Combination of mean residual life order with reliability applications

Statistical Methodology 29 (2016) 51–69

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Statistical Methodology journal homepage: www.elsevier.com/locate/stamet

Combination of mean residual life order with reliability applications M. Kayid a,b,∗ , S. Izadkhah c , H. Alhalees a a

Department of Statistics and Operations Research, College of Science, King Saud University, Riyadh 11451, Saudi Arabia b

Department of Mathematics and Computer Science, Faculty of Science, Suez University, Suez 41522, Egypt

c

Department of Statistics, Faculty of Mathematics and Computing, Higher Education Complex of Bam, Bam, Kerman, Iran

article

info

Article history: Received 28 September 2014 Received in revised form 14 September 2015 Accepted 23 October 2015 Available online 31 October 2015 MSC: 62N05 62E10 60K20

abstract The purposes of this paper are to introduce a new stochastic order and to study its reliability properties. Some characterizations and preservation properties of the new order under reliability operations of monotone transformation, mixture, weighted distributions and shock models are discussed. In addition, a new class of life distributions is proposed, and some of its reliability properties are investigated. Finally, to illustrate the concepts, some applications in the context of reliability theory and life testing are presented. © 2015 Elsevier B.V. All rights reserved.

Keywords: Mean residual life Hazard rate Characterization Preservation Mixture Shock models Excess lifetime models Aging notions Hypothesis testing applications

∗ Corresponding author at: Department of Statistics and Operations Research, College of Science, King Saud University, Riyadh 11451, Saudi Arabia. E-mail address: [email protected] (M. Kayid). http://dx.doi.org/10.1016/j.stamet.2015.10.001 1572-3127/© 2015 Elsevier B.V. All rights reserved.

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M. Kayid et al. / Statistical Methodology 29 (2016) 51–69

1. Introduction and motivations In the context of reliability and survival analysis, the goal of explicitly calculating a probability distribution is rarely attainable. This makes alternative methods of analysis attractive. One approach is the notion of stochastic orders. Two well-known stochastic orders that have been introduced and studied in reliability theory are the hazard rate (HR) and the mean residual life (MRL) orders (see, Shaked and Shanthikumar [46] and Müller and Stoyan [40] for a broad perspective of the theory and utility of stochastic orderings). Throughout this paper X and Y are two non-negative random variables with distribution functions F and G, respectively. Denote by f the density function of X and by F = 1 − F the corresponding survival function. We use a similar notation for all other distribution functions. If

¯ (t )/F¯ (t ) is non-decreasing in t ≥ 0, G then X is said to be smaller than Y in the HR order (denoted as X ≤HR Y ). If ∞



¯ (u)du/ G t

∞



F¯ (u)du is non-decreasing in t ≥ 0, t

then X is said to be smaller than Y in the MRL order (denoted by X ≤MRL Y ). Another partial ordering, known as combination convexity (CCX) order has been considered and studied by Alzaid [8] and Sekeh et al. [45]. If ∞



xF¯ (x)dx ≤

∞



t

¯ (x)dx, xG

for all t ≥ 0,

t

then X is said to be smaller than Y in the CCX order (denoted by X ≤CCX Y ). In renewal theory, the equilibrium distribution arises as the limiting distribution of the forward recurrence time in a renewal process. Formally, let {Xk , k = 1, 2, . . .} be a sequence of mutually independent and identically distributed (i.i.d.) nnon-negative random variables with common distribution function F . For n ≥ 1, denote by Sn = i=1 Xi the time of the nth arrival and let S0 = 0. Let N (t ) = Sup {n : Sn ≤ t } represent the number of arrivals during the interval [0, t ]. Then, N = {N (t ), t ≥ 0} is a renewal process with underlying distribution F (Ross [44]). Let γ (t ) be the excess lifetime at time t ≥ 0, that is, γ (t ) = SN (t )+1 − t. Denote by M (t ) = E [N (t )] the renewal function which satisfies the following equation M (t ) = F (t ) +

t



F (t − y)dM (y),

t ≥ 0.

0

According to Barlow and Proschan [12], for all t ≥ 0 and x ≥ 0: P [γ (t ) > x] = F¯ (t + x) +



t

F¯ (t + x − u)dM (u).

0

By the elementary renewal theorem, it is straightforward to conclude

 F (x) = lim P (γ (t ) ≤ x) t →∞  x 1 = F¯ (u)du, x ≥ 0. µ 0

(1.1)

The distribution function given in (1.1) is the equilibrium distribution of X , where µ = E (X ) < ∞ (cf. Abouammoh et al. [2], Abouammoh et al. [3] and Mugdadi and Ahmad [39]). The MRL order is characterized via the HR order as (Hu et al. [21]) X ≤MRL Y ⇔  X ≤HR  Y.

(1.2)

M. Kayid et al. / Statistical Methodology 29 (2016) 51–69

53

Let v be a non-negative weight function such that E [v(X )] < ∞ and let Xv be the weighted version of X having density function (Rao [43]) fv (x) =

v(x)f (x) , E [v(X )]

for all x ∈ R.

In statistical modeling, a weighted distribution can cover more situations than just the parent distribution. The renewal process could therefore be more flexible if the inter-failure times follow the weighted distribution Fv . The resulted limiting distribution of the excess lifetime in such a renewal process then has density fv∗ (x) =

F¯v (x) E (Xv )

=

E [v(X ) | X > x]F¯ (x) E (Xv )E (v(X ))

,

(1.3)

where the second equality follows from Jain et al. [26]. Motivated by (1.3), the limiting distribution of the excess lifetime in a renewal process where the inter-failure times come from the weighted distribution Fv is the weighted version of the equilibrium distribution  F with weight function w(x) = E [v(X ) | X > x]. A special case of interest arises when the weight function is of the form w(x) = xα , for some α ≥ 0. These kinds of distributions are known in the literature as the size-biased distributions of order α . The most common cases of size-biased distributions occur when α = 1 or 2. These special cases are termed as length- and area-biased, respectively. Denote by  Xw and  Yw the weighted versions of  X and  Y , respectively, that have density functions

w(x)F¯ (x)  fw (x) =  ∞ , w(u)F¯ (u)du 0

x ≥ 0,

w(x)G(x)  gw (x) =  ∞ , w(u)G(u)du 0

x ≥ 0.

and

To compare  Xw and  Yw by HR order, we have

∞  Xw ≤HR  Yw ⇔

t

w(x)F¯ (x)dx ≤ F¯ (t )

∞ t

w(x)G¯ (x)dx , ¯ (t ) G

for all t ≥ 0.

For an increasing weight function w , Theorem 9(a) in Bartoszewicz and Skolimowska [13] states that X ≤HR Y implies Xw ≤HR Yw . Here, if we assume that w is an increasing weight function, then

 X ≤HR  Y ⇒ Xw ≤HR  Yw .

(1.4)

By considering w(x) = x in (1.4), two lifetime variables are compared according to the HR order of the length-biased versions of their equilibrium distributions. By (1.1) and (1.4), X ≤MRL Y implies  Xw ≤HR  Yw . We propose the combination mean residual life (CMRL) function of X as the reciprocal hazard rate of the length-biased equilibrium distribution given by

∞ mX (t ) =

t

xF¯ (x)dx

t F¯ (t )

,

t ≥ 0.

