Evaluations of the mean residual lifetime of an m -out-of- n system

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Statistics and Probability Letters 80 (2010) 333–342

Contents lists available at ScienceDirect

Statistics and Probability Letters

journal homepage: www.elsevier.com/locate/stapro

Evaluations of the mean residual lifetime of anm-out-of-n systemMohammad Z. Raqab ∗Department of Mathematics, University of Jordan, Amman 11942, Jordan

a r t i c l e i n f o

Article history:Received 13 February 2009Accepted 13 November 2009Available online 26 November 2009

MSC:62G3062E15

a b s t r a c t

Themean residual lifetime is an importantmeasure in the reliability theory and in studyingthe lifetime of a living organism. This paper presents sharp upper bounds on the deviationsof the mean residual lifetime of an m-out-of-n system from the mean of a residual liferandomvariable Xt = (X−t|X > t), for any arbitrary t > 0 in various scale units generatedby central absolutemoments. The results are derived byusing the greatest convexminorantapproximation combined with the Hölder inequality. We also determine the distributionsfor which the bounds are attained. The optimal bounds are numerically evaluated andcompared with other classical rough bounds.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Suppose that X1, X2, . . . , Xn are identically independent distributed (iid) lifetimes of n components connected in an m-out-of-n system having a common continuous distribution function (cdf) F , a probability density function (pdf) f and aquantile F−1 defined by

F−1(u) = supu∈(0,1)

{x : F(x) ≤ u}.

We assume that the expected value and rth absolute moment of a single observation X1,

µ = EF (X1) =∫ 1

0F−1(u)du,

and

σ rr = EF |X1 − µ|r =∫ 1

0|F−1(u)− µ|r du, 1 ≤ r <∞,

exist and are finite. Moreover, if the support of X is bounded, we denote the essential supremum of its deviation from themean by

σ∞ = ess sup |X1 − µ| = sup{|F−1(u)− µ| : 0 < u < 1}.

Let X1:n, X2:n, . . . , Xn:n be the order statistics from the sample of lifetimes X1, X2, . . . , Xn. Actually Xn−m+1:n represents thelife length of an m-out-of-n system consisting of n components. As special cases, X1:n and Xn:n represent the life lengths ofseries (n-out-of-n) and parallel (1-out-of-n) systems, respectively.The mean residual lifetime (MRL) at time t > 0 is defined as

m(t) = E(X − t|X > t) =

∫∞

t F̄(x) dx

F̄(t),

∗ Fax: +962 6 5348932.E-mail address:[email protected].

0167-7152/$ – see front matter© 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.spl.2009.11.007


334 M.Z. Raqab / Statistics and Probability Letters 80 (2010) 333–342

where F̄(x) denotes the survival function of X . The MRL function can be considered as the conditional tail measure giventhat the lifetime X did not fail in (0, t). In the spirit of order statistics, Li and Chen (2004) have discussed the aging propertiesof the residual life length of anm-out-of-n system with independent (not necessarily identical) components given that the(n−m)th failure has occurred at time t > 0. The aging properties of the parallel system have been studied by Abouammohand El-Neweihi (1986). Poursaeed and Nematollahi (2008) have studied the mean past and mean residual life functions ofthe components of parallel system under double monitoring.Generally, we define the MRL of anm-out-of-n system as follows:

Mm,n(t) = E(Xn−m+1:n − t|Xn−m+1:n > t). (1)

For a detailed study of the MRL of other related systems, we refer, for example, Bairamov et al. (2002) and Asadi andBayramoglu (2005, 2006).The subject of bounds for the moments of order statistics and record values received a great attention by several

