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Properties of aging intensity function
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Statistics & Probability Letters 77 (2007) 365–373 www.elsevier.com/locate/stapro
Properties of aging intensity function Asok K. Nanda, Subarna Bhattacharjee, S.S. Alam Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, West Bengal, India Received 6 January 2006; received in revised form 4 May 2006; accepted 7 August 2006 Available online 5 September 2006
Abstract Recently, the concept of aging intensity (AI) function has been introduced in the literature for evaluating the aging property of a unit (that may be a system or a living organism) quantitatively. In this paper, we discuss the properties of AI function and study its nature for various distributions. The closure properties of the aging classes defined in terms of AI function are also presented. We define an ordering, called aging intensity ordering, and study its closure properties under different reliability operations, viz., formation of k-out-of-n system, and increasing transformations. r 2006 Elsevier B.V. All rights reserved. Keywords: Aging property; k-out-of-n system; Spare allocation; Stochastic orders
1. Introduction An important phenomenon in reliability theory is ‘aging’ which is an inherent property of a unit that may be a living organism or a system of components. By aging we generally mean positive aging. Intuitively, aging means an increase of failure risk as a function of time. To be more specific, by aging (respectively, antiaging) we mean a mathematical specification of degradation (respectively, upgradation) of a unit over time. Antiaging is also known as negative aging. Failure rate, defined as the ratio of the density to its survival function, is one such measure of aging. It plays an important role in analyzing the failure pattern of a unit. Jiang et al. (2003) have pointed out that a unimodal failure rate can be effectively viewed as either approximately decreasing or approximately increasing or approximately constant. Clearly, such representation is qualitative. In their paper, a quantitative measure, called aging intensity (AI) function, defined as the ratio of the failure rate rðtÞ to a baseline failure rate, has been studied. One choice of the baseline failure rate Rt could be the average failure rate HðtÞ ¼ ð 0 rðuÞ duÞ=t. Thus, the AI function becomes LðtÞ ¼
rðtÞ ; HðtÞ
t40.
Here, in the present study, we explore the different properties of the AI function. Throughout this paper, the words ‘increasing (decreasing)’ and ‘nondecreasing (nonincreasing)’ are used interchangeably. We write Rþ ¼ ½0; 1Þ. The variables considered in this paper are all nonnegative. For a Corresponding author. Tel.: +91 3222 283686; fax: +91 3222 255303.
E-mail address: [email protected] (A.K. Nanda). 0167-7152/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.08.002
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random variable Z, we write its density as f Z ðÞ, survival function as F¯ Z ðÞ, failure rate as rZ ðÞ, distribution function as F Z ðÞ, and AI function LZ ðÞ. We conclude this section by giving a few well known aging properties from Bryson and Siddiqui (1969) (see also Barlow and Proschan (1975)) followed by a brief discussion on partial orders available in Shaked and Shanthikumar (1994) and Fagiuoli and Pellerey (1993). Definition 1.1. A random variable X is said to be (i) increasing in failure rate (IFR) if rX ðtÞ is increasing in tX0; Rt (ii) increasing in failure rate average (IFRA) if 0 rX ðuÞ du=t is increasing in t40; (iii) decreasing in mean residual life (DMRL) if mX ðtÞ ¼ E½X tjX 4t is decreasing in tX0. If, in the above definition, the word ‘increasing’ is replaced by ‘decreasing’ and conversely, then we get the respective dual classes known as decreasing in failure rate (DFR), decreasing in failure rate average (DFRA), and increasing in mean residual life (IMRL), respectively. Let us recall a few well-known stochastic orders. Definition 1.2. A nonnegative random variable X is said to be smaller than another nonnegative random variable Y in (i) (ii) (iii) (iv) (v) (vi)
likelihood ratio order (denoted by X plr Y ) if f Y ðtÞ=f X ðtÞ is increasing in tX0; failure rate order (denoted by X pfr Y ) if rX ðtÞXrY ðtÞ, for all tX0; starting failure rate order (denoted by X psfr Y ) if rX ð0ÞXrY ð0Þ; stochastic order (denoted by X pst Y ) if F¯ X ðtÞpF¯RY ðtÞ, for all tX0; R1 1 increasing convex order (denoted by X picx Y ) if t F¯ X ðuÞ dup t F¯ Y ðuÞ du, for all tX0; mean residual life order (denoted by X pmrl Y ) if Z
1
F¯ X ðuÞ du
Z F¯ X ðtÞ p
t
1
F¯ Y ðuÞ du
F¯ Y ðtÞ
for all tX0;
t
(vii) variance residual life order (denoted by X pvrl Y ) if Z t
1
Z
1
F¯ X ðvÞ dv du=
u
Z
1 t
Z
1
F¯ Y ðvÞ dv du
is decreasing in tX0.