Similarly, denote by mY the CMRL function of Y . Next, we propose a new stochastic order as follows. Definition 1.1. The random variable X is said to be smaller than Y in the CMRL order (denoted as X ≤CMRL Y ) if mX (t ) ≤ mY (t ), for all t ≥ 0. In the sequel, it is shown that the CMRL order lies in the framework of the MRL and the CCX orders. As a result, the study of the CMRL order is meaningful because it throws an important light on the understanding of the properties of the MRL and the CCX orders, and of the relationships between these two orders and other related stochastic orders. Furthermore, the CMRL order enjoys several reliability properties which provide some applications in reliability, renewal theory and life testing.

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M. Kayid et al. / Statistical Methodology 29 (2016) 51–69

The rest of the paper is organized as follows. In Section 2, some characterizations and implications regarding the CMRL order are provided. Preservation properties under some reliability operations such as monotone transformation and mixture are discussed in Section 3. In that section, we will also discuss the preservation property of CMRL order under weighted distributions. In Section 4, some applications in the context of reliability, renewal theory and life testing are included. Finally in Section 5, we conclude the paper with some remarks. Throughout the paper, all the random variables are assumed to have absolutely continuous distributions with support [0, ∞) and the corresponding density is assumed to be positive on the support unless otherwise stated. 2. Characterizations and implications The objective of this section is to concentrate on the relationships between the CMRL order and other well-known stochastic orders. Some relevant characterization results are also discussed. First, for ease of reference, we present some definitions that will be used in the sequel (cf. Shaked and Shanthikumar [46] and Karlin [27]). Definition 2.1. A non-negative function β(x, y) is said to be totally positive of order 2 (TP2 ) in (x, y) ∈ χ × γ , if β(x1 , y2 )β(x2 , y1 ) ≤ β(x1 , y1 )β(x2 , y2 ), for all x1 ≤ x2 ∈ χ and y1 ≤ y2 ∈ γ , in which χ and γ are two real subsets of R. Definition 2.2. A probability vector α = (α1 , . . . , αn ) with αi > 0 for i = 1, 2, . . . , n is said to be smaller than another probability vector β = (β1 , . . . , βn ) in the sense of discrete likelihood ratio order (denoted as α ≤DLR β ) if βi /αi ≤ βj /αj for all 1 ≤ i ≤ j ≤ n. In the following result, we provide equivalent conditions for the CMRL order. Theorem 2.1. Let X and Y be two lifetime random variables. The following assertions are equivalent: (i) X ≤CMRL Y . ∞ ¯ xG(x)dx

(ii) t∞ xF¯ (x)dx is non-decreasing in t ≥ 0. t

(iii) E (X 2 | X > t ) ≤ E (Y 2 | Y > t ), for all t ≥ 0. Proof. We first prove that (i) and (ii) are equivalent. We get d dt

∞ t∞ t

xG(x)dx xF (x)dx

=

t

∞ t

x[G(x)F¯ (t ) − G(t )F (x)]dx

 ∞ t

xF (x)dx

2

.

∞

By definition, X ≤CMRL Y if and only if t x[G(x)F¯ (t ) − G(t )F (x)]dx ≥ 0, for all t ≥ 0. We now prove that the statements (i) and (iii) are equivalent. Note that mX (t ) = E (Xt ) +



1 2t

E (Xt )2

= E (X − t ) + =

1 2t

1 2t

(X − t ) | X > t

E (X 2 − t 2 | X > t ),

Similarly, mY (t ) =

1 E 2t

2



for all t ≥ 0.

(Y 2 − t 2 | Y > t ). Now,

X ≤CMRL Y ⇔ mX (t ) ≤ mY (t ), for all t ≥ 0 1 1 ⇔ E (X 2 − t 2 | X > t ) ≤ E (Y 2 − t 2 | Y > t ), for all t ≥ 0 2 2 ⇔ E (X 2 | X > t ) ≤ E (Y 2 | Y > t ), for all t ≥ 0. 

M. Kayid et al. / Statistical Methodology 29 (2016) 51–69

55

For any real number a, let a+ denote the positive part of a; that is a+ = a if a > 0 and a+ = 0 if a < 0. To show how Theorem 2.1(ii) is significant, one can see that

∞ t

∞ t

¯ (x)dx xG xF¯ (x)dx

 is non-decreasing in t over

∞



t :

 ¯ xF (x)dx > 0 ,

t

if and only if E [(Y 2 − t 2 )+ ] E [(X 2 − t 2 )+ ]

is non-decreasing in t over {t : E [(X 2 − t 2 )+ ] > 0}.

Now, one may wonder whether the CMRL order implies the MRL order. Counter example 2.1 gives a negative answer. Counter example 2.1. Suppose that X and Y have respective survival functions

F¯ (x) =

 1,   

1−

  

x≤0 2

x

2

,

0
0,

x>

√ 2

√

2,

and

 1,    2 ¯ (y) = 1 − y , G  3   0,

y≤0 0

√ 3

√

3.

Let µX = E [X − t | X > t] and µY = E [Y − t | Y > t] denote to the MRL functions associated with X and Y , respectively. It is easy to show that µX 2 (t ) = 1 − t /2, for any t ∈ [0, 2), and µY 2 (t ) = 1 − t /3, for any t ∈ [0, 3) and they are equal to zero in any other interval. Clearly, µX 2 (t )≤ µY 2 (t ), for all 

√

t ∈ [0, ∞) which implies that X ≤CMRL Y . In addition, µX (x) = 2 2/3 − x + x3 /6 / 1 − x2 /2 , for any x ∈ [0,

√

 √

 

2) and µY (x) = 2 3/3 − x + x3 /9 / 1 − x2 /3 , for any x ∈ [0,







√

3), and they are

equal to zero in any other interval. Then, it is observed that µX and µY are not ordered on [0, ∞), and hence X ̸≤MRL Y . By comparing size-biased equilibrium distributions according to their hazard rates, we propose a family of stochastic orders as follows. Definition 2.3. For each fixed α ∈ [0, ∞), the lifetime random variable X is said to be smaller than Y (α) in the combination mean residual life order of order α , denoted by X ≤CMRL Y , if

∞ t

xα F¯ (x)dx F¯ (t )

∞ ≤

t

¯ (x)dx xα G ¯ (t ) G

,

for all t ≥ 0,

(2.1)

or equivalently if

∞ t∞ t

¯ (x)dx xα G xα F¯ (x)dx

is non-decreasing in t ≥ 0.

(2.2)

(α)

It is observed that when α = 0 and α = 1, the X ≤CMRL Y corresponds to X ≤MRL Y and X ≤CMRL Y , respectively. Following the discussion given earlier to motivate the CMRL order, we may note here (α) that X ≤CMRL Y if, and only if  Xw ≤HR  Yw , with w(x) = xα . Some useful implications are given below.

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M. Kayid et al. / Statistical Methodology 29 (2016) 51–69

Theorem 2.2. Let α ≤ β ∈ [0, ∞) be fixed. Then (β)

(α)

(i) X ≤CMRL Y implies X ≤CMRL Y . (α)

(ii) X ≤CMRL Y implies

∞ s

¯ (x) − F¯ (x)]dx ≥ 0, for all s ≥ 0. xα [G

Proof. (i) For each α ≤ β ∈ [0, ∞) and for all t ≥ 0, ∞



x t

β

G(x)

¯

¯ (t ) G

−

F¯ (x)



∞

 dx =

F¯ (t )

β−α

x

  d −

u

α

G(u)

¯

¯ (t ) G

x

t

= t β−α

∞



xα

t

∞



∞

+

G(x)

¯

¯ (t ) G

(β − α)xβ−α−1

t

−

F¯ (x) ∞



F¯ (t )

 du



F¯ (t )



−

F¯ (u)

uα

x

dx G(u)

¯

¯ (t ) G

−

F¯ (u) F¯ (t )





du dx.