researchers. Moriguti (1953) suggested sharp bounds for the expectations of single order statistics using a monotoneapproximation of density functions of standard uniform samples by means of the derivatives of the greatest convexminorants of their antiderivatives. Arnold (1985) and Rychlik (1993) presentedmore general sharp bounds for themaximumand arbitrary combination of order statistics, respectively, of possibly dependent samples in terms of central absolutemoments of various orders based on the Hölder inequality. In the context of record values, Grudzień and Szynal (1985)obtained nonsharp bounds for kthe record values. Raqab (1997) improved their results of using theMoriguti method. Raqab(2000) evaluated bounds on expectations of ordinary record values based on the Hölder inequality. Optimal mean–variancebounds for order and record statistics from various restricted nonparametric populations were presented in Rychlik(2001). Raqab and Rychlik (submitted for publication) developed sharp bounds for the MRL of an m-out-of-n system usingthe greatest convex minorant approximation combined with the Cauchy–Schwarz inequality.In this paper, under the condition that the system is working at time t > 0, we establish general sharp bounds for the

deviations of the MRL Mm,n(t) of an m-out-of-n system from the MRL m(t) of X1. The so obtained bounds here are derivedbased on Lr -projection approach (1 ≤ r ≤ ∞) and expressed in various scale units generated by the central absolutemoments.

2. Preliminaries and auxiliary results

Let X1, . . . , Xn be the iid random variables (r.v.’s) with cdf F , pdf f , and finite first and second moments. Assume thatt = F−1(p) (0 ≤ p < 1) is the 100pth percentile of F . Let us denote the mean and the rth absolute central moment ofXt = X − t|X > t by

m(t) = EF (Xt) =11− p

∫ 1

p[F−1(u)−m(t)] du,

and

σ rr (t) = EF(|X1 − t −m(t) |r |X > t

)=

11− p

∫ 1

p|F−1(u)− t −m(t)|r du, 1 ≤ r <∞,

respectively. In addition, we put

σ∞(t) = supu∈(p,1)

{F−1(1−)− t −m(t), t +m(t)− F−1(p)}.

Indeed, as p→ 0,m(t)→ µ and σ rr (t)→ σ rr .Now, consider the deviation of the MRL of anm-out-of-n system from the MRL function of the parent r.v. X1 as

Mm,n(t)−m(t) = EF (Xn−m+1:n − t|Xn−m+1:n > t)− EF (X1 − t|X1 > t)

=

∫ 1

p[F−1(u)− t]

fn−m+1:n(u)

F n−m+1:n(p)du−

∫ 1

p[F−1(u)− t]

du1− p

=1

F n−m+1:n(p)

∫ 1

p[F−1(u)− t −m(t)] [fn−m+1:n(u)− d] du, (2)

for some constant d, where

fn−m+1:n(u) = m( nm

)un−m(1− u)m−1, 0 < u < 1,

Fn−m+1:n(u) =n∑

j=n−m+1

(nj

)uj(1− u)n−j, 0 < u < 1,

are the pdf and cdf of themth greatest order statistics from a sample of size n from uniform U(0, 1), respectively.


M.Z. Raqab / Statistics and Probability Letters 80 (2010) 333–342 335

To evaluate the MRL Mm,n(t) defined in (1), we use inequalities of the integral of the product of two functions suchthat these functions are proportional. The Moriguti projection method (1953) has been effectively exploited in determiningsharp bounds on values of statistical functionals over general and restricted families of distributions. This method isbased on approximating Fn−m+1:n(u) by its greatest convex minorant (GCM) F̃n−m+1:n(u). The GCM of the antiderivativeFn−m+1:n(u) − Fn−m+1:n(p) of fn−m+1:n(u), p < u < 1, is defined as the supremum of all convex functions dominated byFn−m+1:n(u)− Fn−m+1:n(p), p < u < 1 and its form is defined as follows:

F̃n−m+1:n(u) =

{Fn−m+1:n(u)− Fn−m+1:n(p), if p < u ≤ um,n,Fn−m+1:n(um,n)− Fn−m+1:n(p)+fn−m+1:n(um,n)(u− p), if um,n < u < 1,

for a unique u = um,n ∈ (p, θm,n)with θm,n = (n−m)/(n− 1) satisfying

1− Fn−m+1:n(u) = fn−m+1:n(u)(1− u). (3)

If p ≥ um,n, then

F̃n−m+1:n(u) =F n−m+1:n(p)1− p

(u− p), p < u < 1.

