 2010 Available online 29 June 2010
In this work we discuss the properties of the reversed percentile residual life function and its relationship with the reversed hazard function. Some models with simple functional forms for both reversed hazard rate and reversed percentile residual life function are proposed. A method of distinguishing decreasing (increasing) reversed hazard rates (reversed percentile lives) is also presented. © 2010 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.
AMS 2000 subject classifications: primary 62N05 secondary 90B25 Keywords: Reversed percentile residual life Reversed hazard rate Characterization
1. Introduction The concept of residual life and various related measures such as mean, variance, coefficient of variation, median and percentiles of residual life have been extensively used to model and analyze lifetime data and in other disciplines such as actuarial science, biometry, survival analysis and economics. Of these, the percentile residual life function occupies an important role, as revealed from the works of Arnold and Brocket (1983), Csorgo and Csorgo (1987), Csorgo and Viharos (1992), Gupta and Langford (1984), Joe (1985), Joe and Proschan (1984a,b), Lillo (2005), Schmittlein and Morrison (1981) and Song and Cho (1995). Gupta and Langford (1984) established that a single percentile residual life function does not characterize a life distribution and provided a comprehensive solution to the problem. Let F (t ) be the distribution function of a lifetime random variable X (such that F (0−) = 0). Then the α th percentile residual life of X for 0 < α < 1 is defined as Rα (t ) = Q {1 − (1 − α) (1 − F (t ))} − t ,
t ≥0
where Q (u) = F −1 (u) = inf {x : F (x) ≥ u} ,
0≤u≤1
is the quantile function corresponding to F . Song and Cho (1995) have shown that if F (t ) is continuous, strictly increasing log(1−α) and for 0 < α, β < 1, log(1−β) is irrational, then F (t ) is uniquely determined by Rα (t ) and Rβ (t ). When F (t ) has a density f (t )
f (t ), the hazard function h(t ) = 1−F (t ) satisfies the equation t +Rα (t )
∫
h(y)dy = − log (1 − α) ,
t ≥ 0.
t
∗
Corresponding author. E-mail address: [email protected] (B. Vineshkumar).
1226-3192/$ – see front matter © 2010 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jkss.2010.06.001
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Recently in a parallel theoretical framework some concepts in reversed time, (to analyze the behaviour of the lifetime random variable X given that failure has occurred in [0, t]) have been successfully employed in various applications. Two important concepts arising in this context that are widely discussed in the literature are the reversed hazard rate (RHR) defined as f (t )
λ(t ) =
(1.1)
F (t )
and the reversed mean residual life (RMRL) r (t ) = E (t − X |X ≤ t ) =
t
∫
1 F (t )
F (y)dy,
(1.2)
0
for all t such that F (t ) > 0. Nanda, Singh, Misra, and Paul (2003) and a review of the literature in Nair and Asha (2004) on reversed hazard rates contain the following properties and applications of λ(t ) and r (t ). These two functions determine the distribution uniquely through the inversion formulae ∞
∫
F (t ) = exp −
λ (y) dy
(1.3)
t
and F (t ) = lim
x→∞
r (x) r (t )
x
∫ exp − t
1 r (y)
dy .
(1.4)
Further,
λ(t ) =
1 − r ′ (t ) r (t )
(1.5)
where the prime denotes the derivative with respect to t. If X has support (a, b), −∞ ≤ a < b ≤ ∞, then
λ−X (t ) = hX (−t ). Hence all the properties of the reversed hazard rate of non-negative random variables, with right end point of the support as +∞, cannot be formally obtained from the hazard rate. Thus when the support is (0, ∞), the former provides some distinct results that are difficult to obtain from the latter. From (1.3) and (1.4) one can identify life distributions from the functional forms of λ(t ) and r (t ), which forms the basis of many characterizations of specific distributions. However, one notable distinction from the usual hazard rate (mean residual life) is that there exists no increasing (decreasing) RHR (RMRL) on the entire positive real line. Thus both RHR and RMRL fail to describe patterns of positive and negative ageing and hence are not applicable in distinguishing life distributions on that basis. For example, when X is exponential, the hazard rate is constant, representing no-ageing, while the corresponding RHR decreases exponentially. The random variable (t − X |X ≤ t ) represents the waiting time (also called inactivity time) since the failure of a device, given that failure has occurred in [0, t]. Hence r (t ) becomes the average waiting time (for example, for repair or a replacement) which is vital in evolving repair/maintenance strategies. Another area in which RHR is involved is in the estimation of the survival function and in modeling and analysis of right truncated and left censored data. System reliability and new stochastic orders based on concepts in reversed time are recent areas where RHR is of potential use. Besides the above, Nanda et al. (2003) also introduce the concepts of variance and coefficients of variations of reversed residual life, stochastic order involving RMRL and its relationship with other partial orders. Characterization of some families of distributions by means of identities involving RHR is also established in Nair and Asha (2004). These involve the Burr system with F (x) = (1 + exp[−G(x)])−1 where G(x) = by
∞ x
(1.6)
g (y)dy is a non-decreasing continuous function satisfying G(−∞) = −∞ and G(∞) = ∞ characterized
g (x) = h(x) + λ(x) or G(x) = log (h(x)/λ(x)) . Their result on the Pearson family is subsumed in a later paper (Nair & Sudheesh, 2006) as follows. Let X be a random variable with support (a, b), −∞ ≤ a < b ≤ ∞ with absolutely continuous distribution function F (x) and density function f (x). For B-measurable functions c (x) and g (x) such that c ′ (x) exists on (a, b) satisfying ′ E c (X )g (X ) < ∞, the following statements are equivalent.
(i) E [a(X )|X ≤ x] = µ − σ λ(x)g (x) f ′ (x) g ′ (x) µ − a(x) (ii) =− + f ( x) g ( x) σ g (x) where a(x) is a B-measurable function with mean µ and variance σ 2 . The Pearson family is a particular case with a(x) = x and g (x) = a0 + a1 x + a2 x2 , where a0 , a1 and a2 are real constants. A list of distributions and families of the form (ii) and the applications of the result in unbiased estimation and reliability theory are also presented in Nair and Sudheesh (2006).
