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If the random variable X denotes the lifetime of a unit, then the random variable XðtÞ ¼ ½t X Xj 6 t for a fixed t > 0 is known as the inactivity time. In this paper, based on the random variable X(t), a new class of life distributions, namely increasing variance inactivity time (IVIT) and the concept of inactivity coefficient of variation (ICV), are introduced. The closure properties of the IVIT class under some reliability operations, such as mixing, convolution and formation of coherent systems, are obtained.
Journal of Advanced Research (2012) 3, 29–34 Cairo University Journal of Advanced Research ORIGINAL ARTICLE Characterization and preservations of the variance inactivity time ordering and the increasing variance inactivity time class Mervat Mahdy * Department of Statistics, Mathematics and Insurance, College of Commerce, Benha University, Egypt Received 25 December 2010; revised February 2011; accepted March 2011 Available online 16 April 2011 KEYWORDS Conditional variance; Increasing variance inactivity time; Mixing; Convolution; Formation of coherent system; Erlang distribution Abstract If the random variable X denotes the lifetime of a unit, then the random variable Xtị ẳ ẵt XjX t for a fixed t > is known as the inactivity time In this paper, based on the random variable X(t), a new class of life distributions, namely increasing variance inactivity time (IVIT) and the concept of inactivity coefficient of variation (ICV), are introduced The closure properties of the IVIT class under some reliability operations, such as mixing, convolution and formation of coherent systems, are obtained ª 2011 Cairo University Production and hosting by Elsevier B.V All rights reserved Introduction Let the random variable X denote the lifetime (X > 0, with probability one) of a unit, having an absolutely continuous distribution function F, survival function F ¼ À F and density function f Let the random variable Xtị ẳ ẵt XjX t denote * Tel.: +20 120682460/+20 225077099; fax: +20 13 323 0860 E-mail addresses: drmervat.mahdy@fcom.bu.edu.eg, mervat_em@ yahoo.com 2090-1232 ª 2011 Cairo University Production and hosting by Elsevier B.V All rights reserved Peer review under responsibility of Cairo University doi:10.1016/j.jare.2011.03.001 Production and hosting by Elsevier the time elapsed after failure till time t, given that the unit has already failed at time t, for t > The random variable X(t) is known as the inactivity time of a unit at time t Recently, the random variable X(t) has received considerable attention in the literature, see Ahmad and Kayid [1], Li and Xu [2], Lai and Xie [3], Mahdy [4], and Nair and Sudheesh [5] In the literature, the function r~F xị ẳ fxị=Fxị is known as the reversed (or retro or backward) hazard rate function (cf Shaked and Shanthikumar [6]) In the analysis of left-censored data, the reversed hazard rate function plays the same role as that of the hazard rate function in the analysis of right-censored data (cf Anderson et al [7]) The reversed hazard rate ordering is related to the random variable X(t) Ahmad and Kayid [1] characterized the decreasing reversed hazard rate (DRHR) based on variability ordering of the inactivity time of k-out-of-n system given that the time of the (n À k + 1)th failure occurs at or sometimes before time t P In this paper, we focus our attention on nonparametric classes of life distributions defined in terms of the variance 30 M Mahdy of X(t).These classes are the increasing variance inactivity time (IVIT) and inactivity coefficient of variation time (ICV) Section ‘preliminaries’ contains definitions, notation