u
The present paper is arranged as follows. In Section 2, we present the AI function of various distributions, introduce the aging class based on AI function and study the closure properties of the aging class. In Section 3 of this paper, we define an ordering called AI ordering and give some of its properties. The closure of the AI order is studied here under different reliability operations, viz., formation of k-out-of-n system, and increasing transformations. 2. Properties of AI function Jiang et al. (2003) have developed a new notion, called AI, to quantitatively evaluate the aging property of a unit which may be a system or a component or a living being. AI function of a random variable X, denoted by LX ðtÞ, is defined as the ratio of the instantaneous failure rate to the failure rate average, i.e., rX ðtÞ H X ðtÞ tf X ðtÞ ; ¼ F¯ X ðtÞ ln F¯ X ðtÞ
LX ðtÞ ¼
t40.
ð1Þ
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The larger the value of LX ðtÞ, the stronger the tendency of aging of the random variable X . The failure rate function uniquely determines the AI function but not conversely. In fact, two distributions having proportional failure rate functions do have the same AI function. It is to be noted that LX ðtÞ ¼ 1 for all t40 if and only if the failure rate function rX ðtÞ is constant (i.e., X follows exponential distribution). Thus, ‘LX ðtÞ 1 for t40,’ characterizes exponential distribution. Further, LX ðtÞ41 if rX ðtÞ is increasing in t (i.e., X is IFR); and LX ðtÞo1 if rX ðtÞ is decreasing in t (i.e., X is DFR). The next theorem shows that a constant AI function characterizes Weibull distribution. The proof is not difficult and hence is not included here. Theorem 2.1. For a random variable X, LX ðtÞ ¼ c, for t40, c being a constant if and only if X follows Weibull distribution with shape parameter c. On the basis of the monotonicity of the AI function, we define the following nonparametric family of distributions. Definition 2.1. A random variable X is said to be increasing in aging intensity (IAI) if the corresponding AI function LX ðtÞ is increasing in t40. We call the random variable X as decreasing in aging intensity (DAI) if LX ðtÞ is decreasing in t40. Further, it is seen that the monotonic behavior of the failure rate function is not, in general, transmitted to the monotonicity of the AI function as is evident from the following two counterexamples. Counterexample 2.1. Let X be a random variable having Erlang distribution with density function f X ðtÞ ¼ 4te2t ; tX0. Then, rX ðtÞ ¼ 4t=ð1 þ 2tÞ is increasing in tX0. So, X is IFR. Here, LX ðtÞ ¼ 4t2 =½ð1 þ 2tÞf2t lnð1 þ 2tÞg, which decreases in t40. Counterexample 2.2. Let X be a random variable having uniform distribution over ½a; b; 0paobo1. Then, the failure rate is given by rX ðtÞ ¼ ðb tÞ1 , aotob, which is increasing in t. So, X is IFR. It can be proved that LX ðtÞ ¼ t=½ðb tÞfln b lnðb tÞg increases with t, for aotob. From the foregoing two counterexamples it is observed that an IFR random variable can be IAI or DAI. For the random variable X having normal distribution with mean 4 and standard deviation unity, define Y ¼ ðX jX X4Þ, which is IFR. Then LY ðtÞ is nonmonotone in t. Thus, for an IFR random variable, the AI function could be nonmonotonic as well. Similar trend is also observed for a DFR random variable. For example, let a random variable X have the failure rate function ( 0:8 3t for 0ptp0:06; rX ðtÞ ¼ 0:62 for tX0:06: The corresponding AI function is given by 8 0:8 3t > > for 0otp0:06; < 0:8 1:5t LX ðtÞ ¼ 0:62t > > for tX0:06: : 0:0426 þ 0:62ðt 0:06Þ One can check that LX ðtÞ is decreasing for 0otp0:06, and increasing for tX0:06. 2.1. Closure properties of IAI and DAI classes In this section, we study whether IAI and DAI classes are closed under different reliability operations, viz., mixture of distributions, convolution of distributions and formation of k-out-of-n system. By k-out-of-n system we mean a system which operates as long as at least k out of the n components forming the system work. The following counterexamples show that IAI class is not closed under the aforesaid operations. It is to be mentioned here that a distribution with linear failure rate rX ðtÞ ¼ a þ bt with a40, bX0 is IAI.