The assertion is now clear. (ii) By (2.1) and (2.2), for all 0 ≤ t ≤ s,

∞ t∞ t

¯ (x)dx xα G xα F¯ (x)dx

∞ ≤ s∞ s

¯ (x)dx xα G xα F¯ (x)dx

,

(2.3)

and

¯ (x) G F¯ (x)

∞ ≤ t∞ t

¯ (x)dx xα G xα F¯ (x)dx

.

(2.4)

By taking t = 0 in (2.3) and (2.4), for all s ≥ 0,

∞ s

∞ s

¯ (x)dx xα G xα F¯ (x)dx

∞ ≥ 0∞ 0

≥

¯ (x)dx xα G xα F¯ (x)dx

¯ (0) G F¯ (0)

which proves the result.

= 1,



As a consequence of Theorem 2.2, the next corollary is immediate. Corollary 2.1. Let X and Y be two lifetime random variables. Then (β)

(i) X ≤MRL Y implies X ≤CMRL Y , for all β ≥ 0. (β)

(ii) X ≤CMRL Y implies X ≤CMRL Y , for all β ≥ 1. (iii) X ≤CMRL Y implies X ≤CCX Y . In many reliability engineering problems, it is interesting to study the residual life of X at random time Y given by XY = [X − Y | X > Y ]. The residual life at random time (RLRT) represents the actual working time of the standby unit if X is the total random life of a warm standby unit with age Y . For more details about RLRT we refer to Stoyan [47], Li and Zuo [36], Misra et al. [38] and Kayid and Izadkhah [30]. Suppose that X and Y are independent. Then, the survival function of XY , for any x ≥ 0, is given by

∞ P (XY > x) =

0

F¯ (x + y)dG(y) ∞ . F¯ (y)dG(y) 0

The following result provides a characterization of the CMRL order based on the concept of the RLRT. Theorem 2.3. XY ≤CMRL X for any Y that is independent of X if and only if Xt ≤CMRL X for all t ≥ 0.

M. Kayid et al. / Statistical Methodology 29 (2016) 51–69

57

Proof. Suppose Xt ≤CMRL X for all t ≥ 0. Then, for all s > 0,



∞

xF (t + x)dx ≤

F (t + s) F ( s)

s

∞



xF (x)dx.

(2.5)

s

By integrating both sides of (2.5) over t ∈ (0, ∞) with respect G, it is deduced that ∞



∞



xF (t + x)dxdG(t ) ≤ s

0

∞





F (t + s) F (s)

0

xF (x)dx dG(t ) s

∞

∞



F (t + s)dG(t )

=



∞



0

s

xF (x)dx F (s)

,

for all s ≥ 0,

which implies that XY ≤CMRL X , for all Y that are independent of X . Suppose that XY ≤CMRL X for any non-negative random variable Y . By taking Y as a degenerate variable, we get Xt ≤CMRL X , for all t ≥ 0.  3. Preservation properties The problem of preservation of stochastic orders under well-known reliability operations has received considerable attention in the literature (cf. Alzaid [7], Bartoszewicz and Skolimowska [13], Misra et al. [37], Izadkhah et al. [22], Izadkhah et al. [24], Izadkhah et al. [23], Izadkhah et al. [25] and Kayid et al. [28]). In this section, we get some preservation properties of the CMRL order under several reliability operations. We also obtain some conditions under which the CMRL order is preserved by weighted distributions. The next result shows that the CMRL order is preserved under increasing convex transformations. Theorem 3.1. Let φ be strictly increasing and convex such that φ(0) = 0. Then, X ≤CMRL Y implies φ(X ) ≤CMRL φ(Y ). Proof. First, assume without loss of generality that φ is differentiable and denote its first derivative by φ ′ . Notice that X ≤CMRL Y implies, for all t ≥ 0, that



∞



φ −1 (t )

¯ (x) xG

−

¯ φ −1 (t ) G 

xF¯ (x)

  dx ≥ 0.

F¯ φ −1 (t )



On the other hand, φ(X ) ≤CMRL φ(Y ), if, and only if

∞

xP [φ (Y ) > x] dx

t

P [φ (Y ) > t]

∞ ≥

t

xP [φ (X ) > x] dx P [φ (X ) > t]

,

for all t ≥ 0,

which is equivalent to





∞

φ −1 (t )

γ (x)

¯ (x) xG

−

¯ φ −1 (t ) G 



xF¯ (x) F¯ φ −1 (t )





dx ≥ 0,

for all t ≥ 0,

φ(x)φ ′ (x)

where γ (x) = . It is well-known that if φ is non-negative and convex with φ(0) = 0, then x φ(x)/x is non-decreasing. Thus, by assumption, γ (x) is the product of two non-negative and nondecreasing functions, and hence γ (x) is non-decreasing. The result now follows from Lemma 7.1(a) of Barlow and Proschan [12].  Mixture models are widely used as computational convenient representations for modeling complex probability distributions. In practical situations, it often happens that data from several populations are mixed and information about which sub-population gave rise to individual data points is unavailable. Mixture models are used to model such data sets in nature. For example, measurements of life lengths of a device may be gathered without regard to the manufacturer, or data may be

58

M. Kayid et al. / Statistical Methodology 29 (2016) 51–69

gathered on humans without regard, say, to blood type. If the ignored variable (manufacturer or blood type) has a bearing on the characteristic being measured, then the data are said to come from a mixture. Actually, it is hard to find data that are not some kind of a mixture because there is almost always some relevant covariate that is not observed (cf. Kayid and Izadkhah [31] and the references cited therein). Let Xi , i = 1, . . . , n be a collection of independent random variables. Suppose that Fi is the distribution function of Xi . Let α = (α1 , . . . , αn ) and β = (β1 , . . . , βn ) be two probability vectors

¯ defined by and let X and Y be two random variables having respective survival functions F¯ and G F¯ (x) =

n 

αi F¯i (x),

(3.1)

βi F¯i (y).

(3.2)

i=1

and

¯ (y) = G

n  i=1

Theorems 3.2–3.4 establish that the CMRL order is preserved under the mixture of life distributions, when appropriate assumptions are imposed. Theorem 3.2. Let X , Y , and Θ be random variables such that [X | Θ = θ ] ≤CMRL Y | Θ = θ ′ for all θ and θ ′ in the support of Θ . Then X ≤CMRL Y .





¯ (· | θ ), F¯ · | θ ′ and G¯ · | θ ′ be the survival Proof. Select θ and θ ′ in the support of Θ . Let F¯ (· | θ ), G     functions of [X | Θ = θ ] , [Y | Θ = θ ] , X | Θ = θ ′ and Y | Θ = θ ′ , respectively. The proof is similar to that of Theorem 1.B.8 in Shaked and Shanthikumar [46]. It is sufficient to show that for each ν ∈ (0, 1) and for all t > 0 we have 

ν

∞ t

≤

uF¯ (u | θ )du + (1 − ν)

∞ t







uF¯ (u | θ ′ )du

ν F¯ (t | θ ) + (1 − ν)F¯ (t | θ ′ ) ∞  ¯ (u | θ )du + (1 − ν) ∞ uG¯ (u | θ ′ )du ν uG t

t

ν G¯ (t | θ ) + (1 − ν)G¯ (t | θ ′ )

.

