Mean Inactivity Time Function, Associated Orderings, and Classes of Life Distributions

IEEE TRANSACTIONS ON RELIABILITY, VOL. 63, NO. 2, JUNE 2014 593

Mean Inactivity Time Function, AssociatedOrderings, and Classes of Life Distributions

M. Kayid and S. Izadkhah

Abstract—The concept of mean inactivity time plays an impor-tant role in reliability and life testing. In this investigation, basedon the comparison of mean inactivity times of a certain function oftwo lifetime random variables, we introduce and study a new sto-chastic order. This new order lies between the reversed hazard rateand the mean inactivity time orders. Several characterizations andpreservation properties of the new order under reliability opera-tions of monotone transformation, mixture, and shock models arediscussed. In addition, a new class of life distributions called strongincreasing mean inactivity time is proposed, and some of its relia-bility properties are investigated. Finally, to illustrate the concepts,some applications in the context of reliability theory are included.

Index Terms—Mean inactivity time order, reversed hazard rateorder, preservation, mixture, shock models, strong increasingmean inactivity time class.

ACRONYMS AND ABBREVIATIONS

MIT mean inactivity time

RHR reversed hazard rate

SMIT strong mean inactivity time

SIMIT strong increasing mean inactivity time

DRHR decreasing reversed hazard rate

totally positive of order two

RLRT residul life at random time

D-SMIT discrete strong mean inactivity time

HR hazard rate

NOTATIONS

random life

pdf, cdf, and sf of lifetime r.v.

pdf, cdf, and sf of lifetime r.v

mean inactivity time function of

Manuscript received May 05, 2013; revised September 27, 2013; acceptedDecember 14, 2013. Date of publication April 11, 2014; date of current versionMay 29, 2014. This work was supported by the Deanship of Scientific Researchfrom King Saud University through the Research Group Project RGP-VPP-022.Associate Editor: M. J. Zuo.M. Kayid is with the Department of Statistics and Operations Research, King

Saud University, Riyadh, Saudi Arabia, and also with the Department of Math-ematics, Faculty of Science, Suez University, Egypt (e-mail: [email protected]).S. Izadkhah is with the School of Mathematical Sciences, Ferdowsi Univer-

sity of Mashhad, Mashhad, Iran (e-mail: [email protected]).Digital Object Identifier 10.1109/TR.2014.2315954

mean inactivity time order

reversed hazard rate function of

set of positive real line

reversed hazard rate order

strong mean inactivity time order

set of real line

discrete likelihood ratio order

residual life at random time

i.i.d -independent and identical distribution

excess life time of renewal process

renewal function

residual life at the time

maximum order statistics among

hazard rate order

indicator function of the set

inverse of

derivative of

set of natural numbers

I. INTRODUCTION AND MOTIVATIONS

T HE mean inactivity time (MIT) function, also knownas the mean past lifetime and the mean waiting time

functions, is a well-known reliability measure which has appli-cations in many disciplines such as reliability theory, survivalanalysis, and actuarial studies. Let be a lifetime randomvariable with distribution function . The MIT function ofis defined by

An interpretation of the MIT is as follows. Consider a situa-tion in which we have realized a device already has failed at timesay. The MIT is a useful measure to predict the actual time at

which the failure of the device occurs. Apart from applicationsin reliability engineering, reliability concepts are also heavilyused in biomedicine. In cases of diseases that can recur, effi-ciency of a treatment is determined by analyzing the remission

0018-9529 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

594 IEEE TRANSACTIONS ON RELIABILITY, VOL. 63, NO. 2, JUNE 2014

period, i.e., disease free survival time. Often the true remissionperiod is unknown due to an inability to continuously monitorpatients because of the high cost and effort involved. In suchcircumstances, the true remission period can be estimated usingan MIT function. Furthermore, in actuarial science, these con-cepts are used to calculate optimal premiums for life insurancepolicies.The MIT function was studied by Kayid and Ahmad [1], and