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Although theoretically there is analogy in the works relating to residual and reversed residual life functions, the properties and models relating to them differ substantially to merit the study of the latter. The relevance of various existing concepts in reversed time and the enormous literature on percentile residual lifetime mentioned above motivate us to study the properties of the reversed version of the percentile residual life function in the present paper. Such a study, along with the relationships that reversed percentile residual life has with other concepts used in this connection, does not appear to have been discussed in the literature. In Section 2 we define RPRL, discuss the problem of characterizing life distributions using RPRL and point out the relationship between RPRL and RHR. Some models with simplified expressions for RPRL and RHR and methods to distinguish them are presented in Section 3. 2. Definition and properties Let F (x) be the distribution function of a lifetime random variable (such that F (0−) = 0) with quantile function Q (u). Then the reversed residual life Xt = (t − X |X ≤ t ) has distribution function Ft (x) = 1 −
F (t − x) F (t )
,
0 ≤ x ≤ t.
(2.1)
Accordingly for 0 < α < 1, the α th reversed percentile residual life function of X is defined as Pα (t ) = Ft−1 (α) = inf (x : Ft (x) ≥ α) ,
= inf (x : F (t − x) ≤ (1 − α)F (t )) , = t − F −1 ((1 − α)F (t )) ,
0 ≤ t < Q (1),
(2.2)
with Q (1) = sup (x : F (x) < 1) as the right hand end point of the support of F . From (2.2) we see that the functional equation that solves for Pα (t ) is t − Pα (t ) = F −1 ((1 − α)F (t )) or F (t − Pα (t )) = (1 − α)F (t ).
(2.3)
In terms of the quantile function Q (u) the α -th RPRL, as a function of u, is obtained as pα (u) = Pα (Q (u)) = Q (u) − Q ((1 − α)u) . This definition is quite useful in situations where the quantile functions exist in the simple forms but whose distribution functions do not have closed forms to utilize (2.1) and (2.2). For example, the lambda distribution of Hankin and Lee (2006) specified by Q (u) = cuλ1 (1 − u)−λ2 ,
c , λ1 , λ2 > 0, 0 < u < 1,
has pα (u) = cuλ1 (1 − u)−λ2 − (1 − α)λ1 (1 − (1 − α)u)−λ2 .
There are situations when lifetime data can be modeled by quantile functions of simple forms for which the distribution functions have no explicit form, but have to be evaluated numerically. The study of properties of Pα (t ) using (2.1) and (2.2) becomes difficult and the expression for pα (u) which has an algebraic form becomes handy in such cases. For several examples of such quantile functions and applicability of quantile based reliability concepts, we refer to Nair and Sankaran (2009) and Nair and Vineshkumar (2010). We now discuss some properties of RPRL. First is the problem of characterizing F by the functional form of Pα (t ). We demonstrate through the following example that the RPRL for a given α does not determine F uniquely. In other words the functional equation (2.3) is satisfied by more than one distribution function for a given Pα (t ). Example 2.1. Assuming that X follows the power distribution F (t ) = t a ,
0 ≤ t ≤ 1, a > 0.
From (2.3), for the choice of α = 1 − e−2π a
1
Pα (t ) = 1 − (1 − α) a
t = 1 − e−2π t .
Now consider the distribution specified by G(t ) = t a
1+
1 2
sin log t
,
0 ≤ t ≤ 1, a > 0,
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so that G(t ) ̸= F (t ). Then for the above choice of Pα (t ) and α , G (t − Pα (t )) = (t − Pα (t ))a
−2 π a
1+
= te
1+
1 2
1 2
sin log (t − Pα (t ))
sin log te−2π
1
= t a e−2π a 1 + sin (log t − 2π) 2 1 a −2π a =t e 1 + sin log t
2
= (1 − α)G(t ), so that both G and F have the same RPRL satisfying (2.3). Thus we are lead to the search for some general conditions under which F is determined uniquely. Eq. (2.3) is a particular case of Schroder’s functional equation S (φ(t )) = uS (t ),
0≤t ≤∞
(2.4)
discussed in Gupta and Langford (1984), where 0 < u < 1 and φ(t ) is a continuous and strictly increasing function on (0, ∞) which satisfies φ(t ) > t for all t. The general solution of the equation is S (t ) = S0 (t )K (log S0 (t )) , where K (.) is a periodic function with period − log u and S0 (.) is a particular solution which is continuous and strictly decreasing and satisfies S0 (0) = 1. In our case, in analogy with (2.4), we have from (2.3), φ(t ) = t − Pα (t ), which does not satisfy the requirement φ(t ) > t for the above solution. Therefore we seek the conditions for two distributions to have the same RPRL for a given α . Theorem 2.1. Let F and G be two continuous and strictly increasing distribution functions with corresponding RPRL’s Pα (t ) and Qα (t ). Then a necessary and sufficient condition that Pα (t ) = Qα (t ) for all t ≥ 0 is that F (t ) = G(t ).K (− log G(t )),
(2.5)
where K (.) is a periodic function with period − log(1 − α), 0 < α < 1. Proof. Assume that for a given 0 < α < 1, Pα (t ) = Qα (t ) for all t. Then from (2.2) F −1 ((1 − α)F (t )) = G−1 ((1 − α)G(t )) .
(2.6)
Setting G(t ) = u, 0 < u < 1,
(1 − α)F G−1 (u) = F G−1 (1 − α) u ,
0 < u < 1.
(2.7)
For the function K (t ) = et FG−1 e−t ,
(2.8)
G(t )K (− log G(t )) = F G−1 G(t ) = F (t ),
showing that (2.8) solves (2.5). Further K (t − log(1 − α)) = et −log(1−α) FG−1 elog(1−α)−t
= (1 − α)−1 et FG−1 (1 − α)e−t = (1 − α)−1 et (1 − α)FG−1 e−t = K (t ).