This is an inequality of the form a+b c+d

≥

w+x , y+z

where all eight variables are non-negative and by the assumptions of the theorem they satisfy a c

≥

w y

,

a c

x

b

z

d

≥ ,

≥

w y

,

and

b d

x

≥ . z

It is easy to verify that the latter four inequalities imply the former one, completing the proof of the theorem.  In Theorem 3.3, we discuss the preservation property of the CMRL order under finite mixtures. For a similar kind of result, we refer to Ahmed [6]. Theorem 3.3. Let X1 , . . . , Xn be a collection of independent random variables with corresponding survival functions F¯1 , . . . , F¯n , such that X1 ≤CMRL X2 ≤CMRL . . . ≤CMRL Xn and let α = (α1 , . . . , αn ) and β =

(β1 , . . . , βn ) be such that α ≤DLR β . Suppose X and Y have survival functions F¯ and G¯ defined in (3.1) and (3.2), respectively. Then X ≤CMRL Y .

M. Kayid et al. / Statistical Methodology 29 (2016) 51–69

59

Proof. Because of Theorem 2.1(ii), we need to establish that, for all 0 < x < y

∞ 0

∞ 0

(x + u) (x + u)

n  i =1 n



∞

βi F i (x + u)du

0

(y + v)

≥ ∞

αi F i (x + u)du

0

i=1

(y + v)

n  i=1 n



βi F i (y + v)dv .

(3.3)

αi F i (y + v)dv

i =1

By simple calculations, (3.3) can be written as: n  n  

βj αi

(x + u)F j (x + u)du

n  n  

∞



0

i=1 j=1 i̸=j

≥

∞



(y + v)F i (y + v)dv



0

∞



βj αi

(y + u)F j (y + u)du 0

i=1 j=1 i̸=j

∞





(x + v)F i (x + v)dv , 0

or n  n  

βj αi

∞



(x + u)F j (x + u)du

+ βi αj

∞



(x + u)F i (x + u)du

n  n  

βj αi

(y + v)F j (y + v)dv

∞



(y + v)F j (y + v)dv 0

i=1 j=1 i
+ βi αj

∞



∞



(x + u)F i (x + u)du 0

∞





0

0

≥

(y + v)F i (y + v)dv 0

0

i=1 j=1 i
∞



(y + v)F i (y + v)dv 0

∞





(x + u)F j (x + u)du . 0

Now, for each fixed pair (i, j) with i < j we have

  βj αi

∞



∞

(y + v)F j (y + v)dv (x + u)F i (x + u)du 0 0   ∞  ∞ + βi αj (y + v)F i (y + v)dv (x + u)F j (x + u)du 0 0   ∞  ∞ − βj αi (x + u)F j (x + u)du (y + v)F i (y + v)dv 0 0   ∞  ∞ (y + v)F j (y + v)dv + βi αj (x + u)F i (x + u)du 0 0  ∞  ∞   = βj αi − βi αj (y + v)F j (y + v)dv (x + u)F i (x + u)du 0 0   ∞  ∞ − (x + u)F j (x + u)du (y + v)F i (y + v)dv , 0

0

which is non-negative because both terms are non-negative by the assumptions. This completes the proof.  To demonstrate the usefulness of Theorem 3.3 in recognizing CMRL ordered random variables, we present the following examples.

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Example 3.1. Suppose that Xi , i = 1, . . . , n are independent exponential random variables with means λi , i = 1, . . . , n. Let X and Y be α and β mixtures of Xi . An application of Theorem 3.3, immediately yields X ≤CMRL Y for every two probability vectors α and β such that α ≤DLR β provided that λ1 ≤ · · · ≤ λn . Example 3.2. Let Xλ , and Xµ denote the convolution of n exponential distributions with parameters

λ1 , . . . , λn , and µ1 , . . . , µn , respectively. Assume without loss of generality that λ1 ≤ · · · ≤ λn , and µ1 ≤ · · · ≤ µn . For 0 ≤ q ≤ p ≤ 1 in which p + q = 1, according to Theorem 3.3, pXλ + qXµ ≤CMRL qXλ + pXµ , whenever λi ≤ µi for i = 1, 2, . . . , n. Consider a family of survival functions {F¯θ , θ ∈ χ} where χ is a subset of the real line R. Let X (θ) = [X | Θ = θ ] denote a random variable with survival function F¯θ . For any random variable Θi with support in χ , and with distribution function Λi , we denote by X (Θi ) the random variable that has survival function F¯i (x) =

 χ

F¯θ (x)dΛi (θ ),

x ∈ R, i = 1, 2.

In this case, X (Θi ) is called a mixture of X (θ ) or of the family {F¯θ , θ ∈ χ} with respect to Θi for each i = 1, 2. Theorem 3.4 states another kind of preservation property of CMRL order under mixture. Theorem 3.4. Let X (Θ1 ) and X (Θ2 ) be as described above. If X (θ1 ) ≤CMRL X (θ2 ),

for all θ1 ≤ θ2 ∈ χ ,

(3.4)

and if

Θ1 ≤HR Θ2 ,

(3.5)

then X (Θ1 ) ≤CMRL X (Θ2 ). Proof. To prove the result we need to show that Hi ( t ) =

∞



xF¯i (x)dx

is TP2 in (i, t ) ∈ {1, 2} × [0, ∞).

t

For all t ≥ 0, and i = 1, 2, Hi ( t ) =

∞

 t

∞



xF¯θ (x)dΛi (θ )dx 0

∞



φ(t , θ )dΛi (θ ),

= 0

∞

where φ(t , θ ) = t xF¯θ (x)dx. By Corollary 2.1(iii), (3.4) states that φ(t , θ ) is increasing in θ , and from Theorem 2.1(ii), (3.4) is equivalent to saying that φ(t , θ ) is TP2 in (t , θ ) ∈ [0, ∞) × χ . By (3.5), 1 − Λi (θ ) is TP2 in (i, θ ) ∈ {1, 2} × [0, ∞). Now, Lemma 4.2 of Li and Xu [35] is applicable and completes the proof.  In the rest of this section, we discuss the preservation property of the CMRL order under weighting. Let w1 and w2 be two weight functions and let Xw1 , and Yw2 be weighted versions of X and Y having density functions f 1 ( x) =

w1 (x)f (x) for x ≥ 0, µ1

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61

and g1 (x) =

w2 (x)g (x) for x ≥ 0, µ2

where 0 < µ1 = E [w1 (X )] < ∞, and 0 < µ2 = E [w2 (Y )] < ∞. Let B1 (x) = E [w1 (X ) | X > x], and B2 (x) = E [w2 (Y ) | Y > x]. Then, the corresponding survival functions are given by F¯1 (x) =

B1 (x)F¯ (x)

µ1

,

for x ≥ 0,

(3.6)

for x ≥ 0.