Ahmad et al. [2] to establish several properties of stochasticcomparisons based on the MIT function under the reliabilityoperations of convolution and mixture. Badia and Berrade [3]gave an insight into properties of the MIT in mixtures of dis-tributions. What is more important is the inclusion of some in-teresting characterizations based on the reliability properties ofthe MIT function. In Finkelstein [4], the MIT function has beenused for describing different maintenance policies in reliability.Asadi [5] obtained the MIT of the components of a parallelsystem. Several properties of the MIT for discrete random vari-ables are studied in Goliforushani and Asadi [6]. The MIT hasalso been applied to risk theory and econometrics (cf. Eeck-houdt and Gollier [7], Kijima and Ohnishi [8]). Kundu et al.[9] considered the MIT function for characterizations of quitea few distributions, and Kundu and Nanda [10] discussed itshigher-order moments. Ortega [11] obtained some characteri-zations for comparison of lifetime random variables concerningthe MIT function. Recently, Izadkhah and Kayid [12] used theharmonic mean average of the MIT function to propose a newstochastic order. It is thus seen that the results which discuss thebehavior of the MIT function can be very useful. Therefore, itis not surprising that recently several researchers have devotedtheir efforts to obtain such results.The comparison of the MIT functions of two lifetime random

variables provides a stochastic order between those randomvariables which has been defined and studied in the literature.Let , and have distribution functions and respec-tively. The lifetime random variable is said to be smallerthan in the MIT order (denoted as ) if f

The reversed hazard rate (RHR) function of is given bywhere is the density function of

which is closely related to the MIT function. The random vari-able is said to be smaller than in the RHR order (denotedas ) if

Applications, properties, and interpretations of the RHR ordercan be found in Shaked and Shanthikumar [13]. Recently, Liand Xu [14] proved that, for any strictly increasing, concavetransformation with

Because the inverse of a strictly increasing convex function isstrictly increasing and concave, one can also develop that, ifis strictly increasing and convex such that then

One important choice for a strictly increasing convex function iswhich proposes a new stochastic order called strong

mean inactivity time (SMIT) order.Definition 1: The lifetime random variable is said to be

smaller than in the SMIT order (denoted as ) if

or equivalently, iff

(1)

As demonstrated in Theorem 1 of Section II below, the SMITorder lies in the framework of the RHR and the MIT orders. Asa result, the study of the SMIT order is meaningful because itthrows an important light on the understanding of the propertiesof the MIT and the RHR orders, and of the relationships amongthese two orders and other related stochastic orders. Further-more, the SMIT order enjoys several reliability properties whichprovide some applications in reliability and survival analysis.On the other hand, statisticians and reliability analysts have

shown a growing interest in modeling survival data using classi-fications of life distributions by means of various stochastic or-ders. These categories are useful for modeling situations, main-tenance, inventory theory, and biometry. Consider the situationwherein denotes the risk that the direct insurer faces, andthe corresponding reinsurance contract. One important rein-

surance agreement is quota-share treaty defined as, for . The stochastic comparison between the

quata-share treaty and the risk in the MIT order is equivalent tosay that is non-decreasing in . It is well-knownthat is non-decreasing in implies that is alsonon-decreasing in . As a result, a new class of life distributionscalled strong increasing mean inactivity time (SIMIT) is pro-posed as follows.Definition 1: The lifetime random variable is said to be in

the SIMIT class if

The purpose of this paper is to achieve two goals. The firstgoal is to provide some characterizations, preservation results,and applications for the SMIT order. The second goal is to studysome reliability properties of the SIMIT class, and to providesome applications of it in the context of reliability.The paper is organized as follows. In Section II, some

characterizations and implications regarding the SMIT orderare provided. Preservation properties under some reliabilityoperations such as monotonic transformation and mixture are