and basic properties used through the paper In this section, we study some properties of the IVIT class and the ICV class The main results and their proofs are provided in Section ‘preservation properties’, where we establish closure properties of the classes under relevant reliability operations such as mixing, convolution and formation of coherent systems; we show, for example, that the class IVIT is closed under convolution, mixing and the formation of coherent systems The variance inactivity time of parallel systems is provided in Section ‘variance inactivity time of parallel systems’ d ẵr tị ẳ ~ rtịẵm2F tị r2F tị: dt F The following definition is essential to our work: Definition 2.1 A random variable X having distribution function F has increasing variance inactivity time life, which we denote as IVIT, if FðtÞ Z Let X be a random variable with distribution R function F(t), survival function F ¼ À F, mean life l ¼ FðuÞdu and variance r2 = Var(X) So the mean inactivity time (MIT), mF(t), and variance inactivity time, r2F ðtÞ, respectively, can be dened as follows: mF tị ẳ Et XjX tÞ; Rt FðuÞdu ; X t; t P 0; ẳ Ftị 2:1ị and ẳ Varẵt XjX t: 2:2ị The following denitions extend the increasing mean inactivity time, IMIT, and IVIT classes into the orderings between variables Let X and Y be two non-negative and absolutely continuous random variables, having distribution functions F and G, reversed hazard rate functions ~ rF and r~G , the mean inactivity time functions mF(t) and mG(t), and the variance inactivity time functions r2F ðtÞ and r2G ðtÞ, respectively The mean and the variance inactivity time orderings can be defined as follows: Definition 2.2 X is said to be smaller than or equal to Y in mean inactivity time ordering (X 6mit Y) if Rt Note that r2F tị uyịdy m2F tị; Ry where uyị ẳ FðxÞdx.Equivalently, X IVIT if, and only if, Z t uðyÞdy u2 ðtÞ: FðtÞ Preliminaries r2F ðtÞ t 2 ẳ Eẵt Xị jX t ẵmF tị : Fuịdu P Ftị Rt Guịdu ; for all t P 0: GðtÞ Clearly r2F ðtÞ ẳ 2tmF tị m2F tị Consider EẵU2 jt ẳ one has r2F tị ỵ m2F tị ẳ R1 FðtÞ FðtÞ Z t xFðxÞdx: 0 u2 dF½ujtdt, using integration by parts Z t Z y Definition 2.3 X is said to be smaller than or equal to Y in variance inactivity time ordering (X 6vit Y) if FðxÞdxdy; 0 so, Eq (2.2) is equivalent to Z tZ y r2F tị ẳ Fxịdxdy m2F tị: Ftị 0 Ry Let uyị ẳ Fxịdx, then Z t r2F tị ỵ m2F tị ẳ uðyÞdy; FðtÞ Rt Rx uðtÞ ; FðtÞ Rt Rx ð2:3Þ Differentiating (2.2) with respect to t, we have Rt 2ẵutịFtị ftị uyịdy d ẵrF tị ¼ À 2mF ðtÞmnF ðtÞ: dt F2 ðtÞ and using (2.3)–(2.5) in (2.6), we obtain that R 0t R 0x 2:4ị also from (2.4), we get m0F tị ẳ À r~ðtÞmF ðtÞ: FðuÞdudx P FðtÞ Rt Rx 0 GðuÞdudx ; for all t P 0: GðtÞ It can be written as and from (2.1), we get mF tị ẳ It can be written as Rt Fuịdu is increasing in t P 0: R 0t GðuÞdu ð2:5Þ ð2:6Þ FðuÞdudx GðuÞdudx is increasing in t P 0: In probability theory and statistics, the coefficient of variation (CV) is a normalized measure of dispersion of a probability distribution It is defined as the ratio of the standard deviation r to the mean l: r CV ¼ : l This is only defined for a non-zero mean, and it is most useful for variables that are always positive It is also known as unitized risk, also it is used in some applied probability fields such as renewal theory, queueing theory, and reliability theory Now, we can define the coefficient of variation of the random variable X(t) as follows: Characterization of variance inactivity time cF tị ẳ 31 rF ðtÞ : mF ðtÞ By using Bool’s rule (Mathews and Fink [8]), we can show that The Erlang distribution is a continuous probability distribution with wide applicability primarily due to its relation