(by (2.7))
Thus K (.) is periodic with period − log(1 − α) and therefore, the condition is necessary. Conversely if (2.5) holds for all t, FG−1 ((1 − α)G(t )) = G G−1 (1 − α)G(t ) K − log GG−1 ((1 − α)G(t ))
= (1 − α)G(t )K (− log(1 − α)G(t )) = (1 − α)G(t )K (− log G(t )) = (1 − α)F (t ). Since F is strictly increasing, G−1 ((1 − α)G(t )) = F −1 ((1 − α)F (t )) and hence Pα (t ) = Qα (t ), as desired.
N. Unnikrishnan Nair, B. Vineshkumar / Journal of the Korean Statistical Society 40 (2011) 85–92
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Remark 2.1. In Eq. (2.7), if we set A(u) = FG−1 (u), we have A ((1 − α)u) = (1 − α)A(u), which is particular case of Schroder’s equation. Theorem 2.2. If F is strictly increasing and continuous and Pα (t ) and Pβ (t ).
log(1−α) log(1−β)
is irrational, then F is uniquely determined by the RPRL’s
Proof. Since F (t ) is a particular solution, from (2.5) another distribution function satisfying (2.5) can be expressed as G(t ) = F (t )K (− log F (t )) . Thus from the condition that Pα (t ) and Pβ (t ) are RPRL’s we have G(t ) = F (t )K1 (− log F (t )) = F (t )K2 (− log F (t )) where K1 and K2 are periodic functions with periods − log(1 − α) and − log(1 − β) respectively. The condition of the irrationality of the periods ensures that G(t ) = cF (t ) where c is a constant. As t → Q (1), c = 1 and hence G(t ) = F (t ). For deducing further properties of RPRL, we find a relationship it has with RHR. From (2.2), F (t − Pα (t )) = (1 − α)F (t ). Using (1.3),
− log F (t ) =
Q (1)
∫
λ(x)dx
t
and hence
∫
t t −Pα (u)
λ(x)dx = − log(1 − α).
Differentiating with respect to t, when Pα (t ) is differentiable, we have
λ (t − Pα (t )) 1 − Pα′ (t ) = λ(t ) or Pα′ (t ) = 1 −
λ(t ) . λ (t − Pα (t ))
(2.9)
From the definition (2.2) we conclude that 0 ≤ Pα (t ) ≤ t, since F −1 ((1 − α)F (t )) < t for every 0 < α < 1. Hence as t tends to zero from above, Pα (t ) = 0. If we assume that Pα (t ) is decreasing we should have Pα (t ) ≤ Pα (0) = 0 for all t > 0, which is impossible. Thus we have (a) There is no strictly decreasing RPRL on the whole positive real line and (b) Whenever λ(t ) is decreasing, Pα (t ) is increasing. These results show that, unlike the percentile residual life functions, Pα (t ) has limited use in describing ageing classes among various life distributions. 3. Models Eq. (2.9) provides a simple identity that relates RPRL with RHR. For many of the standard lifetime models such as the exponential, Weibull, Pareto etc., which have simple forms for the hazard rate, the expression for RHR is more complicated. Even for such models with simple forms for failure rate it is difficult to deduce properties of RHR from them. Hence it is desirable to have models that have simple functional forms for RHR. In the present section, we discuss a general method for obtaining such models from the following theorem. Theorem 3.1. For a non-negative random variable X with hazard rate h(t ), its reciprocal X −1 has RHR λ(t ) that satisfies t 2 h(t ) = λ
1 t
.
Proof. Let G(x) be the distribution of X −1 . Then h( t ) = −
d dt
= t −2 λ
log (1 − F (t )) = −
1 t
d dt
log G
1 t
,
.
It follows that if X has decreasing RHR,
1 X
has increasing failure rate.
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Table 1 RHR and RPRL of distributions. Distribution
F (t )
RPRL
Power
(t /b)a , 0 ≤ t ≤ b e−λ/t , t > 0 Rt −1 c 1 , 0 1+pt λ ,t > 0 exp − σ1t λ/t exp −θ e − 1 , t > 0 θ 1 − e−λt , t > 0 −k 1 + t −c ,t > 0 θ 1 − t −β , t > 0 exp −θ t −β − 1 , 0 < t < 1
Reciprocal exponential Reciprocal beta Reciprocal Lomax Reciprocal Weibull Reciprocal Gompertz Power exponential Burr Generalized power Negative Weibull
RHR
1 − (1 − α)1/a t
at −1
t 2 log(1 − α)/ (t log(1 − α) − λ)
λt −2
1 − (1 − α)1/c t (Rt − 1)
c t (Rt −1)
(1 − α)−1/c − 1 t (pt + 1)
c t (pt +1)
(
t λ+1 σ λ log(1−α) t λ σ λ log(1−α)−1
λt −λ−1
)
t − λ log eλ/t − θ −1 log(1 − α)
t +λ
−1
1/θ
log 1 − (1 − α)
−1
1−e
−λt
t 1 − (1 − α)1/kc 1 + t c 1 − (1 − α)1/k
t − 1 − (1 − α)
t− t
−β
−θ
−1
1/θ
−β −1/β
(1 − t ) −1/β log (1 − α)
θλe−λ/t t −2 θλ eλt − 1
−1/c
kc [t (1 + t c )]−1
−1 βθ t β − 1 θβ t −β−1
Example 3.1. From (1.3), the family of distributions characterized by
−1 λ(t ) = at + bt 2 has distribution function F (t ) =
1a
bt a + bt
,
t > 0.
For b < 0, there is no proper distribution function. When a > 0, b > 0 we have with appropriate reparametrization, F (t ) =
c
pt 1 + pt
,
t > 0, p, c > 0
,
1
while for a < 0, b > 0 F (t ) =
Rt − 1
c
Rt
R
< t < ∞, R, c > 0
and as a → 0, b > 0 λ F (t ) = e− t
t , λ > 0, λ = b−1 .