The derivative of F̃n−m+1:n(u) is f̃n−m+1:n(u) = fm:n(min{u, um,n}). Function f̃n−m+1(u) is called the projection of fm:n(u) ontothe convex cone of non-decreasing functions in L2([0, 1], du). The following lemma follows fromMoriguti (1953), Theorem1). We assume that all the integrals treated below are finite.

Lemma 1 (Moriguti’s Inequality). Let h denote an integrable function on an interval [a, b] and let h̃ denote the right derivativeof the greatest convex minorant, say H̃, of H(x) =

∫ xa h(t)dt. Then, for every non-decreasing functionw on [a, b], we have∫ b

aw(x) h(x) dx ≤

∫ b

aw(x) h̃(x) dx. (4)

Equality holds in (4) if and only if w is constant on every open interval where H > H̃. �

Applying the Hölder inequality to (2), we have for 1 ≤ r <∞

Mm,n(t)−m(t) ≤1

F n−m+1:n(p)

{∫ 1

p

[F−1(u)− t −m(t)

]rdu}1/r {∫ 1

p[fn−m+1:n(u)− d]s du

}1/s= σr(t)

(1− p)1/r

F n−m+1:n(p)‖fn−m+1:n − d‖s, (5)

where s = +∞, r/(r − 1), 1 for r = 1, 1 < r < +∞, and r = +∞, respectively, and

‖g‖r =[∫ 1

pg r(u) du

]1/r,

defines the norm of g ∈ Lr([p, 1], du) and ‖g‖s is defined analogously for the conjugate exponent s = r/(r − 1). Settingd = 0, we obtain the following classical bound on

(Mm,n(t)−m(t)

)/σr(t) as

Bcs (m, n, p) =(1− p)

s−1s

F n−m+1:n(p)‖fn−m+1:n‖s. (6)

Since the constant d = 0 does not minimize the norm ‖fn−m+1:n − d‖s, the bound in (6) is not optimal bound. Further,the equality in (5) holds iff

F−1(u)− t −m(t) = gr (fn−m+1:n(u)− d) , (7)

for monotone function gr(u) = α|u|1/(r−1)sgn(u) for some α ≥ 0 and properly chosen d = d∗. The right hand side of (7) isnot monotonic in general. So the bound in (6) is also not attainable. Optimal sharp bounds can be investigated for generaldistribution by projecting fn−m+1:n(u) onto the family of non-decreasing functions in Lr([p, 1], du).



3. Main results

In this section we present sharp upper bounds on MRL Mm,n(t) of an m-out-of-n system from general populationsexpressed in terms of scale units σr(t) generated by the rth central absolute moments. The resulting bounds aremean–variance bounds and they are valid for all parent distribution functions F and any arbitrary t being the 100pthpercentile of F(0 ≤ p < 1).

Theorem 1. For a parent distribution with finite mean µ, rth absolute central moment σ rr , 1 < r < +∞, and arbitrary1 < m < n with p < um,n, we have,

Mm,n(t)−m(t)σr(t)

≤ Bs(m, n, p), (8)

with

Bss(m, n, p) =(1− p)s−1

[F n−m+1:n(p)]s‖fn−m+1:n − fn−m+1:n(c∗)‖ss

=(1− p)s−1

[F n−m+1:n(p)]s

{∫ c∗

p

(fn−m+1:n(c∗)− fn−m+1:n(u)

)s du+

∫ um,n

c∗

(fn−m+1:n(u)− fn−m+1:n(c∗)

)s du+ (1− um,n) (fn−m+1:n(um,n)− fn−m+1:n(c∗))s}where d∗ = fn−m+1:n(c∗) and c∗ ∈ (p, um,n) is the unique solution to the equation∫ c

p(fn−m+1:n(c)− fn−m+1:n(u))s−1 du =

∫ um,n

c(fn−m+1:n(u)− fn−m+1:n(c))s−1 du

+ (1− um,n)(fn−m+1:n(um,n)− fn−m+1:n(c)

)s−1. (9)

Furthermore, under the additional condition

m(t) ≥[fn−m+1:n(c∗)− fn−m+1:n(p)