KAYID AND IZADKHAH: MEAN INACTIVITY TIME FUNCTION, ASSOCIATED ORDERINGS, AND CLASSES OF LIFE DISTRIBUTIONS 595

discussed in Section III. In that section, we provide some ex-amples to demonstrate the usefulness of the obtained results inrecognizing the SMIT ordered random variables. In Section IV,some reliability properties of the SIMIT class including somecharacterization and preservation properties under Poissonshock models are discussed. To illustrate the concepts, someapplications in the context of reliability theory are included inSection V. Finally, in Section VI, we give a brief conclusion,and some remarks of the current and future of this research.In the rest of the paper, we have two important assumptions.• The non-negative random variables and will have

and , respectively as their respec-tive survival functions, and and as their correspondingdensities.

• All integrals, expectations, and derivatives are implicitlyassumed to exist wherever they are given.

Proofs are given in the Appendix.

II. CHARACTERIZATIONS AND IMPLICATIONS

The objective of this section is to concentrate on the relationsbetween SMIT order with other well-known stochastic orders.A characterization result based on the excess lifetime in a re-newal process is also discussed. For an exhaustive monographon the definitions and properties of stochastic orders, we referto Müller and Stoyan [15], and Shaked and Shanthikumar [13].For ease of reference, before stating our main results, we presentsome definitions and basic properties which will be used in thesequel.Definition 2: The lifetime random variable is said to be

decreasing reversed hazard rate (DRHR) if

Definition 3: A non-negative function is said to betotally positive of order 2 in , if

for all , and in which and aretwo real subsets of .For more details and properties about the totally positive

functions, we refer to Karlin [16].Definition 4: A probability vector with

for is said to be smaller than anotherprobability vector in the sense of discretelikelihood ratio order (denoted as ) if

The next theorem shows that the SMIT order lies between RHRand MIT orders.Theorem 1: Let and be two non-negative random vari-

ables. Then,i) andii) .

In the context of Theorem 1 (i), the following counterexampleshows that does not imply .

Counterexample 1: Let , and have distribution functions, and respectively, which are given by

Direct calculations give

which is non-decreasing in and hence . In par-allel, Nanda et al. [17] showed in their Counterexample 1 that

does not hold in this case.In general, the MIT order does not imply the SMIT order.

However, the next theorem presents a sufficient condition underwhich implies .Theorem 2: Let for all

Then, .In many reliability problems, it is interesting to study

, the residual life of with a random age .The residual life at a random time (RLRT) represents the actualworking time of the standby unit if is regarded as the totalrandom life of a warm standby unit with its age . For moredetails about RLRT, see Stoyan [18]. Yue and Cao [19] gavesome stochastic properties of , and applied them to queuingtheory. Also, Li and Zuo [20], and Misra et al. [21] obtainedsome new stochastic orders and aging properties regarding .Recently, Cai and Zheng [22] studied the RLRT in the contextof generalized aging classes. Suppose that and are s-inde-pendent. Then, the distribution function of , for anyis given by

(2)

Theorem 3: for any that is s-independentof iff for all .Let be a sequence of mutually s-inde-

pendent and identically distributed (i.i.d.) non-negative randomvariables with common distribution function . For ,denote as the time of the -th arrival, with

and let represent the numberof arrivals during the interval . Then,is a renewal process with underlying distribution (see Ross[23]). Let be the excess lifetime at time that is,

. In this context, we denote the renewalfunction by which satisfies the well-knownfundamental renewal equation


According to Barlow and Proschan [24], it holds that, for alland ,

(3)

In the literature, several results have been given to charac-terize the stochastic orders by the excess lifetime in a renewalprocess. For more details on definitions and properties, readersare referred to Barlow and Proschan [24]. Chen [25] investi-gated the relationship between the behavior of the renewal func-tion and the aging property of the underlyingdistribution. Ahmad et al. [26] established some stochastic com-parisons of the excess lifetime at different times of a renewalprocess when the inter-arrival times belong to a nonparametricaging class. Later, Belzunce et al. [27] established comparisonsof expected failure times of an age (block) replacement policy toa renewal process with no planned replacements when the life-time of the unit belongs to the new better than used in expecta-tion class. Next, we will investigate the behavior of the excesslifetime of a renewal process with SMIT interarrivals.Theorem 4: If for all then