(3.1)
The distribution defined in (3.1) is called the reciprocal exponential distribution. The reciprocal random variable X −1 has the generalized Pareto distribution with reciprocal linear failure rate discussed in Lai and Xie (2006). The RHR and RPRL of the above distributions and those of others obtained by the same method from some standard life distributions are given in Table 1. The hazard rate properties of these distributions and other reliability aspects are documented in Marshal and Olkin (2007). The fact that λ(t ) (Pα (t )) is non-increasing (non-decreasing) on the entire positive real line leaves little scope for classification or identification of life distributions on the basis of their monotonicity as with the cases of ordinary hazard rate function and percentile residual life. One way of resolving this problem is to compare their growth rates. For the reversed hazard rate we define its growth rate as g (t ) =
1 dλ(t ) , λ(t ) dt
t ≥ 0.
(3.2)
It is easy to see that g (t ) determine λ(t ) up to a constant. Hence the function g (t ) is an appropriate quantity to distinguish a suitable model among the class of decreasing reversed hazard rate distributions. Table 2 exhibits the growth rate and behaviour of some important models. It seems desirable to compare the relative growth rates of one distribution with respect to another distribution to see the extent to which changes are taking place in their reversed hazard rates. Here we study the relative growth rate by comparing the rate of a given distribution with that of the reciprocal exponential. This is motivated by (i) there being no distribution on (0, ∞) with constant growth rate, which would have been the natural choice if one existed (ii) the RHR of reciprocal exponential has a simple form (iii) there are many distributions that have growth rate less or more than that of the reciprocal exponential (iv) X1 has an exponential distribution.
N. Unnikrishnan Nair, B. Vineshkumar / Journal of the Korean Statistical Society 40 (2011) 85–92
91
Table 2 Growth rate of reversed hazard rate. Distribution
g (t )
Power Reciprocal exponential
−t −1 −2t −1
HGR HGR/LGR
(1−2Rt ) t (Rt −1) − t((1pt++pt1))
Reciprocal beta Reciprocal Lomax
LGR HGR
− (λ + 1) t −1 −1 λeλt eλt − 1 −t −1 λt −1 + 2 −1 −β t β−1 t β − 1 − (β + 1) t −1
Reciprocal Weibull Power exponential Reciprocal Gompertz Generalized power Negative Weibull
HGR for 0 ≤ λ ≤ 1, LGR for λ ≥ 1 LGR LGR HGR for t ≥
2 2−β
1/β
and LGR t ≤
2 2−β
1/β
HGR for 0 < β ≤ 1, LGR β ≥ 1
Definition 3.1. A life distribution F is said to have higher growth rate—HGR (lower growth rate—LGR) in reversed hazard rate compared to the reciprocal exponential if gF (t ) ≥ (≤)gRE (t ) for all t > 0 where RE stands for the reciprocal exponential distribution. Using the above definition, we can see that there exist classes of distributions, in the same way as get classes using the monotonicity of failure rates or mean residual life functions. Further when X is HGR (LGR), X1 is DFR (IFR). Example 3.2. From the expressions of g (t ) presented in Table 2 we see that power, reciprocal Lomax, reciprocal Weibull for λ ≤ 1, negative Weibull for β ≤ 1 are HGR distributions, while the reciprocal beta, power exponential, reciprocal Gompertz and Weibull for λ ≥ 1 are LGR. The generalized power law is initially LGR and then HGR with a change point at t =
β1
2 2−β
.
In a similar manner, we define growth rate for the reversed percentile residual life. Definition 3.2. F is said to have higher growth rate in reversed percentile life—HGP (lower growth rate—LGP) if aF (t ) ≥ (≤)aRE (t ) for all t ≥ 0,
(3.3)
where a( t ) =
1
dPα (t )
Pα (t )
dt
.
(3.4)
Although a(t ) determines Pα (t ) up to a constant, the latter does not characterize the corresponding distribution. Example 3.3. The growth rate in reversed percentile residual life of the reciprocal exponential distribution is aRE (t ) =
2λ − t log(1 − α) t (λ − t log(1 − α))
.
For the power distribution the growth rate is ap ( t ) = t − 1 . Hence power distribution has LGP. On the other hand for the reciprocal beta we have aB (t ) =
2Rt − 1 t (Rt − 1)
.
Hence aB (t ) > aRE (t ) for all t < 2λ Rλ + log α ′ fication is well defined.
−1
and hence for this range of t, reciprocal beta has HGP. Thus the classi-
Acknowledgements The authors wish to thank the referees for their valuable comments that enhanced the contents and presentation of the paper. In this work the first author is supported by the University Grant Commission, Government of India and the second author by the Council of Scientific and Industrial Research, Government of India. References Arnold, B. C., & Brocket, P. L. (1983). When does the β -th percentile residual life function determine the distribution. Operations Research, 31, 391–396. Csorgo, M., & Csorgo, S. (1987). Estimation of percentile residual life. Operations Research, 35, 598–606.