(3.7)

and

¯ 1 (x) = G

¯ (x) B2 (x)G µ2

,

Theorem 3.5. Let B1 (x) be non-decreasing in x and let B2 (x)/B1 (x) be non-decreasing in x ≥ 0. If X ≤CMRL Y , then Xw1 ≤CMRL Yw2 . Proof. By (3.6) and (3.7), Xw1 ≤CMRL Yw2 if and only if ∞





¯ ( x) xB2 (x)G ¯ (t ) B2 (t )G

t

−

xB1 (x)F¯ (x)



B1 (t )F¯ (t )

dx ≥ 0,

for all t ≥ 0.

By assumption B2 (x)/B2 (t ) ≥ B1 (x)/B1 (t ), for all t ≤ x. Thus, ∞

 t



¯ (x) xB2 (x)G ¯ (t ) B2 (t )G

−

xB1 (x)F¯ (x) B1 (t )F¯ (t )



 ¯  xG(x) xF¯ (x) dx ≥ − dx ¯ (t ) B1 (t ) G F¯ (t ) t  ∞ = [B1 (t )]−1 h(x)dWt (x), 

∞

B1 (x)

0

where h(x) = B1 (x) is non-decreasing and dWt (x) = wt (x)dx with

  ¯ xF¯ (x) xG(x) − I [x > t ]. wt (x) = ¯ (t ) G F¯ (t ) ∞ As in the proof of Theorem 2.2(i), s dWt (x) ≥ 0, for all t , s ≥ 0. Hence, by Lemma 7.1(a) of Barlow ∞ and Proschan [12], 0 h(x)dWt (x) ≥ 0, for all t ≥ 0, which completes the proof.  To demonstrate the usefulness of Theorem 3.5 in developing CMRL ordered random variables we provide the next example. Example 3.3. Let X and Y be two random variables denoting the lifetimes of two devices. For all x ≥ 0, let

λ1 (x) =

∞



F¯ (u)du/F¯ (x),

x

and

λ2 (x) =

∞



¯ (u)du/G¯ (x), G x

be the MRL functions of X and Y , respectively, such that λ2 (x)/λ1 (x) is non-decreasing and λ1 (x) is ¯ (x)/g (x). Note that non-decreasing. Let X ≤CMRL Y hold and take w1 (x) = F¯ (x)/f (x) and w2 (x) = G Xw1 and Yw2 are equal in distribution to  X and  Y , respectively. Hence, it is clear that B1 (x) = λ1 (x) and B2 (x)/B1 (x) = λ2 (x)/λ1 (x). As a result, an application of Theorem 3.5 provides that  X ≤CMRL  Y.

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4. Some applications In this section, we give various applications of the results that were developed in previous sections in the context of reliability, renewal theory, aging notions and hypothesis testing. Application 4.1 (Reliability). Suppose that a system consisting of n i.i.d. components has an ability to withstand a random number of some shocks, and it is commonly assumed that the number of shocks and the interarrival times of shocks are independent, and they are arriving according to a homogeneous Poisson process. Let N denote the number of shocks survived by the system, and let Xj denote the random interarrival time between the (j − 1)th and jth shocks. Then, the lifetime T of the N system is given by T = j=1 Xj . In particular, if the inter-arrivals are assumed to be independent and exponentially distributed with common parameter λ, then the survival function of T can be written as

¯ (t ) = H

∞  e−λt (λt )k k=0

k!

P¯ k ,

t ≥ 0,

where P¯ k = P [N > k] for all k ∈ N (and P¯ 0 = 1). Shock models of this kind, called Poisson shock models, have been studied extensively. For more details, we refer to Pellerey [41], Alzaid et al. [9], Belzunce et al. [15], Finkelstein [20], Izadkhah and Kayid [22], Kayid and Izadkhah [29] and Kayid et al. [32] and the references therein. Consider now two devices subjected to shocks occurring as events of a Poisson counting process, [2] [1] as above; and let P¯ k , and P¯ k be the survival functions of the random number of shocks N1 and N2 related to the two devices, respectively, which are the probabilities of surviving the first k shocks. Let Ti , i = 1, 2 denote the lifetime of the device with survival function

¯ i (t ) = H

∞  e−λt (λt )k k=0

k!

P¯ k , [i]

t ≥ 0.

(4.1)

A problem that has been extensively studied in the literature is to investigate when a particular stochastic order between the random number of shocks survived by two devices is preserved by the corresponding comparison between random lifetimes of the two devices under the shock model. Such results were proven to hold in Pellerey [42] and Fagiuoli and Pellerey [19] for the HR and the MRL orders, respectively. Here, we study the same problem in the sense of the CMRL order. Before stating Theorem 4.1, we give the following definition which indeed introduces the discrete version of the CMRL order. Definition 4.1. Let Ni be a discrete random variable with survival function P¯ n[i] , i = 1, 2. Then, it is said that N1 is smaller than N2 in combination mean residual life order (denoted by N1 ≤CMRL N2 ) whenever ∞ 

[2]

nP¯ n−1

n=k+1

∞ 

is non-decreasing in k ∈ N . [1]

nP¯ n−1

n=k+1

Theorem 4.1. If N1 ≤CMRL N2 , then T1 ≤CMRL T2 . Proof. By (4.1), for all t ≥ 0, ∞



¯ i (x)dx = λ−2 xH t

∞  e−λt (λt )j j=0

j!



∞ 

k=j+1

 [i] ¯ kPk−1 .

M. Kayid et al. / Statistical Methodology 29 (2016) 51–69

63

∞ ¯ [1] ¯ [2] Because e−λt (λt )j /j! is TP2 in (t , j) and k =j+1 kPk−1 is non-decreasing in j ∈ N, the k=j+1 kPk−1 / ∞ ¯ general composition theorem of Karlin [27] provides that t xHi (x)dx is TP2 in (i, t ) ∈ {1, 2} × R+ , which is equivalent to saying that T1 ≤CMRL T2 .  ∞

Let us now consider another application of the CMRL order in reliability engineering. Suppose that S1 and S2 denote two series systems each consisting of n i.i.d. components. Further, suppose that X1 , X2 , . . . , Xn are i.i.d. lifetime random variables from F and that Y1 , Y2 , . . . , Yn are i.i.d. lifetime random variables from G. Denote by T1 = min{X1 , X2 , . . . , Xn }, and T2 = min{Y1 , Y2 , . . . , Yn }, the lifetimes of S1 and S2 , respectively. The next result shows that if the lifetimes of two series systems with i.i.d. components are CMRL ordered, then their components are CMRL ordered. Theorem 4.2. If T1 ≤CMRL T2 , then Xi ≤CMRL Yi , for all i = 1, 2, . . . , n. Proof. Let T1 ≤CMRL T2 . Then, ∞



¯ n (x)F¯ n (t ) − G¯ n (t )F¯ n (x) dx ≥ 0, x G 



for all t ≥ 0.

t

Note that

¯ (x)F¯ (t ) − G¯ (t )F¯ (x) = G¯ n (x)F¯ n (t ) − G¯ n (t )F¯ n (x) h(x), G 



with

 h( x ) =

 −1

n  

   ¯ (x)F¯ (t ) n−i G¯ (t )F¯ (x) i−1 G

,

i=1

is a non-decreasing function. By Lemma 7.1(a) of Barlow and Proschan [12], for all t ≥ 0, ∞



¯ (x)F¯ (t ) − G¯ (t )F¯ (x) dx ≥ 0, x G 



t

which implies that Xi ≤CMRL Yi , for all i = 1, 2, . . . , n.