to the Exponential and Gamma distributions The distribution is used in the field of stochastic processes The probability density function of the Erlang distribution is kk xkÀ1 expðÀkxÞ fx; k; kị ẳ ; k 1ị! for x; k P 0: k1 X kxịn expkxị: n! nẳ0 (ii) An approximation of the variance inactivity lifetime function is given by: P kiị i2 t2 2t k1 ỵ kiẳ1 ðiÀ1Þ! k Bi k r2F ðtÞ % À m2F ðtÞ; Pk1 ktịn nẳ0 n! expktị where i 512 À1kt t i 96 À1kt t i 512 3kt 3t 28 e e ỵ e ỵ ỵ ekt ti : 45 45 135 45 (iii) The inactivity coefficient of variation of the random variable X(t) is greater than Proof When X has the Erlang distribution and using Eq (2.4), we can show that Rt 1 Pk1 kxịn nẳ0 n! Pk1 ktịn nẳ0 n! expkxịdx ; expktị hP i k i2 i1 ki t k1 ỵ expktị t iẳ1 i1ị! k k ẳ ; t P 0; P ktịn k1 nẳ0 n! expktị where Z t k1 X kxịn expkxịdx n! nẳ0 " # k X k1 k i i2 i1 ỵ expktị k t : ẳt k i 1ị! iẳ1 Hence from Eq (2.7) the result follows y FðxÞdxdy % t2 À 2t k kÀ1 X k À i i2 ỵ k Bi ; k i 1ị! iẳ1 where i 512 À1kt t i 96 À1kt t i 512 3kt 3t 28 ỵ e þ þ eÀkt ti : e e 45 45 135 45 From (2.2), (2.7), and (2.8), we get the complete proof of (ii) Also, by using (i) and (ii), we get the complete proof (iii) A hyper-exponential distribution is a continuous distribution with the probability density function as follows: fX ¼ Proposition 2.1 If X is a non-negative random variable having the Erlang distribution with scale parameter k, and shape parameter k Then (i) The mean inactivity lifetime function is given by: hP i k kiị i2 i1 t k1 ỵ expktị t iẳ1 i1ị! k k ; 2:8ị mF tị ẳ P ktịn k1 nẳ0 n! expktị mF tị ẳ Z ð2:7Þ Now, we can discuss the behavior of ICV in the Erlang distribution Bi ¼ t Bi ¼ where ‘‘exp’’ is the base of the natural logarithm and ‘‘!’’ is the factorial function The parameter k is called the shape parameter and the parameter k is called the rate (scale) parameter The cumulative distribution function of the Erlang distribution can be expressed as Fx; k; kị ẳ Z n X fYi yịpi ; 2:9ị iẳ1 where Yi is an exponentially distributed random variable with rate parameter ki , and pi is the probability that X will take on the form of the exponential distribution with rate ki Now, we can study some of properties of hyper-exponential distributions in terms of the following propositions: Proposition 2.2 If X has a hyper-exponential distribution, then (i) The mean inactivity time function is given by o Pn n 1 i¼1 pi t ỵ ki expki tị ki Pn ; mF tị ẳ iẳ1 pi f1 expki tịg 2:10ị (ii) The variance inactivity time function is given by n o P i tị niẳ1 pi t2 kti expk ỵ k12 k2i i Pn rF tị ẳ iẳ1 pi f1 expki tịg o32 2Pn n 1 iẳ1 pi t ỵ ki expki tị À ki 5; À Pn i¼1 pi f1 À expðÀki tÞg (iii) The coefficient of variation of X(t) is less than Proof By the expression followed from (2.4) for the distribution function given in (2.9), (i) is satisfied Also, by using (2.2) and (2.10) we get the complete the proof of (ii) It is easy to check that maximum value of the inactivity coefficient of variation of the random variable X(t) is less than one; this is the complete proof of (iii) h Preservation properties This section will develop some preservation of VIT order and IVIT Theorem 3.1 X is IVIT if and only if X6vit X ỵ Y for any Y independent of X 32 M Mahdy Proof Necessity: If r2F ðtÞ is increasing in t P 0, then by Fubini’s theorem, we have for any t P 0, Rt Ry Rx 0 Fðx uịdGuịdxdy 2 rXỵYị tị ỵ mXỵYị tị ẳ ; Rt Fðt À uÞdGðuÞ R t R tÀu R yu FxịdxdydGuị ; ẳ 0Rt Ft uịdGuị Rt Ft uịẵr2Xị t uị ỵ m2Xị t uịdGuị ẳ Rt Ft uịdGuị r2Xị tị ỵ m2Xị tị: By Proposition 2.3 in Li and Xu [2] we get that Z t Z Gn ðtÞFn ðuÞdudx À 0 t Z x Fn ðtÞGn ðuÞdudx P 0: Since, for any t P 0, " n X kuị ẳ ẵGni