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Csorgo, S., & Viharos, L. (1992). Confidence bounds for percentile residual life functions. Journal of Statistical Planning and Inference, 30, 327–337. Gupta, R. C., & Langford, E. S. (1984). On the determination of a distribution by its percentile residual function—a functional equation. Journal of Applied Probability, 21, 120–128. Hankin, R. K. S., & Lee (2006). A new family of non-negative distributions. Australian and New Zealand Journal of Statistics, 48, 67–78. Joe, H. (1985). Characterization of life distributions from percentile residual life functions. Annals of the Institute of Statistical Mathematics, 37, 165–172. Joe, H., & Proschan, F. (1984a). Percentile residual life functions. Operations Research, 32, 668–677. Joe, H., & Proschan, F. (1984b). Comparison of two life distributions on the basis of their percentile residual lives. Canadian Journal of Statistics, 12, 91–97. Lai, C. D., & Xie, M. (2006). Stochastic ageing and dependence in reliability. New York: Springer. Lillo, R. E. (2005). Median residual life and its ageing properties—a characterization theorem. Naval Research Logistics, 52, 370–380. Marshal, A. W., & Olkin, I. (2007). Life distributions. New York: Springer. Nair, N. U., & Asha, G. (2004). Characterizations using failure rate and reversed failure rate. Journal of the Indian Society of Probability and Statistics, 8, 222–232. Nair, N. U., & Sankaran, P. G. (2009). Quantile-based reliability analysis. Communications in Statistics: Theory and Methods, 38, 222–232. Nair, N. U., & Sudheesh, K. K. (2006). Characterization of continuous distributions by variance bounds and their implications to reliability modelling and catastrophe theory. Communications in Statistics: Theory and Methods, 15, 1189–1199. Nair, N. U., & Vineshkumar, B. (2010). L-moments of residual life. Journal of Statistical Planning and Inference, doi:10.1016/j.jspi.2010.03.025. Nanda, A., Singh, H., Misra, N., & Paul, P. (2003). Reliability properties of reversed residual lifetime. Communications in Statistics: Theory and Methods, 32, 2031–2042. Schmittlein, D. C., & Morrison, D. G. (1981). Median residual lifetime—a characterization problem and its applications. Operations Research, 29, 392–399. Song, J. K., & Cho, G. (1995). A note on percentile residual life. Sankhya, Series A, 333–335.
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Reversed percentile residual life and related concepts N. Unnikrishnan Nair, B. Vineshkumar ∗ Department of Statistics, Cochin University of Science and Technology, Cochin, India
article
abstract
info
Article history: Received 6 October 2009 Accepted 3 June 2010 Available online 29 June 2010
In this work we discuss the properties of the reversed percentile residual life function and its relationship with the reversed hazard function. Some models with simple functional forms for both reversed hazard rate and reversed percentile residual life function are proposed. A method of distinguishing decreasing (increasing) reversed hazard rates (reversed percentile lives) is also presented. © 2010 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.
AMS 2000 subject classifications: primary 62N05 secondary 90B25 Keywords: Reversed percentile residual life Reversed hazard rate Characterization
1. Introduction The concept of residual life and various related measures such as mean, variance, coefficient of variation, median and percentiles of residual life have been extensively used to model and analyze lifetime data and in other disciplines such as actuarial science, biometry, survival analysis and economics. Of these, the percentile residual life function occupies an important role, as revealed from the works of Arnold and Brocket (1983), Csorgo and Csorgo (1987), Csorgo and Viharos (1992), Gupta and Langford (1984), Joe (1985), Joe and Proschan (1984a,b), Lillo (2005), Schmittlein and Morrison (1981) and Song and Cho (1995). Gupta and Langford (1984) established that a single percentile residual life function does not characterize a life distribution and provided a comprehensive solution to the problem. Let F (t ) be the distribution function of a lifetime random variable X (such that F (0−) = 0). Then the α th percentile residual life of X for 0 < α < 1 is defined as Rα (t ) = Q {1 − (1 − α) (1 − F (t ))} − t ,
t ≥0
where Q (u) = F −1 (u) = inf {x : F (x) ≥ u} ,
0≤u≤1
is the quantile function corresponding to F . Song and Cho (1995) have shown that if F (t ) is continuous, strictly increasing log(1−α) and for 0 < α, β < 1, log(1−β) is irrational, then F (t ) is uniquely determined by Rα (t ) and Rβ (t ). When F (t ) has a density f (t )
f (t ), the hazard function h(t ) = 1−F (t ) satisfies the equation t +Rα (t )
∫
h(y)dy = − log (1 − α) ,
t ≥ 0.
t
∗
Corresponding author. E-mail address: [email protected] (B. Vineshkumar).
1226-3192/$ – see front matter © 2010 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jkss.2010.06.001
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Recently in a parallel theoretical framework some concepts in reversed time, (to analyze the behaviour of the lifetime random variable X given that failure has occurred in [0, t]) have been successfully employed in various applications. Two important concepts arising in this context that are widely discussed in the literature are the reversed hazard rate (RHR) defined as f (t )
λ(t ) =
(1.1)
F (t )
and the reversed mean residual life (RMRL) r (t ) = E (t − X |X ≤ t ) =
t
∫
1 F (t )
F (y)dy,
(1.2)
0
for all t such that F (t ) > 0. Nanda, Singh, Misra, and Paul (2003) and a review of the literature in Nair and Asha (2004) on reversed hazard rates contain the following properties and applications of λ(t ) and r (t ). These two functions determine the distribution uniquely through the inversion formulae ∞
∫
F (t ) = exp −
λ (y) dy
(1.3)
t
and F (t ) = lim
x→∞
r (x) r (t )
x
∫ exp − t
1 r (y)
dy .
(1.4)
Further,
λ(t ) =
1 − r ′ (t ) r (t )
(1.5)
where the prime denotes the derivative with respect to t. If X has support (a, b), −∞ ≤ a < b ≤ ∞, then
λ−X (t ) = hX (−t ). Hence all the properties of the reversed hazard rate of non-negative random variables, with right end point of the support as +∞, cannot be formally obtained from the hazard rate. Thus when the support is (0, ∞), the former provides some distinct results that are difficult to obtain from the latter. From (1.3) and (1.4) one can identify life distributions from the functional forms of λ(t ) and r (t ), which forms the basis of many characterizations of specific distributions. However, one notable distinction from the usual hazard rate (mean residual life) is that there exists no increasing (decreasing) RHR (RMRL) on the entire positive real line. Thus both RHR and RMRL fail to describe patterns of positive and negative ageing and hence are not applicable in distinguishing life distributions on that basis. For example, when X is exponential, the hazard rate is constant, representing no-ageing, while the corresponding RHR decreases exponentially. The random variable (t − X |X ≤ t ) represents the waiting time (also called inactivity time) since the failure of a device, given that failure has occurred in [0, t]. Hence r (t ) becomes the average waiting time (for example, for repair or a replacement) which is vital in evolving repair/maintenance strategies. Another area in which RHR is involved is in the estimation of the survival function and in modeling and analysis of right truncated and left censored data. System reliability and new stochastic orders based on concepts in reversed time are recent areas where RHR is of potential use. Besides the above, Nanda et al. (2003) also introduce the concepts of variance and coefficients of variations of reversed residual life, stochastic order involving RMRL and its relationship with other partial orders. Characterization of some families of distributions by means of identities involving RHR is also established in Nair and Asha (2004). These involve the Burr system with F (x) = (1 + exp[−G(x)])−1 where G(x) = by
∞ x
(1.6)
g (y)dy is a non-decreasing continuous function satisfying G(−∞) = −∞ and G(∞) = ∞ characterized
g (x) = h(x) + λ(x) or G(x) = log (h(x)/λ(x)) . Their result on the Pearson family is subsumed in a later paper (Nair & Sudheesh, 2006) as follows. Let X be a random variable with support (a, b), −∞ ≤ a < b ≤ ∞ with absolutely continuous distribution function F (x) and density function f (x). For B-measurable functions c (x) and g (x) such that c ′ (x) exists on (a, b) satisfying ′ E c (X )g (X ) < ∞, the following statements are equivalent.