Application 4.2 (Renewal Theory). In the literature, several characterizations of the stochastic orders by the excess lifetime in a renewal process have been given. For more details, readers are referred to Chen [18], Ahmad et al. [5] and Belzunce et al. [16]. Next, we investigate the behavior of the excess lifetime of a renewal process with respect to the CMRL order. Theorem 4.3. If Xt ≤CMRL X , for all t ≥ 0, then γ (t ) ≤CMRL γ (0) for all t ≥ 0. Proof. Note that Xt ≤CMRL X , for all t ≥ 0, if and only if for any t ≥ 0 and s > 0,

∞

∞



xF¯ (t + x)dx ≤ F¯ (t + s) s

s

xF¯ (x)dx F¯ (s)

.

(4.2)

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M. Kayid et al. / Statistical Methodology 29 (2016) 51–69

By (1.2) and (4.2), ∞



xP [γ (t ) > x] dx =

∞



xF¯ (t + x)dx +

0

s

s

= ≤ =

xF¯ (t − u + x)dxdM (u) s

  t¯  F ( t − u + s) ∞ ¯ xF (x)dx dM (u) F¯ (s) 0 s s   ∞ ∞ ¯  xF (x)dx  xF¯ (t + x)dx + s P (γ (t ) > s) − F¯ (t + s) ¯ F (s) s ∞ ∞ ¯ (x)dx    x F xF¯ (x)dx  s F¯ (t + s) + s P (γ (t ) > s) − F¯ (t + s) F¯ (s) F¯ (s) ∞ ¯ xF (x)dx s P (γ (t ) > s) . F¯ (s)

 ≤

∞

 t

∞

xF¯ (t + x)dx +

Hence, for all t ≥ 0 and s > 0,

∞ s

xP (γ (t ) > x) dx P (γ (t ) > s)

∞ ≤

s

xF¯ (x)dx F¯ (s)

,

which means that γ (t ) ≤CMRL γ (0) for all t ≥ 0.



Application 4.3 (Aging Notions). Aging notions play a central role in survival analysis, reliability theory, maintenance policies and many other actuarial science, engineering, economics, biometry and applied probability areas. They are also useful in obtaining fundamental inequalities of estimates and test procedures. Many classes of life distributions are categorized or defined in the literature according to their aging properties. ‘No aging’ means that the age of a component has no effect on the distribution of the residual lifetime of the component. ‘Positive aging’ describes the situation where residual lifetime tends to decrease, in some probabilistic sense, with increasing age of a component. This situation is common in reliability engineering as components tend to become worse with time due to increased wear and tear (see, Lai and Xie [33] for an exhaustive monograph on this topic). The following definition is due to Barlow and Proschan [12]. Definition 4.2. A lifetime random variable X with an absolutely continuous distribution having density function f and survival function F¯ is said to have an increasing hazard rate (denoted as IFR) if its hazard rate is non-decreasing in t ∈ R+ . As another application of the obtained results in the context of aging notions, we propose a new class of distributions exhibiting a notion of positive aging on the basis of the CMRL function. Definition 4.3. The lifetime random variable X is said to be decreasing in combination mean residual lifetime (denoted as DCMRL) if its CMRL function mX is non-increasing in t ∈ R+ . To characterize the IFR class, in terms of the DCMRL aging notion we need the following lemma (Barlow and Proschan [11]). Lemma 4.1. Let K (t , x) > 0 be TP2 and let g (x) change sign once at most. Let h( t ) =



K (t , x)g (x)dµ(x),

be an absolutely convergent integral with µ a σ -finite measure. Then h(t ) changes sign once at most and the sequence of sign changes in h is the same as in g. Theorem 4.4. If the lifetime random variable X is IFR, then X is DCMRL.

M. Kayid et al. / Statistical Methodology 29 (2016) 51–69

65

Proof. In view of Definition 4.3, X is DCMRL if

∞ t

[xf (x) − F¯ (x)]dx ∞ , xF¯ (x)dx t

is non-decreasing in t > 0. Let us, for c > 0, write h(t ) =

∞



[xf (x) − F¯ (x) − cxF¯ (x)]dx t

∞



xI[t ,∞) (x)F¯ (x)g (x)dx,

= 0

where g (x) = [f (x)/F¯ (x) − 1/x − c ] and I[t ,∞) (x) denotes the indicator function of the set [t , ∞). Now, it is clear that K (t , x) = xI[t ,∞) (x)F¯ (x)

is TP2 in t ∈ [0, ∞) and x ∈ [0, ∞).

Since X is IFR, g (x) changes sign in x at most once from negative to positive. Hence, by Lemma 4.1, h(t ) changes sign in t at most once from negative to positive, which is equivalent to saying that h(t ) is increasing in t ≥ 0. This completes the proof.  Application 4.4 (Hypothesis Testing Application). The exponential distribution represents the lifetime of the units that never ages due to wear and tear. An assumption of exponentially distributed lifetimes implies that a used item is stochastically as good as new, so there is no reason to replace a functioning item. One of the oldest problems in aging distributions is testing exponentiality against some classes, such as IFR and others. Many authors studied this problem (cf. Belzunce et al. [14], Ahmad [4] and Anis [10]). Since the exponential distribution has a constant hazard rate function, it can be considered as an IFR distribution. In order to detect possible departures from exponentiality in the data, the problem of testing exponentiality against DCMRL alternatives may be of interest. First, in Theorem 4.5, we establish a moment inequality for the DCMRL distributions. Then, we use this inequality to construct a new test for exponentiality versus DCMRL. Theorem 4.5. Let X1 and X2 be two i.i.d. copies from F . If F is DCMRL, then E [min(X1 , X2 )]4 ≥

1 4

E [X 1 X 2 ]2 .

(4.3)

Proof. By definition, F is DCMRL if and only if t 2 F¯ 2 (t ) ≥ −

d dt

{t F¯ (t )}

∞



uF¯ (u)du,

for all t ≥ 0.

(4.4)

t

Integrating both sides of (4.4) over [x, ∞), gives ∞



t 2 F¯ 2 (t )dt ≥ −

x

∞



{t F¯ (t )}

dt

x

= xF¯ (x)

d

∞

 t

∞





uF¯ (u)du dt

t F¯ (t )dt −

∞



x

t 2 F¯ 2 (t )dt ,

for all x ≥ 0,

x

which is equivalent to ∞



t 2 F¯ 2 (t )dt ≥ xF¯ (x)

2



x

∞

t F¯ (t )dt ,

for all x ≥ 0.

(4.5)

x

Now, by integrating both sides of (4.5) over x ∈ [0, ∞), we get ∞



∞



t 2 F¯ 2 (t )dtdx ≥

2 0

x

∞



xF¯ (x) 0

∞



t F¯ (t )dtdx. x

(4.6)

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M. Kayid et al. / Statistical Methodology 29 (2016) 51–69

For the left hand side of (4.6), we get ∞



∞



t 2 F¯ 2 (t )dtdx = 2

2 0

∞



t 2 F¯ 2 (t )dxdt 0

0

x

t



∞



1

t 3 F¯ 2 (t )dt =

=2

2

0

E [min(X1 , X2 )]4 .