tịFni uịẵFi1 tịGi1 uị #1 ; iẳ1 is non-negative and decreasing in u P 0, by Theorem 3.1 of Li and Xu [2] we have, Z tZ x ẵGtịFuị FtịGuịdudx m2XỵYị tị m2Xị tị: x Z ẳ Z Thus t Z x kuịẵGn tịFn uị À Fn ðtÞGn ðuÞdudx P 0; h X 6vit X ỵ Y: which states that X 6vit Y Theorem 3.2 Assume that / is strictly increasing and concave Variance inactivity time of parallel systems /0ị ẳ 0: We consider a parallel system consisting of n identical components with independent lifetimes having a common distribution function F It is assumed that at time t the system failed Under these conditions, Asadi [9], Asadi and Bayramoglu [10] and Bairamov et al [11] introduced MIT of the components of this system Also, they mention some of its properties such as recovered distribution function by application of MIT, and comparison between MITs of two parallel systems On the basic of the structure of parallel systems, when a component with lifetime Tr:n = r = 1, 2, , n À fails the system is continuing to work until Tn:n fails In fact, the system can be considered as a black box in the sense that the exact failure time of Tr:n is unknown Motivated by this, we assume that at time t the system is not working and in fact, it has failed at time t or sometime before time t Let If X 6vit Y then /ðXÞ 6vit uðYÞ Proof Without loss of generality, assume that / is differentiable with derivative /n Thus X 6vit Y implies that for any t P 0, Z Z /À1 ðtÞ /À1 ðx:Þ FðuÞ GðuÞ À dudx P 0: Fð/À1 ðtÞÞ Gð/À1 ðtÞÞ Since /n ðtÞ is non-negative and decreasing, by Theorem 3.1 of Li and Xu [2] it holds that Z Z /À1 ðtÞ /À1 ðx:Þ /n ðtÞ FðuÞ GðuÞ À dudx Fð/À1 ðtÞÞ Gð/À1 ðtÞÞ P 0; for any t > 0: Equivalently, Z Z /1 tị ITr:n;tị ẳ ẵt Tr:n jTn:n t; t > 0; r ¼ 1; 2; ; n: /À1 ðxÞ Z /À1 ðtÞ Z n / ðtÞFðuÞ dudx Fð/À1 ðtÞÞ /À1 ðxÞ P 0 where ITr:n,(t) shows, in fact, the time that has passed from the failure of the component with lifetime Tr:n in the system given that the system has failed at or before time t If we denote the expectation of ITr:n,(t) by Mrn ðtÞ and variance of ITr:n,(t) by Vrn ðtÞ, i.e /n ðtÞGðuÞ dudx; Gð/À1 ðtÞÞ that is, for any t > 0, Z t Z x Fð/À1 ðuÞÞ dudx P Fð/À1 ðtÞÞ Z t Z Mrn tị ẳ EITr:n;tị ị; t > 0; r ¼ 1; 2; ; n; x Gð/À1 ðuÞÞ dudx; Gð/À1 ðtÞÞ which shows for any t P that /ðXÞ 6vit uðYÞ and h Theorem 3.3 Let X1, , Xn and Y1, , Yn be independent and identically distributed (i.i.d) copies of X and Y, respectively If maxfX1 ; ; Xn g 6vit maxfY1 ; ; Yn g then X 6vit Y Proof maxfX1 ; ; Xn g6vit maxfY1 ; ; Yn g implies that Rt Rx 0 Fn ðuÞdudx P Fn ðtÞ that is Rt Rx 0 Gn ðuÞdudx for any t > 0; Gn ðtÞ Vrn tị ẳ VarITr:n;tị ị; t > 0; r ẳ 1; 2; ; n: Then Mrn ðtÞ measures the MIT from the failure of the component with lifetime Tr:n given that the system has a lifetime less than or equal to t Also, Vrn ðtÞ measures the VIT from the failure of the component with lifetime Tr:n given that the system has a lifetime less than or equal to t Let a parallel system with n non-negative independent components having a common continuous distribution function F with left extremity a = inf {t:F(t) > 0} and right extremity b = sup {t:F(t) < 1} In the following, we derive the distribution of ITr:n,(t) Let RðxjtÞ denote the reliability function of ITr:n,(t), for x < t and x, t (a, b) Then Characterization of variance inactivity time Rxjtị ẳ PrITr:n;tị P xị; Pn n i F t xịFtị Ft xịịni iẳr i ; 4:1ị ẳ Fn tị n X X n i Ft xị iỵj n ni ¼ ðÀ1Þj ð Þ ; for r ¼ 1; .; n: Ftị j i jẳ0 iẳr Using the survival function given in (4.1), Asadi [9] obtains the MIT of Tk:k as follows: Mk tị ẳ Eẵt Tk:k P xjTk:k t; k ¼ 1; 2; ; Rt k F uịdu : ẳ k F tị Now, let us define the second non-central moment of the