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
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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
596 IEEE TRANSACTIONS ON RELIABILITY, VOL. 63, NO. 2, JUNE 2014
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 .
KAYID AND IZADKHAH: MEAN INACTIVITY TIME FUNCTION, ASSOCIATED ORDERINGS, AND CLASSES OF LIFE DISTRIBUTIONS 597
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
598 IEEE TRANSACTIONS ON RELIABILITY, VOL. 63, NO. 2, JUNE 2014
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 ,
KAYID AND IZADKHAH: MEAN INACTIVITY TIME FUNCTION, ASSOCIATED ORDERINGS, AND CLASSES OF LIFE DISTRIBUTIONS 599
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
600 IEEE TRANSACTIONS ON RELIABILITY, VOL. 63, NO. 2, JUNE 2014
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 .
KAYID AND IZADKHAH: MEAN INACTIVITY TIME FUNCTION, ASSOCIATED ORDERINGS, AND CLASSES OF LIFE DISTRIBUTIONS 601
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 .
602 IEEE TRANSACTIONS ON RELIABILITY, VOL. 63, NO. 2, JUNE 2014
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.
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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.
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