The right hand side of (4.6) is obtained as ∞



xF¯ (x)

∞



t F¯ (t )dtdx = x

0

=

1 2 1 8

2

∞



xF¯ (x)dx 0

E [X1 X2 ]2 .

It is now plain to get (4.3) and therefore the proof is completed.



Suppose now that the lifetime X of a device has a distribution function F which is unknown. We have at our disposal a random sample X1 , X2 , . . . , Xn of independent observations from F . Let exp(µ) denote an exponential distribution with mean µ. We want to test the null hypothesis H0 against its alternative H1 , where: H0 : F is exp(µ) in which µ is unspecified, versus H1 : F belongs to the DCMRL class and F is not exponential. From Theorem 4.5 a measure of departure from H0 in favor of H1 is ∆ = E [φ(X1 , X2 )], where 1

φ(X1 , X2 ) = [min(X1 , X2 )]4 − [X1 X2 ]2 . 4

Under H0 , ∆0 = 23µ4 /16, while under H1 , we have | ∆ − ∆0 |> 0. To test against H1 , let X1 , X2 , . . . , Xn be a random sample from F so that n > 2. We estimate ∆ by its empirical counterpart

ˆ = ∆

1



n(n − 1) i= ̸ j

φ(Xi , Xj ).

Also, to make ∆ scale free, we only need to consider δ = ∆/µ4 which is a scale free testing measure ˆ /X¯ 4 is asymptotically distribution free as we show in the against H1 , and the test statistic δˆ = ∆ following result.

√

Theorem 4.6. As n → ∞, n(δˆ − δ) is asymptotically normal with zero mean and variance σ 2 given . in (4.7). Under H0 , the variance σ02 is equal to 1679 972 Proof. Using the general theory of U-statistics and von Mises statistics (see Lee [34]) as n → ∞, √ n(δˆ − δ) is asymptotically normal with mean 0 and variance σ 2 , where

σ 2 = Var {E [φ(X1 , X2 ) | X1 ] + E [φ(X2 , X1 ) | X1 ]} = 4Var {E [φ(X1 , X2 ) | X1 ]} . One can see that E [φ(X1 , X2 ) | X1 ] = E [min(X1 , X2 )]4 − X1

 = 0

1

E [(X1 X2 )2 | X1 ] 4  ∞ X2 4x3 F¯ (x)dx − 1 xF¯ (x)dx. 2 0

Thus,

σ = 4Var 2

X1

 0

X2 4x F¯ (x)dx − 1 3

2

 0

∞

 ¯ xF (x)dx .

(4.7)

M. Kayid et al. / Statistical Methodology 29 (2016) 51–69

67

Table 1 The upper percentile of δˆ with 25 000 replications. n

90%

95%

99%

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

0.30838 0.21854 0.17871 0.15351 0.13701 0.12596 0.11541 0.10762 0.10347 0.09745 0.09231 0.09035 0.08583 0.08208 0.07981 0.07787 0.07582 0.07273 0.07061 0.06916

0.40114 0.27651 0.22653 0.19327 0.17448 0.15924 0.14836 0.13748 0.13227 0.12494 0.11871 0.11406 0.10982 0.10483 0.10146 0.09883 0.09557 0.09301 0.09038 0.08813

0.55696 0.38541 0.31788 0.27051 0.24457 0.22546 0.20672 0.19083 0.18206 0.17414 0.16547 0.15862 0.15390 0.14743 0.14311 0.13874 0.13208 0.13001 0.12833 0.12445

Under H0 , F in the above identity is exponential with mean 1. After some calculation, the asserted follows.  In practice, put δ0 = ∆0 /µ4 and evaluate

√

n | δˆ − δ0 |

σ0

.

Then, reject H0 if the observed value exceeds the 1 −α/2 quantile of the standard normal distribution. To assess the goodness, we evaluate the Pitman’s asymptotic efficacy of the test, PAE (δθ ) =

972 1679



d dθ

δθ

2 θ→θ0

.

Three most commonly used alternatives are: (i) Linear failure rate (LFR) distribution: F¯1 (x) = exp{−x − θ2 x2 } for x, θ ≥ 0; θ

(ii) Weibull distribution: F 2 (x) = e−X , for x, θ ≥ 0. (iii) Makeham distribution: F¯3 (x) = exp{−x − θ (e−x + x − 1)} for x, θ ≥ 0; The null is at θ = 0 in (i), (iii) and at θ = 1 in (ii). Direct calculation for the above three alternatives gives the values 0.7145, 0.4362 and 0.5261, respectively. In practice, simulated percentiles for small samples are commonly used by applied statisticians and reliability analyst. Table 1 gives the upper percentile values of the statistic δˆ given earlier for 90%, 95%, 99%, and the calculations are based on 5000 simulated samples of sizes n = 5(5)100. The power of the proposed test at a significance level α with respect to the alternatives F1 , F2 and F3 is calculated based on simulation data. In such simulation, 25 000 samples were generated with sizes n = 10, 20 and 30 from the alternatives. Table 2 shows the power of test at different values of θ and the significance level α = 0.05. From Table 2, it is noted that the power of the test increases as the values of the parameter θ and the sample size n increase. To demonstrate the test method above, we apply it to two data sets. The first data set in Bryson and Siddiqui [17] which are survival times in days from diagnosis of 43 patients suffering from chronic granulocytic leukemia. The second one in Abouammoh et al. [1] which represents ordered lifetimes (in days) of 40 patients suffering from blood cancer in one of the Ministry of Health Hospitals in Saudi