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Counterexample 2.3. Let X 1 and X 2 be two random variables having respective failure rate functions rX 1 ðtÞ ¼ 1 þ 2t and rX 2 ðtÞ ¼ 2rX 1 ðtÞ; t40, and X be another random variable having survival function F¯ X ðtÞ ¼ 0:2F¯ X 1 ðtÞ þ 0:8F¯ X 2 ðtÞ. Then, for t40, 2
LX ðtÞ ¼
2tf4 þ 8t þ etþt ð0:5 þ tÞg . 2 tþt ð4 þ e Þf 0 ð8 þ 16u þ euþu2 ð1 þ 2uÞÞ=ð4 þ euþu2 Þ dug Rt
Although both X 1 and X 2 are IAI, the AI function of X is nonmonotone. Thus, IAI is not closed with respect to mixture. Next, we establish the fact that IAI class is not closed under convolution. Counterexample 2.4. Let X 1 and X 2 be X 2 and X 1 , respectively of Counterexample 2.3, then the AI function of the random variable X c ¼ X 1 þ X 2 is LX c ðtÞ ¼ f ðtÞ=gðtÞ where h n pffiffiffiffiffiffi 2 2 f ðtÞ ¼ 2t 3ðe1=6 ð1 þ 2tÞ þ e1=6þtþt ð1 þ 4tÞÞ þ e1=4þ2t=3þ4t =3 3pð1 þ 8t þ 4t2 Þ pffiffiffi pffiffiffiffiffiffi pffiffiffi oi 2 erfðð1 2tÞ=ð2 3ÞÞ e1=4þ2t=3þ4t =3 3pð1 þ 8t þ 4t2 Þ erfðð1 þ 4tÞ=2 3Þ , pffiffiffi 2 pffiffiffiffiffiffi 2 pffiffiffiffiffiffi 2 gðtÞ ¼ 3e1=6 ð3 þ 6etþt 2eð1=12Þð1þ4tÞ 3pð1 þ tÞ erfðð1 2tÞ=2 3 þ 2eð1=12Þð1þ4tÞ 3pð1 þ tÞ pffiffiffi pffiffiffi 2 pffiffiffiffiffiffi 1 2 erfðð1 þ 4tÞ=2 3ÞÞ ln e2tð1þtÞ ð3 þ 6etþt 2eð1=12Þð1þ4tÞ 3pð1 þ tÞ erfðð1 2tÞ=2 3Þ 9 pffiffiffi 2 pffiffiffiffiffiffi þ 2eð1=12Þð1þ4tÞ 3pð1 þ tÞ erfðð1 þ 4tÞ=2 3ÞÞ , and erfðzÞ is the ‘error function’ defined by erfðzÞ ¼ 2ð
Rz 0
pffiffiffi 2 et dtÞ= p. Here LX c ðtÞ, for t40, is nonmonotone.