Bs(m, n, r)F n−m+1:n(p)

]1/(r−1)σr(t), (10)

the distribution attaining the bound (8) has cdf of the form

F(x) =

F0(x), if x < t,p, if t ≤ x < x1,

f −1n−m+1:n

(d∗ + sgn(x− t −m(t))Bs(m, n, p)

F n−m+1:n(p)1− p

∣∣∣∣x− t −m(t)σr(t)

∣∣∣∣r−1), if x1 ≤ x < x2,

1, if x ≥ x2,

(11)

where F0 is an arbitrary continuous function with F0(t) = p, and

x1 = t +m(t)−[(1− p) (fn−m+1:n(c∗)− fn−m+1:n(p))

Bs(m, n, r)F n−m+1:n(p)

]1/(r−1)σr(t),

x2 = t +m(t)+

[(1− p)

(fn−m+1:n(um,n)− fn−m+1:n(c∗)

)Bs(m, n, r)F n−m+1:n(p)

]1/(r−1)σr(t).

Proof. By (2), (4) and the Hölder inequality, we have

Mm,n(t)−m(t) =1

F n−m+1:n(p)

∫ 1

p

[F−1(u)− t −m(t)

][fn−m+1:n(u)− d] du

≤1

F n−m+1:n(p)

∫ 1

p

[F−1(u)− t −m(t)

] [̃fn−m+1:n(u)− d

]du,

≤(1− p)(s−1)/s

F n−m+1:n(p)‖̃fn−m+1:n − d‖s σr(t), (12)



for arbitrary real d. Now we find d that minimizes ‖̃fn−m+1:n − d‖s. Indeed, we can confine our attention to the points d’ssuch that d = fn−m+1:n(c), p ≤ c ≤ um,n, of the range of the projection function f̃n−m+1:n. It is easily checked that the norm‖̃fn−m+1:n− fn−m+1:n(c)‖s is minimized by choosing c = c∗, the unique solution to (9). In consequence, the optimal bound isreadily obtained in (8) and can be computed based on the form given (9) with c = c∗ ∈ (p, um,n). The latter equality in (12)holds if

F−1(u)− t −m(t) = α|̃fn−m+1:n(u)− fn−m+1:n(c)|1/(r−1)sgn(̃fn−m+1:n(u)− fn−m+1:n(c)

), (13)

for some α ≥ 0. The rth absolute central moment condition implies

α =

(1− p

F n−m+1:n(p) Bs(m, n, p)

)1/(r−1)σr(t).

Plugging the value of α in (13), we have

F−1(u)− t −m(t)σr(t)

=(1− p)1/(r−1)|fn−m+1:n(u)− d∗|1/(r−1) sgn (fn−m+1:n(u)− d∗)

[F n−m+1:n(p) Bs(m, n, p)]1/(r−1). (14)

The former equality in (12) is attained iff F−1(u)−t−m(t) is constant on (um,n, 1), which is already satisfied by the conditionin (14). For u ∈ (p, 1), the quantile function (14) can be rewritten as

F−1(u)− t −m(t)σr(t)

=

−

[(1− p)(d∗ − fn−m+1:n(u))


]1/(r−1), if p ≤ u < c∗,[

(1− p)(fn−m+1:n(u)− d∗)


]1/(r−1), if c∗ ≤ u < um,n,[

(1− p)(fn−m+1:n(um,n)− d∗)


]1/(r−1), if um,n ≤ u < 1.

(15)

Combining (15) and the assumption

F−1(u) ≤ t, 0 < u < p,

we describe the distribution function attaining the resulting bound as (11). The Condition (10) is required to assure thenon-negativity jump at t . �

Remark 1. The distribution function in (11) is a location-scale family of distributions consisting any arbitrary continuousfunction supported on the left of t with probability p, constant supported on (t, x1) and an inverse of polynomial functionsupported on right tail of x1 with probability 1− p. By the monotonicity of fn−m+1:n(u) on (p, um,n), then the right hand sideof (10) is positive. The arbitrary distribution function F0 with F0(t) = p defined on the left tail will appear frequently inall other distributions attaining the bounds in this section. Further, this distribution is not absolutely continuous function.It has an atom of measure 1 − um,n on the right end of the support interval. However, this distribution function can beapproximated by sequences of absolutely continuous functions attaining the bound asymptotically.