(i) E [a(X )|X ≤ x] = µ − σ λ(x)g (x) f ′ (x) g ′ (x) µ − a(x) (ii) =− + f ( x) g ( x) σ g (x) where a(x) is a B-measurable function with mean µ and variance σ 2 . The Pearson family is a particular case with a(x) = x and g (x) = a0 + a1 x + a2 x2 , where a0 , a1 and a2 are real constants. A list of distributions and families of the form (ii) and the applications of the result in unbiased estimation and reliability theory are also presented in Nair and Sudheesh (2006).
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Although theoretically there is analogy in the works relating to residual and reversed residual life functions, the properties and models relating to them differ substantially to merit the study of the latter. The relevance of various existing concepts in reversed time and the enormous literature on percentile residual lifetime mentioned above motivate us to study the properties of the reversed version of the percentile residual life function in the present paper. Such a study, along with the relationships that reversed percentile residual life has with other concepts used in this connection, does not appear to have been discussed in the literature. In Section 2 we define RPRL, discuss the problem of characterizing life distributions using RPRL and point out the relationship between RPRL and RHR. Some models with simplified expressions for RPRL and RHR and methods to distinguish them are presented in Section 3. 2. Definition and properties Let F (x) be the distribution function of a lifetime random variable (such that F (0−) = 0) with quantile function Q (u). Then the reversed residual life Xt = (t − X |X ≤ t ) has distribution function Ft (x) = 1 −
F (t − x) F (t )
,
0 ≤ x ≤ t.
(2.1)
Accordingly for 0 < α < 1, the α th reversed percentile residual life function of X is defined as Pα (t ) = Ft−1 (α) = inf (x : Ft (x) ≥ α) ,
= inf (x : F (t − x) ≤ (1 − α)F (t )) , = t − F −1 ((1 − α)F (t )) ,
0 ≤ t < Q (1),
(2.2)
with Q (1) = sup (x : F (x) < 1) as the right hand end point of the support of F . From (2.2) we see that the functional equation that solves for Pα (t ) is t − Pα (t ) = F −1 ((1 − α)F (t )) or F (t − Pα (t )) = (1 − α)F (t ).
(2.3)
In terms of the quantile function Q (u) the α -th RPRL, as a function of u, is obtained as pα (u) = Pα (Q (u)) = Q (u) − Q ((1 − α)u) . This definition is quite useful in situations where the quantile functions exist in the simple forms but whose distribution functions do not have closed forms to utilize (2.1) and (2.2). For example, the lambda distribution of Hankin and Lee (2006) specified by Q (u) = cuλ1 (1 − u)−λ2 ,
c , λ1 , λ2 > 0, 0 < u < 1,
has pα (u) = cuλ1 (1 − u)−λ2 − (1 − α)λ1 (1 − (1 − α)u)−λ2 .
There are situations when lifetime data can be modeled by quantile functions of simple forms for which the distribution functions have no explicit form, but have to be evaluated numerically. The study of properties of Pα (t ) using (2.1) and (2.2) becomes difficult and the expression for pα (u) which has an algebraic form becomes handy in such cases. For several examples of such quantile functions and applicability of quantile based reliability concepts, we refer to Nair and Sankaran (2009) and Nair and Vineshkumar (2010). We now discuss some properties of RPRL. First is the problem of characterizing F by the functional form of Pα (t ). We demonstrate through the following example that the RPRL for a given α does not determine F uniquely. In other words the functional equation (2.3) is satisfied by more than one distribution function for a given Pα (t ). Example 2.1. Assuming that X follows the power distribution F (t ) = t a ,
0 ≤ t ≤ 1, a > 0.
From (2.3), for the choice of α = 1 − e−2π a
1
Pα (t ) = 1 − (1 − α) a
t = 1 − e−2π t .
Now consider the distribution specified by G(t ) = t a
1+
1 2
sin log t
,
0 ≤ t ≤ 1, a > 0,
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so that G(t ) ̸= F (t ). Then for the above choice of Pα (t ) and α , G (t − Pα (t )) = (t − Pα (t ))a
−2 π a
1+
= te
1+
1 2
1 2
sin log (t − Pα (t ))
sin log te−2π
1
= t a e−2π a 1 + sin (log t − 2π) 2 1 a −2π a =t e 1 + sin log t
2
= (1 − α)G(t ), so that both G and F have the same RPRL satisfying (2.3). Thus we are lead to the search for some general conditions under which F is determined uniquely. Eq. (2.3) is a particular case of Schroder’s functional equation S (φ(t )) = uS (t ),
0≤t ≤∞
(2.4)
discussed in Gupta and Langford (1984), where 0 < u < 1 and φ(t ) is a continuous and strictly increasing function on (0, ∞) which satisfies φ(t ) > t for all t. The general solution of the equation is S (t ) = S0 (t )K (log S0 (t )) , where K (.) is a periodic function with period − log u and S0 (.) is a particular solution which is continuous and strictly decreasing and satisfies S0 (0) = 1. In our case, in analogy with (2.4), we have from (2.3), φ(t ) = t − Pα (t ), which does not satisfy the requirement φ(t ) > t for the above solution. Therefore we seek the conditions for two distributions to have the same RPRL for a given α . Theorem 2.1. Let F and G be two continuous and strictly increasing distribution functions with corresponding RPRL’s Pα (t ) and Qα (t ). Then a necessary and sufficient condition that Pα (t ) = Qα (t ) for all t ≥ 0 is that F (t ) = G(t ).K (− log G(t )),
(2.5)
where K (.) is a periodic function with period − log(1 − α), 0 < α < 1. Proof. Assume that for a given 0 < α < 1, Pα (t ) = Qα (t ) for all t. Then from (2.2) F −1 ((1 − α)F (t )) = G−1 ((1 − α)G(t )) .