parallel system lifetime, which is denoted by Sk(t) as follows: Rt Rx 0 Fk ðuÞdudx ; Sk ðtÞ ¼ Fk ðtÞ Rt k uF ðuÞdu ; ð4:2Þ ¼ 2tMk ðtÞ À k F ðtÞ ¼ 2tMk tị Uk tị; 33 Rxjtị ẳ iỵj t n X X n nÀi nÀi e À ex 1ịj ; et i jẳ0 j iẳr x < t; t > 0: By Asadi [4], we get that: Rt uFiỵj uịdu ; Siỵj tị ẳ 2tMiỵj tị iỵj F tị iỵj Rt iỵj P 2t 1ịk ekt ị uFiỵj uịdu k k kẳ0 ẳ : et ịiỵj Since Z t uFiỵj uịdu ẳ iỵj X iỵj 1ịk ẵ1 ekt tk ỵ 1ị; k k kẳ0 so that Piỵj Siỵj tị ẳ where Rt iỵj 1 ekt ị k k Uiỵj tị; et ịiỵj k kẳ0 1ị whereas uFk uịdu : Fk ðtÞ Also, by using (4.2), we get the VIT of Tk:k as follows: P k iỵj iỵj ẵ1 ekt tk ỵ 1ị kẳ0 1ị k2 k : Uiỵj tị ẳ et ịiỵj Vk tị ẳ Varẵt Tk:k P xjTk:k t; So, we can show that for r = 1, , n, Uk tị ẳ k ẳ 1; 2; ; ẳ 2tMk tị M2k ðtÞ À Uk ðtÞ: Furthermore, Asadi [9] mentioned that MIT of Tr:n is Z t Rxjtịdx; Mrn tị ẳ n X X n nÀi nÀi Miỵj tị: 1ịj ẳ i jẳ0 j iẳr Now, we can obtain the VIT of Tr:n as follows: Z t Vrn tị ẳ t xịRxjtịdx ẵMrn tị2 : Piỵj k iỵj 1ị ẵ1 ekt n ni kẳ0 X n X k k j nÀi r Vn ðtÞ ẳ 2t 1ị i jẳ0 j et ịiỵj i¼r n X X n nÀi nÀi 1ịj i jẳ0 j iẳr Piỵj i ỵ j k kt tk ỵ 1ị ẵ1 e kẳ0 1ị k k iỵj t e ị 32 P iỵj X iỵj 1ịk ẵ1 ekt n nÀi k¼0 k X n nÀi k 7: 1ịj i jẳ0 j et ịiỵj iẳr But note that Z t Urn tị ¼ xRðxjtÞdx; n X X n ni j ni 1ị ẳ Uiỵj tị; i jẳ0 j i¼r so we can define the second non-central moment of Tr:n as follows: Acknowledgement The author is grateful to Dr Ibrahim Ahmad (Professor and Head, Department of Statistics, Oklahoma State University, USA) for reading preliminary versions of this paper and making many useful comments Srn tị ẳ 2tMrn tị Urn tị: Consequently Vrn tị ẳ 2tMrn tị Urn tị ẵMrn tị2 : Example Let T0i s, i ¼ 1; ; n, n P 1, be an independent exponential with mean Then References [1] Ahmad IA, Kayid M Characterizations of the RHR and MIT orderings and the DRHR and IMIT classes of life distributions Probab Eng Inf Sci 2005;19(4):447–61 [2] Li X, Xu M Some results about MIT order and IMIT class of life distributions Probab Eng Inf Sci 2006;20(3):481–96 34 [3] Lai CD, Xie M Stochastic aging and dependence for reliability 1st ed New York: Springer; 2006 [4] Mahdy M On some new stochastic orders and their properties in the statistical reliability theory Benha University; 2009 [5] Nair NU, Sudheesh KK Characterization of continuous distributions by properties of conditional variance Stat Methodol 2010;7(1):30–40 [6] Shaked M, Shanthikumar JG Stochastic orders 1st ed New York: Springer; 2007 [7] Andersen PK, Borgan O, Gill RD, Keiding N Statistical models based on counting processes New York: Springer; 1993 M Mahdy [8] Mathews JH, Fink KD Numerical methods using MATLAB Upper Saddle River N.J.: Prentice Hall; 1999 [9] Asadi M On the mean past lifetime of the components of a parallel system J Stat Plan Infer 2006;136(4):1197–206 [10] Asadi M, Bayramoglu I A note on the mean residual life function of a parallel system Commun Stat Theory Methods 2005;34(2):475–84 [11] Bairamov I, Ahsanullah M, Akhundov I A residual life function of a system having parallel or series structures J Stat Theory Appl 2002;1(2):119–32 ... renewal theory, queueing theory, and reliability theory Now, we can define the coefficient of variation of the random variable X(t) as follows: Characterization of variance inactivity time cF tị... get the complete the proof of (ii) It is easy to check that maximum value of the inactivity coefficient of variation of the random variable X(t) is less than one; this is the complete proof of. .. M Mahdy of X(t).These classes are the increasing variance inactivity time (IVIT) and inactivity coefficient of variation time (ICV) Section ‘preliminaries’ contains definitions, notation and basic