Next, we present optimal bounds for the extreme cases r = 1 and r = ∞ using different techniques. The distributionsattaining the bounds are also different from those obtained for 1 < r <∞.

Theorem 2. Let a parent distribution with finite mean µ and first central absolute moment σ1 < +∞. If 1 ≤ m < n withp < um,n, then

Mm,n(t)−m(t)σ1(t)

≤ B∞(m, n, p) =(1− p)

[fn−m+1:n(um,n)− fn−m+1:n(p)

]2 F n−m+1:n(p)

. (16)

The equality in (16) is attained in the limit by the family of distributions of the form

F(x) =

F0(x), if x < t,p, if t ≤ x < t +m(t),

um,n, if t +m(t) ≤ x < t +m(t)+(1− p)σ1(t)2(1− um,n)

,

1, if x ≥ t +m(t)+(1− p)σ1(t)2(1− um,n)

.

(17)

where F0 is an arbitrary continuous function except for an atom at (1− p)σ1(t)/2ε with ε→ 0 and F0(t) = p.



Proof. By (2), and (4), we have

Mm,n(t)−m(t) ≤1

F n−m+1:n(p)

∫ 1

p

[F−1(u)− t −m(t)

] [̃fn−m+1:n(u)− d

]≤

1

F n−m+1:n(p)supu∈(p,1)

|̃fn−m+1:n(u)− d|∫ 1

p|F−1(u)− t −m(t)| du

=1− p

F n−m+1:n(p)supx∈(p,1)

|̃fn−m+1:n(u)− d| σ1(t), (18)

for arbitrary d. From the fact that f̃n−m+1:n is non-decreasing function, the minimized value of the supremum is obtainedwhen d = d∗ =

(fn−m+1:n(p)+ fn−m+1:n(um,n)

)/2, p < um,n and then

supu∈(p,1)

|̃fn−m+1:n − d∗| =fn−m+1:n(um,n)− fn−m+1:n(p)

2.

As a result, the bound in (16) follows immediately.Now, suppose that p < um,n for 1 ≤ m < n. The former equality in (18) is attained iff F−1(u) − t − m(t) is constant

on (um,n, 1). The condition for the equality in the latter one is that F−1(u) − t − m(t) is almost surely negative, zero, andpositive on the sets {0}, (p, um,n), and (um,n, 1), respectively. Therefore,

F−1(u)− t −m(t) = −ξ1 (say) with probability ε= 0 with probability um,n − p= +ξ2 (say) with probability 1− um,n,

where ε → 0. Using the moment conditions E(X − t − m(t)|X > t) = 0 and E ((|X − t −m(t)|)|X > t) = σ1(t), theformer equality in (18), and the assumption F−1(u) ≤ t for u ∈ (0, p) we describe the distribution function attaining thebound as given in (17). �

Remark 2. The distribution function (17) is consisting of a continuous function with an atom at t+m(t)− (1− p)σ1(t)/2εof jump ε, ε→ 0 on the left end with probability p and two other atoms at t+m(t), t+m(t)+ (1− p)σ1(t)/2(1−um,n) ofjumps um,n − p and 1− um,n, respectively. Form = 1, the bound and its respective attaining distribution are obtained from(16) and (17), respectively, by setting um,n = 1.

We now turn to the case r = ∞.

Theorem 3. Assume that X1 is bounded almost surely. Let 1 ≤ m < n and r = ∞. If um,n is the unique solution to (3). Then

Mm,n(t)−m(t)σ∞(t)

≤ B1(m, n, p) =

1+ Fn−m+1(p)− 2Fn−m+1(

p+12 )

F n−m+1(p), if um,n ≥

p+ 12

,

(1− p)F n−m+1(um,n)− F n−m+1(p)(1− um,n)

(1− um,n)F n−m+1(p), if um,n <

p+ 12

.