(2.6)
Setting G(t ) = u, 0 < u < 1,
(1 − α)F G−1 (u) = F G−1 (1 − α) u ,
0 < u < 1.
(2.7)
For the function K (t ) = et FG−1 e−t ,
(2.8)
G(t )K (− log G(t )) = F G−1 G(t ) = F (t ),
showing that (2.8) solves (2.5). Further K (t − log(1 − α)) = et −log(1−α) FG−1 elog(1−α)−t
= (1 − α)−1 et FG−1 (1 − α)e−t = (1 − α)−1 et (1 − α)FG−1 e−t = K (t ).
(by (2.7))
Thus K (.) is periodic with period − log(1 − α) and therefore, the condition is necessary. Conversely if (2.5) holds for all t, FG−1 ((1 − α)G(t )) = G G−1 (1 − α)G(t ) K − log GG−1 ((1 − α)G(t ))
= (1 − α)G(t )K (− log(1 − α)G(t )) = (1 − α)G(t )K (− log G(t )) = (1 − α)F (t ). Since F is strictly increasing, G−1 ((1 − α)G(t )) = F −1 ((1 − α)F (t )) and hence Pα (t ) = Qα (t ), as desired.
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Remark 2.1. In Eq. (2.7), if we set A(u) = FG−1 (u), we have A ((1 − α)u) = (1 − α)A(u), which is particular case of Schroder’s equation. Theorem 2.2. If F is strictly increasing and continuous and Pα (t ) and Pβ (t ).
log(1−α) log(1−β)
is irrational, then F is uniquely determined by the RPRL’s
Proof. Since F (t ) is a particular solution, from (2.5) another distribution function satisfying (2.5) can be expressed as G(t ) = F (t )K (− log F (t )) . Thus from the condition that Pα (t ) and Pβ (t ) are RPRL’s we have G(t ) = F (t )K1 (− log F (t )) = F (t )K2 (− log F (t )) where K1 and K2 are periodic functions with periods − log(1 − α) and − log(1 − β) respectively. The condition of the irrationality of the periods ensures that G(t ) = cF (t ) where c is a constant. As t → Q (1), c = 1 and hence G(t ) = F (t ). For deducing further properties of RPRL, we find a relationship it has with RHR. From (2.2), F (t − Pα (t )) = (1 − α)F (t ). Using (1.3),
− log F (t ) =
Q (1)
∫
λ(x)dx
t
and hence
∫
t t −Pα (u)
λ(x)dx = − log(1 − α).
Differentiating with respect to t, when Pα (t ) is differentiable, we have
λ (t − Pα (t )) 1 − Pα′ (t ) = λ(t ) or Pα′ (t ) = 1 −
λ(t ) . λ (t − Pα (t ))
(2.9)
From the definition (2.2) we conclude that 0 ≤ Pα (t ) ≤ t, since F −1 ((1 − α)F (t )) < t for every 0 < α < 1. Hence as t tends to zero from above, Pα (t ) = 0. If we assume that Pα (t ) is decreasing we should have Pα (t ) ≤ Pα (0) = 0 for all t > 0, which is impossible. Thus we have (a) There is no strictly decreasing RPRL on the whole positive real line and (b) Whenever λ(t ) is decreasing, Pα (t ) is increasing. These results show that, unlike the percentile residual life functions, Pα (t ) has limited use in describing ageing classes among various life distributions. 3. Models Eq. (2.9) provides a simple identity that relates RPRL with RHR. For many of the standard lifetime models such as the exponential, Weibull, Pareto etc., which have simple forms for the hazard rate, the expression for RHR is more complicated. Even for such models with simple forms for failure rate it is difficult to deduce properties of RHR from them. Hence it is desirable to have models that have simple functional forms for RHR. In the present section, we discuss a general method for obtaining such models from the following theorem. Theorem 3.1. For a non-negative random variable X with hazard rate h(t ), its reciprocal X −1 has RHR λ(t ) that satisfies t 2 h(t ) = λ
1 t
.
Proof. Let G(x) be the distribution of X −1 . Then h( t ) = −
d dt
= t −2 λ
log (1 − F (t )) = −
1 t
d dt
log G
1 t
,
.
It follows that if X has decreasing RHR,
1 X
has increasing failure rate.
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Table 1 RHR and RPRL of distributions. Distribution
F (t )
RPRL
Power
(t /b)a , 0 ≤ t ≤ b e−λ/t , t > 0 Rt −1 c 1 ,
Reciprocal exponential Reciprocal beta Reciprocal Lomax Reciprocal Weibull Reciprocal Gompertz Power exponential Burr Generalized power Negative Weibull
RHR
1 − (1 − α)1/a t
at −1
t 2 log(1 − α)/ (t log(1 − α) − λ)
λt −2
1 − (1 − α)1/c t (Rt − 1)
c t (Rt −1)
(1 − α)−1/c − 1 t (pt + 1)
c t (pt +1)
(
t λ+1 σ λ log(1−α) t λ σ λ log(1−α)−1
λt −λ−1
)
t − λ log eλ/t − θ −1 log(1 − α)
t +λ
−1
1/θ
log 1 − (1 − α)
−1
1−e
−λt
t 1 − (1 − α)1/kc 1 + t c 1 − (1 − α)1/k
t − 1 − (1 − α)
t− t
−β
−θ
−1
1/θ
−β −1/β
(1 − t ) −1/β log (1 − α)
θλe−λ/t t −2 θλ eλt − 1
−1/c
kc [t (1 + t c )]−1
−1 βθ t β − 1 θβ t −β−1
Example 3.1. From (1.3), the family of distributions characterized by
−1 λ(t ) = at + bt 2 has distribution function F (t ) =
1a
bt a + bt
,
t > 0.