Order statistics have nice applications in the study of the successive times of failures of the components or of systems. The kth order statistic is the lifetime of the ðn k þ 1Þ-out-of-n system. Next, we cite a counterexample to show that IAI class is not closed under the formation of a k-out-of-n system. Counterexample 2.5. Let X be a random variable with failure rate rX ðtÞ ¼ 1 þ 2t, t40, and f X ð2:3Þ ðtÞ be the density function of the 2nd order statistic in a sample of size 3 from the distribution of X . Then 2 2 2 LX ð2:3Þ ðtÞ ¼ ½6tðetþt 1Þð1 þ 2tÞ=½ð3etþt 2Þ lnfe3tð1þtÞ ð2 þ 3etþt Þg, the AI function of X ð2:3Þ , is nonmonotone. Next we deal with DAI class of distributions to study its closure properties under mixture, and formation of k-out-of-n system. In fact, it is also observed that DAI class of distributions is not closed under the aforementioned reliability operations which will be clear from the following counterexamples. Counterexample 2.6. Let X 1 and X 2 be two random variables having gamma distributions with respective probability density functions given by f X 1 ðtÞ ¼ 12t2 et , t40, and f X 2 ðtÞ ¼ 16 t3 et , t40, each of which is DAI. Let the survival function of the mixture random variable X be F¯ X ðtÞ ¼ 0:2F¯ X 1 ðtÞ þ 0:8F¯ X 2 ðtÞ. Then it follows that DAI class is not closed under mixture of distributions as LX ðtÞ ¼ ½t3 ð0:75 þ tÞ=½7:5et þ ð1:88233 þ tÞð3:98443 þ 1:86767t þ t2 Þ, the AI function of X , is nonmonotone for t40. From the counterexample given below, we see that DAI class is not closed under the formation of a k-outof-n system. Counterexample 2.7. Let X be a random variable having gamma distribution with probability density function f X ðtÞ ¼ 16et t3 , tX0, and f X ð2:3Þ ðtÞ be the density function of the 2nd order statistic X ð2:3Þ in a sample of size 3
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from this distribution. Then, the corresponding AI function is LX ð2:3Þ ðtÞ ¼ hðtÞ=kðtÞ, where hðtÞ ¼ 3t4 ð6 6et þ 6t þ 3t2 þ t3 Þ, and kðtÞ ¼ ð6 þ 6t þ 3t2 þ t3 Þð6 9et þ 6t þ 3t2 þ t3 Þ 3t e ln ð6 þ 9et 6t 3t2 t3 Þð6 þ 6t þ 3t2 þ t3 Þ2 . 108 It can be verified that LX ð2:3Þ ðtÞ is nonmonotone, although the underlying distribution is DAI. 3. Some properties of AI order We define a probabilistic order based on the AI function LðtÞ as follows. Definition 3.1. A random variable X is said to be smaller than another random variable Y in the AI order (denoted by X pAI Y ) if LX ðtÞXLY ðtÞ, for all t40. We mention a family of parametric distributions where AI ordering between the random variables is present. Example 3.1. Let X i be a random variable having Weibull distribution with survival function F¯ X i ðtÞ ¼ expðtai =bi Þ, ai ; bi 40; tX0; i ¼ 1; 2. If a1 Xa2 , then X 1 pAI X 2 . Theorem 3.1. For two random variables X and Y, the following conditions are equivalent. (i) RX pAI Y . R t t (ii) 0 rX ðuÞ du= 0 rY ðuÞ du is increasing in t40. (iii) ln F¯ X ðtÞ= ln F¯ Y ðtÞ is increasing in t40. A probabilistic order based on the monotonicity of the ratio rX ðtÞ=rY ðtÞ is studied in Sengupta and Deshpande (1994) and Rowell and Siegrist (1998). The random variable X is said to be aging