(19)

If um,n ≥ (p + 1)/2, then the bound (19) is attained in the limit by continuous distribution functions converging weakly to thedistribution function of the form

F(x) =

F0(x), if x < t,p, if t ≤ x < t +m(t)− σ∞(t),p+ 12

, if t +m(t)− σ∞(t) ≤ x < t +m(t)+ σ∞(t),1, if x ≥ t +m(t)+ σ∞(t).

(20)

Moreover if um,n < (p+ 1)/2, the bound is attained in the limit by the following distribution function

F(x) =

F0(x), if x < t,p, if t ≤ x < t +m(t)− σ∞(t),

um,n, if t +m(t)− σ∞(t) ≤ x < t +m(t)+um,n − p1− um,n

σ∞(t),

1, if x ≥ t +m(t)+um,n − p1− um,n

σ∞(t).

(21)



Proof. Arguments similar to those in the proof of Theorem 1, we have

Mm,n(t)−m(t) ≤1

F n−m+1:n(p)‖̃fn−m+1:n(u)− d‖1 σ∞(t). (22)

Now we look for d∗ = fn−m+1:n(c∗) that minimizes the norm

‖̃fn−m+1:n − d‖1 =∫ c

p(fn−m+1:n(c)− fn−m+1:n(u)) du+

∫ um,n

c(fn−m+1:n(u)− fn−m+1:n(c)) du

+(fn−m+1:n(um,n)− fn−m+1:n(c)

)(1− um,n)

= (2c − p− 1) fn−m+1:n(c)− 2Fn−m+1:n(c)+ Fn−m+1:n(p)+ 1. (23)

The differentiation of the right hand side of the latter identity leads to (2c − p − 1)f ′n−m+1:n(c), with f′

n−m+1:n(c) > 0for c < um,n. Consequently, the optimal constant c = c∗ that minimizes the norm (23) is c∗ = min{um,n, (p + 1)/2}. Forum,n ≥ (p+ 1)/2,

d∗ = fn−m+1:n

(p+ 12

)= m

( nm

)(p+ 12

)n−m (1− p2

)m−1. (24)

By (3) and (24), we have

‖̃fn−m+1:n − d‖1 = 1+ Fn−m+1:n(p)− 2Fn−m+1:n

(p+ 12

).

For um,n < (p+ 1)/2, the norm is computed to be

‖̃fn−m+1:n − d∗‖1 =1− p1− um,n

F n−m+1:n(um,n)− F n−m+1:n(p).

Combination of the norm evaluations for both cases leads to the bound evaluation in (19). If um,n ≥ (p+ 1)/2, the boundis attained if F−1(u) − t − m(t) = sgn(u − (p + 1)/2)σ∞(t). Using the moment conditions E(X − t|X > t) = m(t)and supu∈(p,1) (|X − t −m(t)| |X > t) = σ∞(t) and the fact that F−1(u) ≤ t for u ∈ (0, p), the distribution function (20)follows immediately. When um,n < (p + 1)/2, the bound is attained iff F−1(u) − t − m(t) = −σ∞ for p < u < um,n andF−1(u)− t −m(t) is constant for um,n < u < 1. Precisely, we have

F−1(u)− t −m(t) = −σ∞ with probability um,n − p= ζ (say) with probability 1− um,n.

and then the distribution function attaining the bound is described in (21). �

Let us consider the extreme order statistics for (1 = m < n, n ≥ 2) and (m = n or 1 < m < n with p ≥ um,n). In theformer case, Fn:n(u) is convex function on (p, 1) and the problem is reduced to obtain the optimal constant d∗ = fn:n(c∗),c∗ ∈ (p, 1) that minimizes ‖̃fn:n − fn:n(c)‖ss. In this case, the optimal constant c = c

∗ is determined via solving the followingequation∫ c

p

(cn−1 − un−1

)s−1du =

∫ 1

c

(un−1 − cn−1

)s−1du, (25)

which gives a unique solution c∗. If either m = n or 1 < m < n with p ≥ um,n, then the GCM’s of the antiderivativesFn−m+1:n(u)− Fn−m+1:n(p) amount to (1− p)n−1 and F n−m+1:n(p)/(1− p), respectively. This forces that the right hand sideof (12) is 0. The bounds and their respective attaining distributions are presented in the two subsequent theorems.