III. PRESERVATION PROPERTIES

In this section, we develop some preservation properties ofthe SMIT ordering under some reliability operations such asmonotone transformation andmixture. Some examples for theseresults are mentioned that can be useful in recognizing situationswhen the random variables are SMIT ordered. The next resultshows that the SMIT order is preserved under monotone con-cave transformation.Theorem 5: Assume that is strictly increasing, and concave;

. Then

In general, the SMIT order is not preserved under mixture.However, the following theorem shows that this order may bepreserved under mixture, when appropriate assumptions are sat-isfied.Theorem 6: Let , and be random variables such that

for all , and in the sup-port of . Then .Let be a collection of s-independent random

variables. Suppose that is the distribution function of . Let, and be two probability

vectors; and let , and be two random variables having therespective distribution functions , and defined by

(4)

The next result gives conditions under which and arecomparable with respect to the SMIT order.Theorem 7: Let be a collection of s-independent

random variables with corresponding distribution functionssuch that

, and let and be suchthat . Let , and have distribution functions ,and defined in (4). Then .To demonstrate the usefulness of Theorem 7 in recognizing

SMIT ordered random variables, we consider the following ex-amples.Example 1: Suppose that are s-independent

exponential random variables with means . Letand be and mixtures of . An application of Theorem

7 immediately yields for every two probabilityvector and such that provided that.Example 2: Let , and denote the convolution ofexponential distributions with parameters , and

, respectively. Without loss of generality, assumethat and . Forand according to Theorem 7, we have

whenever for .

IV. THE SIMIT CLASS OF LIFE DISTRIBUTIONS

This section examines some properties and applications ofthe SIMIT class of life distributions. The next result providesa sufficient condition for a probability distribution to have theSIMIT property.Theorem 8: Let be a lifetime random variable with the

RHR function . If is non-increasing in, then is SIMIT.To demonstrate the usefulness of Theorem 8 in recognizing

parametric SIMIT distributions, we consider the following ex-amples.Example 3: Let be an exponential random variable with

pdf for , and . The RHRfunction of is . One can easilycheck that is non-increasing in , and hence, accordingto Theorem 8, is SIMIT.Example 4: Let have a power distribution with distribution

function for . The RHR functionof is , and is non-increasing in .According to Theorem 8, is SIMIT.Example 5: Let have a reciprocal Weibull distribution

with distribution function

and . The RHR function is ,and is non-increasing in . Hence, isSIMIT.The following theorem presents a characterization property

of the SIMIT class.Theorem 9: A lifetime random variable is SIMIT iff

for each random variable with ,which is s-independent of .


Theorem 10: Let be non-negative function onstrictly increasing, log-convex, and differentiable. Then

Finally, we discuss a sufficient condition for the life distri-bution of a device subjected to shocks occurring randomly ac-cording to Poisson process to belong to the SIMIT class. Sup-pose that the system is assumed to have an ability to withstanda random number of these shocks, and it is commonly assumedthat the number of shocks and the interarrival times of shocksare s-independent. Let denote the number of shocks survivedby the system, and let denote the random interarrival timebetween the -th and -th shocks. Then the lifetime ofthe system is given by . Therefore, shock modelsare particular cases of random sums. In particular, if the inter-arrivals are assumed to be s-independent and exponentially dis-tributed (with common parameter ), then the distribution func-tion of can be written as

(5)

where for all ( ) (cf. Pellerey[28]).Definition 5: The discrete probability distribution is

said to be discrete strong increasing in mean inactivity time(D-SIMIT) if

Theorem 11: If is D-SIMIT, then with distribu-tion function as given in (5) is SIMIT.