faster than another random variable Y (written as X pAF Y ) if rX ðtÞ=rY ðtÞ is increasing in t. Based on AF order, a generalized stochastic order has been proved in Hu et al. (2001). The crossing hazards phenomenon has been studied in connection with prognostic studies in the treatment of breast cancer by Pocock et al. (1982). An increasing hazards ratio is a reasonable alternative to the proportional hazards model in this case, see Sengupta and Deshpande (1994), for instance. By using variation diminishing property, it can easily be verified that AF order defined above is stronger than AI order. The reflexive, commutative, and antisymmetric properties of the AI order are given below. The proof is omitted. Proposition 3.1. (i) X pAI X . (ii) If X pAI Y and Y pAI Z, then X pAI Z. (iii) If X pAI Y and Y pAI X , then X and Y have proportional failure rates. Before we study the different properties of the AI order, a natural question arises—what is the relationship between AI order and other existing stochastic orders? Below we take an attempt to answer such a question. The following counterexample shows that AI ordering does not imply increasing convex ordering. Counterexample 3.1. Let X be a random variable having exponential distribution with survival function F¯ X ðtÞ ¼ e0:5t , t40; and Y be another random variable having Weibull distribution with survival function
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¯ Y ðtÞ ¼ et0:6 , t40. Then, we have LX ðtÞ ¼ 1 and LY ðtÞ ¼ 0:6, for all t40. Clearly, X pAI Y . Let gðtÞ ¼ F R1 R1 ¯ ¯ t F X ðuÞ du and hðtÞ ¼ t F Y ðuÞ du. If X picx Y , then gðtÞphðtÞ, for all tX0. But from the values gð1:5Þ ¼ 0:944733, hð1:5Þ ¼ 0:7912, gð4:5Þ ¼ 0:210798, hð4:5Þ ¼ 0:321707, we can conclude that gðtÞ4 / hðtÞ for all tX0, i.e., X 4 / icx Y . This shows that X pAI Y RX picx Y . Since AI ordering does not imply increasing convex ordering, it is obvious that AI ordering implies none of likelihood ratio ordering, failure rate ordering, and stochastic ordering. The following counterexample shows that AI ordering does not imply variance residual life ordering. Counterexample 3.2. Let X be a random variable having exponential distribution with mean 2, and Y be 0:8 t40. Then, we another random variable having Weibull distribution with survival function F¯ Y ðtÞ ¼ et R, for 1 R1 ¯ have L ðtÞ ¼ 1 and L ðtÞ ¼ 0:8, for all t40. Here, X p Y . Let bðtÞ ¼ ð Y AI t u F X ðvÞ dv duÞ= R1 R1 X ð t u F¯ Y ðvÞ dv duÞ. If X pvrl Y , then bðtÞ must be a decreasing function of t, but the values bð2Þ ¼ 3:05293, bð5Þ ¼ 3:51248, bð10Þ ¼ 3:40778 reveal that bðtÞ is not a decreasing function of t, proving that X 4 / vrl Y . Thus, X pAI Y RX pvrl Y . Since AI ordering does not imply variance residual life ordering, it is obvious that AI ordering does not imply mean residual life ordering. The following counterexample shows that likelihood ratio ordering does not imply AI ordering. Counterexample 3.3. Let X and Y be two independent random variables having lognormal distribution with density functions 2 1 f X ðtÞ ¼ pffiffiffiffiffiffi eð1=2Þðln tþ4Þ ; t 2p
t40
2 1 f Y ðtÞ ¼ pffiffiffiffiffiffi eð1=2Þðln t4Þ ; t 2p
t40.