Theorem 4. For m = 1, n ≥ 2 (parallel system) with 1 < r <∞, we have

Mm,n(t)−m(t)σr(t)

≤ Bs(1, n, p) =(1− p)(s−1)/s

F n:n(p)‖fn:n − fn:n(c∗)‖s, (26)

with c∗ satisfying (25). The bound is attained in the limit by continuous distribution function converging weakly to the cdf of theform

F(x) =

F0(x), if x < t,p, if t ≤ x < x3,[1n

(nc∗n−1 + sgn(x− t −m(t))Bs(1, n, p)

1− pn

1− p

∣∣∣∣x− t −m(t)σr(t)

∣∣∣∣r−1)]1/(n−1)

, if x3 ≤ x < x4,

1, if x ≥ x4,

(27)



Table 1Values of Bc

∞(m, n, p) and B∞(m, n, p) for various choices ofm, n and p in standard deviation units σ1(t).

n m p = 0.05 p = 0.1 p = 0.2Bc∞(m, n, p) B∞(m, n, p) Bc

∞(m, n, p) B∞(m, n, p) Bc

∞(m, n, p) B∞(m, n, p)

5 1 4.7500 2.3750 4.5001 2.2498 4.0013 1.99743 1.7833 0.5374 1.7021 0.4334 1.5922 0.18255 6.1387 0.0000 7.6208 0.0000 12.2070 0.0000

10 1 9.5000 4.7500 9.0000 4.5000 8.0000 4.00003 2.9080 1.4833 2.7549 0.9495 2.4490 0.84295 2.4717 0.6630 2.3420 0.6246 2.0948 0.49558 2.9418 0.2000 2.9629 0.0000 3.6129 0.000010 15.8667 0.0000 25.8117 0.0000 74.5058 0.0000

20 1 19.0000 9.5000 18.0000 9.0000 16.0000 8.00003 5.4340 1.9158 5.1480 1.8149 4.5760 1.61335 4.1728 1.2132 3.9532 0.9866 3.5139 1.02178 3.5532 0.8258 3.3662 0.7824 2.9922 0.695110 3.4371 0.7535 3.2562 0.6590 2.8960 0.576012 3.4745 0.6091 3.2918 0.5750 2.9554 0.383815 3.8791 0.5093 3.7155 0.2618 4.0605 0.000020 53.0007 0.0000 148.0550 0.0000 1387.7800 0.0000

where

x3 = t +m(t)−

npn−1(1− p)(( c∗

p )n−1− 1

)(1− pn)Bs(1, n, p)

1/(r−1) σr(t),x4 = t +m(t)+

[n(1− p)(1− c∗n−1)(1− pn)Bs(1, n, p)

]1/(r−1)σr(t).

Theorem 5. If either m = n or 1 < m < n with p ≥ um,n, then for 1 ≤ r ≤ ∞,

Mm,n(t) ≤ m(t), (28)

and the equality is attained in the limit by

F(x) =

{F0(x), if 0 ≤ x ≤ t,p, if t < x < t +m(t),1, if x ≥ t +m(t).

(29)

4. Numerical study and discussion

In this section, we have conducted a numerical study to evaluate the resulting bounds on the MRL of the componentsof an m-out-of-n system (Mm,n(t)− m(t))/σr(t) under the condition that this system did not fail in (0, t). The evaluationsare performed for n = 5, 10, 20 and arbitrary values of t being the percentiles of F of orders p = 0.05, 0.1, 0.2 with someselected values ofm, in terms of central moments σr(t) of orders r = 1, 3, and∞.The values of um,n’s are numerically obtained via solving (3). In consequence, Eq. (9) can be solved to determine the

parameters c∗’s. Eqs. (3) and (9) are numerically solved by means of the Newton–Raphson method. The solutions of theseequations are substituted in (8). Further, from (16) and (19), we compute the bounds for the extreme cases r = 1 andr = ∞, respectively. For comparison purposes, we have also computed the rough bounds Bcs (m, n, p) for r = 1 and r = ∞as follows:

Bc∞(m, n, p) =

1− p

F n−m+1:n(p)supu∈(p,1)

|fn−m+1:n(u)| =1− p

F n−m+1:n(p)fn−m+1:n(θm,n)

and

Bc1(m, n, p) =1

F n−m+1:n(p)‖fn−m+1:n‖1 = 1.