V. RELIABILITY APPLICATIONS

In this section, we discuss some relevant applications in re-liability theory involving the SMIT order, and the SIMIT classof life distributions. Suppose that are i.i.d. life-time random variables from , and that are alsoi.i.d. lifetime random variables from . Denote by

and

the lifetimes of the two associated parallel systems. In the fol-lowing result, we show that if the lifetimes of two parallel sys-tems with i.i.d. components are SMIT ordered, then their com-ponents are also SMIT ordered.Theorem 12: If , then for all

.Let be a sequence of i.i.d. random variables, and

let be a positive integer-valued random variable, which iss-independent of the . Denote by

the maximum extreme order statistic. This random variablearises naturally in reliability theory as the lifetime of a parallelsystem with the random number of identical components withlifetimes . In life testing, if a random cen-soring is adopted, then the completely observed data constitutea sample of random size , say, where

is a random variable of integer values. In actuarialscience, the claims received by an insurer in a certain timeinterval should also be a sample of random size, anddenotes the largest claim amount of the period. Letdenote the maximum order statistic amongwhere is a positive integer-valued random variable whichis s-independent from the sequence of for each

. Below we discuss the SMIT order between twosuch extreme order statistics. To state and prove Theorem 13,we need to recall the definition of the hazard rate (HR) order(Ahmad and Kayid [29], and Kayid et al. [30]). The randomvariable is said to be smaller than in the hazard rate order(denoted as ) if for all

.Theorem 13: Let , then

.Consider two devices with random lifetimes , andsubject to Poisson shocks occurring randomly with re-

spect to random numbers , and respectively. Denotefor all , and for each . It

is well-known that, if has the distribution function then(cf. Pellerey [28])

(6)

The following result presents another application in reliabilitytheory.Theorem 14: If

then .In the next example, we discuss the reversed preservation

property of the SIMIT class under the formation of the parallelsystems.Example 6: Let be i.i.d. random lifetimes

such that belongs to the SIMITclass. Then, we have for all . This resultmeans that, for all

(7)

In view of Theorem 3.2 of Li and Xu [14], and from (7), wededuce that for all .That is, is SIMIT.As an application of Theorem 9 in accelerated life models,

we consider the following example.Example 7: Consider units (not necessarily s-independent)

with lifetimes . Suppose that the units are


working in a common operating environment, which is repre-sented by a random vector , s-indepen-dent of , and has an effect on the units of the form

(8)

If has support on , then the components are workingin a harsh environment; and, if they have support on ,then the components are working in a gentler environment (seeMa [31]). In a harsh environment, let be SIMIT for some

. Then, Theorem 9 tells that, for each with supporton which is s-independent of , we have .Thus, implies that . Hence, by(8), it stands that . With a similar discussion, in agentler environment, if is SIMIT for some ,then we conclude that .

VI. CONCLUSION

Due to economic consequences and safety issues, it is nec-essary for the industry to perform systematic studies using re-liability concepts. There exist plenty of scenarios where a sta-tistical comparison of reliability measures is required in bothreliability engineering and biomedical fields. In this paper, wehave proposed a new stochastic order based on the MIT func-tion, called strong mean inactivity time (SMIT) order. The re-lationship of this new stochastic order with other well-knownstochastic orders such as the RHR and the MIT orders are dis-cussed. It was shown that the SMIT order lies in the frameworkof the RHR and the MIT orders, and hence it enjoys severalreliability properties which provide several applications in re-liability and survival analysis. We discussed several character-ization and preservation properties of this new order under re-liability operations of monotone transformation, mixture, andshock models. To enhance the study, we proposed a new class oflife distributions called strong increasing mean inactivity time(SIMIT) class. Several reliability properties of the new class aswell as a number of applications in the context of reliability andsurvival analysis are included. Our results provide new conceptsand applications in reliability, statistics, and risk theory. Furtherproperties and applications of the new stochastic order and thenew proposed class can be considered in the future of this re-search. In particular, the following topics are interesting, andstill remain as open problems.(i) Closure properties of the SMIT order and the SIMIT classunder convolution, and coherent structures.(ii) Discrete version of the SMIT order, and enhance the ob-tained results related to the D-SIMIT class.(iii) Testing exponentiality against the SIMIT class.