and
It is easy to verify that f Y ðtÞ=f X ðtÞ increases with tX0. But cðtÞ ¼ ln F¯ Y ðtÞ= ln F¯ X ðtÞ is not monotone as evident from the values cð0:2Þ ¼ 0:000113422, cð0:5Þ ¼ 0:0000709368, cð6Þ ¼ 0:000731632. Hence, by Theorem 3.1, we have X 4 / AI Y . This means X plr Y , but X 4 / AI Y . Since likelihood ratio ordering does not imply AI ordering, it can be concluded that none of the other stochastic orderings mentioned in Section 1 implies AI ordering. Given two random variables, we state a few conditions under which there will be AI ordering between them. The proof is omitted. Theorem 3.2. For two nonnegative random variables X and Y, if X is IFRA and Y is DFRA, then X pAI Y . Below we take a counterexample where Y is not DFRA. but X pAI Y . This shows that the conditions that X is IFRA and Y is DFRA are sufficient conditions for X pAI Y to hold. Counterexample 3.4. Let the random variables X and Y have the failure rate functions rX ðtÞ ¼ 2t, t40, and rY ðtÞ ¼ 1:5t0:5 , t40. Then LX ðtÞ ¼ 2 and LY ðtÞ ¼ 1:5, for all t40. Here, X and Y both are IFRA, but X pAI Y . Remark 3.1. If X is DMRL and Y is DFR, then X and Y may be not ordered in AI order, which is justified by taking the distributions of X and Y as the following Counterexample shows. Counterexample 3.5. Consider the survival function of the random variable X as F¯ X ðtÞ ¼ et
for 0ptp1,
¼e
1
¼e
1:22t
for 1ptp1:1, for tX1:1.
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pffi Also take the survival function of the random variable Y as F¯ Y ðtÞ ¼ e t , tX0. Here, X is DMRL (cf. Bryson and Siddiqui, 1969) and Y is IMRL. But
8 pffiffi > > t ln F¯ X ðtÞ < p1ffi ¼ t pffiffi ln F¯ Y ðtÞ > > : 2 t 1:2 pffi t
if 0otp1; if 1ptp1:1; if tX1:1;
which is not monotone. Hence, by Theorem 3.1, we have X 4 / AI Y . We study the closure properties of the AI order under different reliability operations as follows. Here, we give the conditions under which the series systems formed by two different sets of components are ordered in AI order. Theorem 3.3. Let fX i ; i ¼ 1; 2; . . . ; mg, fY j ; j ¼ 1; 2; . . . ; ng, be two sets of independent random variables such that X i pAI Y j for all i; j. Then min1pipm X i pAI min1pjpn Y j . P Pn Proof. Write X ¼ min1pipm X i and Y ¼ min1pjpn Y j . Here, rX ðtÞ ¼ m i¼1 rX i ðtÞ and rY ðtÞ ¼ j¼1 rX j ðtÞ. Since, X i pAI Y j , for i ¼ 1; 2; . . . ; m and j ¼ 1; 2; . . . ; n, we have, for t40, Z t Z t rX i ðtÞ rY j ðuÞ duXrY j ðtÞ rX i ðuÞ du, 0
0
for i ¼ 1; 2; . . . ; m and j ¼ 1; 2; . . . ; n. This gives m n Z t n m Z t X X X X rX i ðtÞ rY j ðuÞ du rY j ðtÞ rX i ðuÞ duX0. i¼1
j¼1
0
j¼1
i¼1
(2)
0
Now, "
# Pn Pm r ðtÞ r ðtÞ Y j X j¼1 LX ðtÞ LY ðtÞ ¼ t Pm i¼1 Pn R t . Rt i i¼1 0 rX i ðuÞ du j¼1 0 rY j ðuÞ du X0. The second equality follows from the fact that the failure rate function of a series system is the sum of the component failure rates, whereas the inequality follows from (2). Hence, the result follows. & If fX i ; i ¼ 1; 2; . . . ; mg and fY j ; j ¼ 1; 2; . . . ; ng, be two sets of independent random variables such that X i pAI Y j for all i; j 2 f1; 2; . . . ; minðm; nÞg, then Theorem 3.3 may not hold as the following counterexample shows. Counterexample 3.6. Let X i be a random variable having Rayleigh distribution with failure rate function rX i ðtÞ ¼ t, i ¼ 1; 2. Further, let Y j be a random variable having failure rate rY j ðtÞ ¼ 1, for j ¼ 1; 2, and rY 3 ðtÞ ¼ 3t2 . Then LX i ðtÞ ¼ 2; i ¼ 1; 2, LY j ðtÞ ¼ 1, for j ¼ 1; 2, and LY 3 ðtÞ ¼ 3. Here, LX i ðtÞXLY j ðtÞ, for all t40 and i; j 2 f1; 2g. Thus, X i pAI Y j , i; j 2 f1; 2g. But, one can easily verify that LX ðtÞ LY ðtÞ ¼
2 t2 X0 2 þ t2 p0