For 1 < r <∞, Bcs (m, n, p) can be computed from (6).The so obtained results for r = 1, 3, and∞ are presented in Tables 1–3, respectively. The bounds Bcs (m, n, p) are not

presented in Table 3 for r = ∞ which are equal 1 for all m and n. From Tables 1–3, it is observed that Bs(m, n, p) comparewell with the bounds Bcs (m, n, p) derived directly by applying the Hölder inequality only. Actually, the bounds decreases



Table 2Values of Bcs (m, n, p) and Bs(m, n, p) for various choices ofm, n and p in standard deviation units σ3(t).

n m p = 0.05 p = 0.1 p = 0.2Bcs (m, n, p) Bs(m, n, p) Bcs (m, n, p) Bs(m, n, p) Bcs (m, n, p) Bs(m, n, p)

5 1 1.3432 1.1338 1.3192 1.0955 1.2688 1.01873 1.0949 0.2389 1.0816 0.1680 1.0719 0.07485 1.1470 0.0000 1.3664 0.0000 1.3664 0.0000

10 1 1.6532 1.5741 1.6237 1.5387 1.5612 1.46543 1.2631 0.8786 1.2405 0.8315 1.1928 0.73735 1.2006 0.4625 1.1793 0.3944 1.1400 0.26038 1.2737 0.0170 1.2896 0.0000 1.3517 0.000010 1.6817 0.0000 1.68172 0.0000 1.6817 0.0000

20 1 2.0593 2.0331 2.0225 1.9943 1.9447 1.91283 1.5389 1.3687 1.5114 1.3335 1.4532 1.26165 1.4154 1.0801 1.3901 1.0388 1.3366 0.95588 1.3437 0.7401 1.3197 0.6845 1.2690 0.572610 1.3293 0.5405 1.3056 0.4748 1.2560 0.338712 1.3340 0.3626 1.3103 0.2859 1.2702 0.128115 1.3828 0.1289 1.3698 0.0333 1.4388 0.000020 2.0948 0.0000 2.09480 0.0000 2.0948 0.0000

Table 3Values of Bc1(m, n, p) and B1(m, n, p) for various choices ofm, n and p in standard deviation units σ∞(t).

n m B1(m, n, 0.05) B1(m, n, 0.10) B1(m, n, 0.20)

5 1 0.9202 0.8994 0.84513 0.1392 0.0873 0.01715 0.0000 0.0000 0.0000

10 1 0.9968 0.9949 0.98793 0.8508 0.8009 0.66565 0.3263 0.2566 0.12408 0.0041 0.0000 0.000010 0.0000 0.0000 0.0000

20 1 1.0000 1.0000 0.99993 0.9991 0.9981 0.99285 0.9784 0.9623 0.89818 0.6517 0.5647 0.390910 0.3913 0.3180 0.172212 0.2182 0.1542 0.036215 0.0522 0.0079 0.000020 0.0000 0.0000 0.0000

as r increases when m is held fixed. This observation can be explained by the fact that the bounds become tighter for thenarrower classes when treating σr(t) as a general scale parameter. On the other hand, while the non-monotonic behavior isobserved for the rough bounds Bcs (m, n, p), the bounds Bs(m, n, p) expressed in various scale units, get sharper asm increaseswhen n and p are held fixed.

Acknowledgements

The author would like to thank the University of Jordan for supporting this research work. Thanks are also due to thereferees for their helpful comments and suggestions.

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Documents

Evaluations of the mean residual lifetime of an m -out-of- n system