APPENDIX

Proof of Theorem 1: (i) Note that impliesis non-increasing in or equivalently

This simply gives

which means .(ii) Suppose that , and have the MIT functions , and

, respectively. Note that for all we get

where is a non-negative non-increasing function,and

where stands for the indicator function of the set. For all we have

Note also that implies that, for all ,

(9)

In addition, from (1) it holds that, for all ,

(10)

Combining (9) and (10), one gets, for all ,

which provides that, for all ,

Therefore, implies that forall . Finally, appealing to Lemma 7.1(b) of Barlow andProschan [24], it is concluded that for all

and hence the proof is completed.Proof of Theorem 2: We can write, for all ,


Similarly,

Therefore, using the assumptions, for all ,

which completes the proof.Proof of Theorem 3: First, let for all .

It follows that, for all ,

(11)

Now, we have

In view of (2), this gives . On the other hand,suppose that holds for any non-negative randomvariable . Then for all follows by takingas a degenerate variable.Proof of Theorem 4: First note that for any and

based upon (2), (3), (11) we have

Hence, it holds that, for all and

that is, for all .Proof of Theorem 5: Without loss of generality, assume

that is differentiable, and denote its first derivative by .implies that, for all ,

On the other hand, iff, for all ,

which is equivalent to, for all

where . Its well-known that, if is non-negative and concave with then is non-in-creasing. Thus, due to the assumption, is the product oftwo non-negative non-increasing functions, and hence isnon-increasing. Finally, Lemma 7.1(b) of Barlow and Proschan[24] can be used to conclude the proof.

Proof of Theorem 6: Select and in the support of .Let , and be the distri-

bution functions of ,

and respectively. The proof is similar to that ofTheorem 1.B.8 in Shaked and Shanthikumar [13]. It is sufficientto show that, for each and for all , we have

This is an inequality of the form

where all eight variables are non-negative, and by the assump-tions of the theorem they satisfy


It is easy to verify that the latter four inequalities imply theformer one, completing the proof of the theorem.

Proof of Theorem 7: Because of (1), we need to establishthat, for all ,

(12)

After simple calculations, (12) can be written in the form

Now, for each fixed pair with , we have

which is non-negative because both terms are non-negative bythe assumptions. This completes the proof.

Proof of Theorem 8: It suffices to verify thatis non-decreasing in . Note that

Define now

where , and

Due to the assumption, is in forand . Also, it is easy to observe that is in

for and . From the general compositiontheorem of Karlin [16] it is deduced that is infor and and thus the proof is completed.

Proof of Theorem 9: To prove the “if” part, note thatfor each , and any .

Then, put implying for allwhich means is SIMIT. For the “only if” part, assume thathas distribution function . From the assumption, and the

well-known Fubini’s theorem, for all it follows that

That is .


Proof of Theorem 10: It suffices to prove that

is non-decreasing in . We know that

Define

(13)

for and for all where

and . Easily, is in . By as-sumption, and are non-increasing and non-negative.Hence is in . On applying the general com-position theorem of Karlin [16] to the identity given in (13),

is in . This completes theproof.

Proof of Theorem 11: It is enough to show thatis non-decreasing in . To this end, first

we get, for all that

Note then that

for each positive integer . Hence, is SIMIT if

or equivalently if

for and where

By the assumption, is in for and. It is also evident that is in forand . The result now follows from general

composition theorem of Karlin [16].Proof of Theorem 12: Let hold. Then, we

have

Due to fact that

where , andapplying Lemma 7.1(b) of Barlow and Proschan [24], it obtainsthat, for all ,

which means that for all .Proof of Theorem 13: First, notice that in-

dicates the hazard rate order between and . Denote bythe distribution function of .

where is the common distribution of the andfor all is the pmf of .