68

M. Kayid et al. / Statistical Methodology 29 (2016) 51–69 Table 2 Alternative distributions: LFR, Weibull, Makeham. n 10

20

30

θ

LFR

Weibull

Makeham

2 3 4 2 3 4 2 3 4

0.78110 0.81986 0.83922 0.92110 0.94710 0.96138 0.96512 0.98201 0.98837

0.43858 0.89518 0.99354 0.79158 0.99882 1.00000 0.94018 1.00000 1.00000

0.68062 0.72138 0.75354 0.81510 0.86116 0.89138 0.86874 0.91418 0.94450

Arabia. For the data sets in Bryson and Siddiqui [17] and Abouammoh et al. [1], the corresponding values of δˆ are 0.0072 and 0.0035, respectively. According to Table 1, this suggests to reject H0 in favor of H1 . 5. Conclusion The concept of the length-biased distribution is very important in statistics, reliability and survival analysis. In this investigation, a new stochastic order based on the hazard rate of length-biased equilibrium distribution is introduced and studied. Since this new order lies in the framework of the mean residual life and the combination convexity orders, we called it combination mean residual life order. Relationships of this new order with other well-known stochastic orders are given. It was shown that the new order enjoys several reliability properties, which provide some applications in reliability, renewal theory, aging notions and hypothesis testing. As a consequence, on the basis of the combination mean residual life function, a new class of lifetime distributions called decreasing combination mean residual life was proposed and studied. It was shown that the increasing failure rate class is a subclass of the proposed class. Testing exponentiality against this class was addressed and the asymptotic normality of the proposed statistic was established. In addition, the Pitman asymptotic efficacy, the power and the critical values of the proposed statistic were calculated. Further properties and applications of the new order and the proposed class can be considered in the future of this research. In particular, preservation properties of the new order under convolution and formation of coherent structure are interesting topics, and they still remain as open problems. Acknowledgments The authors are deeply grateful to the Editor, Associate Editor and four anonymous referees for their careful detailed remarks, which helped improve both content and presentation of the paper. The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this Research Group No (RG-1435-036). References [1] A.M. Abouammoh, S.A. Abdulghani, I.S. Qamber, On partial orderings and testing of new better than renewal used classes, Reliab. Eng. Syst. Saf. 43 (1994) 37–41. [2] A.M. Abouammoh, A.N. Ahmad, A.M. Barry, Shock models and testing for the renewal mean remaining life, Microelectron. Reliab. 33 (1993) 729–740. [3] A.M. Abouammoh, R. Ahmad, A. Khalique, On new renewal better than used classes of life distributions, Statist. Probab. Lett. 48 (2000) 189–194. [4] I.A. Ahmad, Moment inequalities of ageing families of distributions with hypothesis testing application, J. Statist. Plann. Inference 92 (2001) 121–132. [5] I.A. Ahmad, A. Ahmed, I. Elbatal, M. Kayid, An ageing notion derived from the increasing convex ordering: the NBUCA class, J. Statist. Plann. Inference 136 (2006) 555–569. [6] A. Ahmed, Preservation properties for the mean residual life ordering, Statist. Papers 29 (1988) 143–150. [7] A.A. Alzaid, Mean residual life ordering, Statist. Papers 29 (1988) 35–43. [8] A.A. Alzaid, Length-biased orderings with applications, Probab. Engrg. Inform. Sci. 2 (1988) 329–341.

M. Kayid et al. / Statistical Methodology 29 (2016) 51–69 [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47]

69

A.A. Alzaid, J.S. Kim, F. Proschan, Laplace ordering and its applications, J. Statist. Plann. Inference 28 (1991) 116–130. M.Z. Anis, A family of tests for exponentiality against IFR alternatives, J. Statist. Plann. Inference 143 (2013) 1409–1415. R.E. Barlow, F. Proschan, Mathematical Theory of Reliability, Wiley, New York, 1965. R.E. Barlow, F. Proschan, Statistical Theory of Reliability and Life Testing, Silver Spring, Maryland, 1981. J. Bartoszewicz, M. Skolimowska, Preservation of classes of life distributions and stochastic orders under weighting, Statist. Probab. Lett. 56 (2006) 587–596. F. Belzunce, J. Candel, J.M. Ruiz, Testing the stochastic order and the IFR, DFR, NBU, NWU ageing classes, IEEE Trans. Reliab. 47 (1998) 285–296. F. Belzunce, E.M. Ortega, J.M. Ruiz, The Laplace order and ordering of residual lives, Statist. Probab. Lett. 42 (1999) 145–156. F. Belzunce, E.M. Ortega, J.M. Ruiz, Comparison of expected failure times for several replacement policies, IEEE Trans. Reliab. 55 (2006) 490–495. M.C. Bryson, M.M. Siddiqui, Some criteria for ageing, J. Amer. Statist. Assoc. 64 (1969) 1472–1483. Y. Chen, Classes of life distributions and renewal counting process, J. Appl. Probab. 31 (1994) 1110–1115. E. Fagiuoli, F. Pellerey, Mean residual life and increasing convex comparison of shock models, Statist. Probab. Lett. 20 (1994) 337–345. M. Finkelstein, Failure Rate Modelling for Reliability and Risk, Springer, New York, 2008. T. Hu, A. Kundu, A.K. Nanda, On generalized orderings and ageing properties with their implications, in: Y. Hayakawa, T. Irony, M. Xie (Eds.), System and Bayesian Reliability, World Scientific, New Jersey, 2001, pp. 199–228. S. Izadkhah, M. Kayid, Reliability analysis of the harmonic mean inactivity time order, IEEE Trans. Reliab. 62 (2013) 329–337. S. Izadkhah, A.H. Rezaei, M. Amini, G.R.M. Borzadaran, A general approach for preservation of some ageing classes under weighting, Comm. Statist.-Theory Methods 42 (2013) 1899–1909. S. Izadkhah, A.H.R. Roknabadi, G.R.M. Borzadaran, On properties of reversed mean residual life order for weighted distributions, Comm. Statist.-Theory Methods 42 (2013) 835–851. S. Izadkhah, A.H.R. Roknabadi, G.R.M. Borzadaran, Aspects of the mean residual life order for weighted distributions, Statistics 48 (2014) 851–861. K. Jain, H. Singh, I. Bagai, Relations for reliability measures of weighted distributions, Comm. Statist.-Theory Methods 18 (1989) 4393–4412. S. Karlin, Total Positivity, Stanford University Press, 1968. M. Kayid, I.A. Ahmad, S. Izadkhah, A.M. Abouammoh, Further results involving the mean time to failure order, and the decreasing mean time to failure class, IEEE Trans. Reliab. 62 (2013) 670–678. M. Kayid, S. Izadkhah, Mean inactivity time function, associated orderings and classes of life distributions, IEEE Trans. Reliab. 63 (2014) 593–602. M. Kayid, S. Izadkhah, Characterizations of the exponential distribution by the concept of residual life at random time, Statist. Probab. Lett. 107 (2015) 164–169. M. Kayid, S. Izadkhah, A new extended mixture model of residual lifetime distributions, Oper. Res. Lett. 43 (2015) 183–188. M. Kayid, S. Izadkhah, H. Alhalees, Reliability analysis of the proportional mean residual life order, Math. Probl. Eng. (2014) 1–8. C.D. Lai, M. Xie, Stochastic Ageing and Dependence for Reliability, Springer, New York, NY, 2006. A.J. Lee, U-Statistics, Marcel Dekker, New York, 1989. X. Li, M. Xu, Some results about MIT order and IMIT class of life distributions, Probab. Engrg. Inform. Sci. 20 (2006) 481–496. X. Li, M.J. Zuo, Stochastic comparison of residual life and inactivity time at a random time, Stoch. Models 20 (2004) 229–235. N. Misra, N. Gupta, I.D. Dhariyal, Preservation of some ageing properties and stochastic orders by weighted distributions, Comm. Statist.-Theory Methods 37 (2008) 627–644. N. Misra, N. Gupta, I.D. Dhariyal, Stochastic properties of residual life and inactivity time at a random time, Stoch. Models 24 (2008) 89–102. A.R. Mugdadi, I.A. Ahmad, Moments inequalities derived from comparing life with its equilibrium form, J. Statist. Plann. Inference 134 (2005) 303–317. A. Müller, D. Stoyan, Comparison Methods for Stochastic Models and Risks, John Wiley, New York, 2002. F. Pellerey, Partial orderings under cumulative shock models, Adv. Appl. Probab. 25 (1993) 939–946. F. Pellerey, Shock models with underlying counting process, J. Appl. Probab. 31 (1994) 156–166. C.R. Rao, On discrete distributions arising out of methods of ascertainment, Sankhya¯ Ser. A 27 (1965) 311–324. S. Ross, Stochastic Processes, John Wiley, New York, 1996. S.Y. Sekeh, G.R.M. Borzadaran, A.H.R. Roknabadi, New stochastic orders based on the inactivity time, Soc. Sci. Res. Netw. (2013). M. Shaked, J.G. Shanthikumar, Stochastic Orders, Springer, New York, 2007. D. Stoyan, Comparison Methods for Queues and other Stochastic Models, Wiley, New York, 1991.