For b < 0, there is no proper distribution function. When a > 0, b > 0 we have with appropriate reparametrization, F (t ) =
c
pt 1 + pt
,
t > 0, p, c > 0
,
1
while for a < 0, b > 0 F (t ) =
Rt − 1
c
Rt
R
< t < ∞, R, c > 0
and as a → 0, b > 0 λ F (t ) = e− t
t , λ > 0, λ = b−1 .
(3.1)
The distribution defined in (3.1) is called the reciprocal exponential distribution. The reciprocal random variable X −1 has the generalized Pareto distribution with reciprocal linear failure rate discussed in Lai and Xie (2006). The RHR and RPRL of the above distributions and those of others obtained by the same method from some standard life distributions are given in Table 1. The hazard rate properties of these distributions and other reliability aspects are documented in Marshal and Olkin (2007). The fact that λ(t ) (Pα (t )) is non-increasing (non-decreasing) on the entire positive real line leaves little scope for classification or identification of life distributions on the basis of their monotonicity as with the cases of ordinary hazard rate function and percentile residual life. One way of resolving this problem is to compare their growth rates. For the reversed hazard rate we define its growth rate as g (t ) =
1 dλ(t ) , λ(t ) dt
t ≥ 0.
(3.2)
It is easy to see that g (t ) determine λ(t ) up to a constant. Hence the function g (t ) is an appropriate quantity to distinguish a suitable model among the class of decreasing reversed hazard rate distributions. Table 2 exhibits the growth rate and behaviour of some important models. It seems desirable to compare the relative growth rates of one distribution with respect to another distribution to see the extent to which changes are taking place in their reversed hazard rates. Here we study the relative growth rate by comparing the rate of a given distribution with that of the reciprocal exponential. This is motivated by (i) there being no distribution on (0, ∞) with constant growth rate, which would have been the natural choice if one existed (ii) the RHR of reciprocal exponential has a simple form (iii) there are many distributions that have growth rate less or more than that of the reciprocal exponential (iv) X1 has an exponential distribution.
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91
Table 2 Growth rate of reversed hazard rate. Distribution
g (t )
Power Reciprocal exponential
−t −1 −2t −1
HGR HGR/LGR
(1−2Rt ) t (Rt −1) − t((1pt++pt1))
Reciprocal beta Reciprocal Lomax
LGR HGR
− (λ + 1) t −1 −1 λeλt eλt − 1 −t −1 λt −1 + 2 −1 −β t β−1 t β − 1 − (β + 1) t −1
Reciprocal Weibull Power exponential Reciprocal Gompertz Generalized power Negative Weibull
HGR for 0 ≤ λ ≤ 1, LGR for λ ≥ 1 LGR LGR HGR for t ≥
2 2−β
1/β
and LGR t ≤
2 2−β
1/β
HGR for 0 < β ≤ 1, LGR β ≥ 1
Definition 3.1. A life distribution F is said to have higher growth rate—HGR (lower growth rate—LGR) in reversed hazard rate compared to the reciprocal exponential if gF (t ) ≥ (≤)gRE (t ) for all t > 0 where RE stands for the reciprocal exponential distribution. Using the above definition, we can see that there exist classes of distributions, in the same way as get classes using the monotonicity of failure rates or mean residual life functions. Further when X is HGR (LGR), X1 is DFR (IFR). Example 3.2. From the expressions of g (t ) presented in Table 2 we see that power, reciprocal Lomax, reciprocal Weibull for λ ≤ 1, negative Weibull for β ≤ 1 are HGR distributions, while the reciprocal beta, power exponential, reciprocal Gompertz and Weibull for λ ≥ 1 are LGR. The generalized power law is initially LGR and then HGR with a change point at t =
β1
2 2−β
.
In a similar manner, we define growth rate for the reversed percentile residual life. Definition 3.2. F is said to have higher growth rate in reversed percentile life—HGP (lower growth rate—LGP) if aF (t ) ≥ (≤)aRE (t ) for all t ≥ 0,
(3.3)
where a( t ) =
1
dPα (t )
Pα (t )
dt
.
(3.4)
Although a(t ) determines Pα (t ) up to a constant, the latter does not characterize the corresponding distribution. Example 3.3. The growth rate in reversed percentile residual life of the reciprocal exponential distribution is aRE (t ) =
2λ − t log(1 − α) t (λ − t log(1 − α))
.
For the power distribution the growth rate is ap ( t ) = t − 1 . Hence power distribution has LGP. On the other hand for the reciprocal beta we have aB (t ) =
2Rt − 1 t (Rt − 1)
.
Hence aB (t ) > aRE (t ) for all t < 2λ Rλ + log α ′ fication is well defined.
−1
and hence for this range of t, reciprocal beta has HGP. Thus the classi-
Acknowledgements The authors wish to thank the referees for their valuable comments that enhanced the contents and presentation of the paper. In this work the first author is supported by the University Grant Commission, Government of India and the second author by the Council of Scientific and Industrial Research, Government of India. References Arnold, B. C., & Brocket, P. L. (1983). When does the β -th percentile residual life function determine the distribution. Operations Research, 31, 391–396. Csorgo, M., & Csorgo, S. (1987). Estimation of percentile residual life. Operations Research, 35, 598–606.
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