One can easily verify, for all and each that

where , and . Denote

for each and . It can easilybe checked that (cf. Shaked and Shanthikumar [13]) ifthen is in . On the other hand,is in . Appealing to Lemma 2.1 in

Ortega [11] gives is in whichis equivalent to .


Proof of Theorem 14: In view of (5), and the proof of The-orem 11, we have for all

Because is in and by assuming

the general composition theorem of Karlin [16] provides thatis in which implies

that .

ACKNOWLEDGMENT

The authors like to thank the Associate Editor Dr. M. J. Zuo,the Managing Editor Dr. J. W. Rupe, and the referees for theircareful reading and constructive comments which helped to im-prove the paper.

REFERENCES

[1] M. Kayid and I. A. Ahmad, “On the mean inactivity time orderingwith reliability applications,” Probability Eng. Inf. Sci., vol. 18, pp.395–409, 2004.

[2] I. A. Ahmad, M. Kayid, and F. Pellerey, “Further results involving theMIT order and the IMIT class,” Probability Eng. Inf. Sci., vol. 19, pp.377–395, 2005.

[3] F. Badia and M. Berrade, “On the reversed hazard rate and mean in-activity time of mixture,” in Advances in Mathematical Modeling forReliability, T. Bedford, Ed. et al. Amsterdam, The Netherlands: DelftUniv. Press, 2008, pp. 103–110.

[4] M. S. Finkelstein, “On the reversed hazard rate,” Rel. Eng. Syst. Safety,vol. 78, pp. 71–75, 2002.

[5] M. Asadi, “On the mean past lifetime of the components of a parallelsystem,” J. Statist. Plann. Inference, vol. 136, pp. 1197–1206, 2006.

[6] S. Goliforushani and M. Asadi, “On the discrete mean past lifetime,”Metrika, vol. 68, pp. 209–217, 2008.

[7] L. Eeckhoudt and C. Gollier, “Demand for risky assets and the mono-tone probability ratio order,” J. Risk Uncertainty, vol. 11, pp. 113–122,1995.

[8] M. Kijima and M. Ohnishi, “Stochastic orders and their applicationsin financial optimization,” Math. Methods Oper. Res., vol. 50, pp.351–372, 1999.

[9] C. Kundu, A. K. Nanda, and S. S. Maiti, “Some distributional re-sults through past entropy,” J. Statist. Plan. Inference, vol. 140, pp.1280–1291, 2010.

[10] C. Kundu and A. K. Nanda, “Some reliability properties of the inac-tivity time,” Commun. Statist.-Theory Methods, vol. 39, pp. 899–911,2010.

[11] E. M. Ortega, “A note on some functional relationships involvingthe mean inactivity time order,” IEEE Trans. Rel., vol. 58, no. 1, pp.172–178, Mar. 2009.

[12] S. Izadkhah and M. Kayid, “Reliability analysis of the harmonic meaninactivity time order,” IEEE Trans. Rel., vol. 62, no. 2, pp. 329–337,Jun. 2013.

[13] M. Shaked and J. G. Shanthikumar, Stochastic Orders. New York,NY, USA: Springer-Verlag, 2007.

[14] X. Li and M. Xu, “Some results about MIT order and IMIT class of lifedistributions,” Probability Eng. Inf. Sci., vol. 20, pp. 481–496, 2006.

[15] A. Müller and D. Stoyan, Comparison Methods for Stochastic Modelsand Risks. New York, NY, USA: Wiley, 2002.

[16] S. Karlin, Total Positivity. Stanford, CA, USA: Stanford Univ. Press,1968.

[17] A. K. Nanda, S. Bhattacharjee, and S. S. Alam, “On upshifted reversedmean residual life order,” Commun. Statist.—Theory Methods, vol. 35,pp. 1513–1523, 2006.

[18] D. Stoyan, Comparison Methods for Queues and Other StochasticModels. New York, NY, USA: Wiley, 1983.

[19] D. Yue and J. Cao, “Some results on the residual life at random time,”Acta. Math. Appl. Sinica, vol. 16, pp. 436–443, 2000.

[20] X. Li and M. J. Zuo, “Stochastic comparison of residual life and in-activity time at a random time,” Stoch. Models, vol. 20, pp. 229–235,2004.

[21] N. Misraa, N. Guptaa, and I. D. Dhariyal, “Stochastic properties ofresidual life and inactivity time at a random time,” Stoch. Models, vol.24, pp. 89–102, 2008.

[22] N. N. Cai and Y. Zheng, “Preservation of generalized ageing classes onthe residual life at random time,” J. Statist. Plan. Inference, vol. 142,pp. 148–154, 2012.

[23] S. Ross, Stochastic Processes. New York, NY, USA: Wiley, 1996.[24] R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life

Testing. Montgomery , MD, USA: Silver Spring, 1981.[25] Y. Chen, “Classes of life distributions and renewal counting process,”

J. Appl. Probability, vol. 31, pp. 1110–1115, 1994.[26] I. A. Ahmad, A. Ahmed, I. Elbatal, and M. Kayid, “An aging notion

derived from the increasing convex ordering: The NBUCA class,” J.Statist. Plan. Inference, vol. 136, pp. 555–569, 2006.

[27] F. Belzunce, E. Ortega, and J. M. Ruiz, “Comparison of expectedfailure times for several replacement policies,” IEEE Trans. Rel., vol.55, no. 3, pp. 490–495, Sep. 2006.

[28] F. Pellerey, “Shock models with underlying counting process,” J. Appl.Probability, vol. 31, pp. 156–166, 1994.

[29] I. A. Ahmad and M. Kayid, “Characterizations of the RHR and MITorderings and the DRHR and IMIT classes of life distributions,” Prob-ability Eng. Inf. Sci., vol. 19, pp. 447–461, 2005.

[30] M. Kayid, I. A. Ahmad, S. Izadkhah, and A. M. Abouammoh, “Furtherresults involving the mean time to failure order, and the decreasingmean time to failure class,” IEEE Trans. Rel., vol. 62, no. 3, pp.670–678, Sep. 2013.

[31] C. Ma, “Uniform stochastic ordering on a system of components withdependent lifetimes induced by a common environment.,” Sankhya:The Indian J. Statist., vol. 61, pp. 218–228, 1999.

M. Kayid is a Professor of Statistics at the Department of Statistics and Oper-ations Research at King Saud University. He obtained the B.Sc. degree (math-ematics, 1991) from Al Menia University, the M.Sc. degree (statistics, 1999)from Al Azhar University, and the Ph.D. degree (statistics, 2003) from SouthValley University. His research interests are in reliability theory and preventivemaintenance, stochastic orders, and their applications; and he has published in avariety of international journals on these issues. His current research is in non-parametric statistics and life testing. He is an elected Reviewer for AmericanMathematical Society, and he is a director for the Accreditation and Assess-ment Unit at King Saud University.

S. Izadkhah is a Ph.D. student of Statistics at the Department of Statistics, Fer-dowsi University of Mashhad, Mashhad, Iran. He received the B.Sc. degree ofStatistics in 2007 from Birjand University, and the M.Sc. degree of Statistics in2010 from Ferdowsi University of Mashhad. His research interests are in relia-bility theory, stochastic orders, life testing, and dependence theory.

Mean Inactivity Time Function, Associated Orderings, and Classes of Life Distributions

Documents

Transcript of Mean Inactivity Time Function, Associated Orderings, and Classes of Life Distributions