pffiffiffi for 0otp 2 pffiffiffi for tX 2,
/ AI Y . where X ¼ min1pip2 X i and Y ¼ min1pjp3 Y j . Thus, X 4 Remark 3.2. The AI ordering is not closed under the formation of parallel system. To show this, let X 1 and X 2 be two independent and identically distributed (iid) random variables having exponential distribution with survival function F¯ X 1 ðtÞ ¼ e2t , tX0. Also, let Y 1 and Y 2 be two iid random variables having Weibull 0:9 distribution with survival function F¯ Y 1 ðtÞ ¼ et , tX0. Here, it is easy to see that LX i ðtÞ ¼ 1 and LY i ðtÞ ¼ 0:9, for t40 and i; j ¼ 1; 2. Thus, X i pAI Y j for all i; j ¼ 1; 2. If we write X ¼ max1pip2 X i and Y ¼ max1pjp2 Y j ,
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then it can be shown that LX ðtÞ ¼
ð2e2t
tð4e2t 4e4t Þ , e4t Þ lnð2e2t e4t Þ
and 0:9
LY ðtÞ ¼
0:9
1:8t0:9 et ð1 et Þ . f1 ð1 et0:9 Þ2 g lnf1 ð1 et0:9 Þ2 g
Write LX ðtÞ LY ðtÞ ¼ LðtÞ. Then we obtain Lð0:2Þ ¼ 0:0907007, Lð3Þ ¼ 0:0211412, and Lð7Þ ¼ 0:03095, showing that X 4 / AI Y . The following theorem shows that, under very mild condition, the AI order is closed under increasing transformation. The proof is easy and therefore omitted. Theorem 3.4. Let X and Y be two continuous random variables. Then X pAI Y if and only if fðX ÞpAI fðY Þ, for any strictly increasing continuous function f : Rþ !Rþ , with fð0Þ ¼ 0. Design engineers are well aware of the fact that under appropriate assumptions, the lifetime of a system where active spare allocation is made at the component level is stochastically larger than that of a system where active spare allocation is made at the system level. Boland and El-Neweihi (1995) have shown that this result cannot be extended to the failure rate ordering even for a series system unless the spares match with the original components. This result has been further strengthened by Singh and Singh (1997) who have shown that if the components and the spares have independent and identically distributed lifetimes, then spare allocation at the component level is superior to that at the system level in likelihood ratio ordering. The question now arises—whether componentwise spare allocation is better than systemwise spare allocation in AI ordering. Here, we cite a counterexample in order to answer the question in negative. Counterexample 3.7. Let X 1 ; X 2 , and X 3 represent the lifetimes of three iid components forming a system with lifetime Z ¼ min½X 1 ; maxðX 2 ; X 3 Þ. Let X 1 , X 2 and X 3 be the lifetimes of independent spare components having identical distributions as that of the original components. Let componentwise and systemwise allocations of the spare components to the original components result in two systems having lifetimes Z1 and Z 2 , respectively. Then, the survival functions of Z 1 and Z 2 are given, respectively, by F¯ Z1 ðtÞ ¼ f1 ð1 F¯ X ðtÞÞ2 gf1 ð1 F¯ X ðtÞÞ4 g and F¯ Z2 ðtÞ ¼ 1 ½1 F¯ X ðtÞf1 ð1 F¯ X ðtÞÞ2 g2 . Now, Z 1 XAI Z 2 , if and only if hðpÞ ¼ ln F¯ Z1 ðtÞ= ln F¯ Z2 ðtÞ is increasing in p ¼ F¯ X ðtÞ 2 ð0; 1. But, hð0:2Þ ¼ 0:784261, hð0:4Þ ¼ 0:725561, hð0:95Þ ¼ 0:913528, which show that hðpÞ is not increasing in p. Hence, Z 1 Z / AI Z 2 . Acknowledgments The authors are very much thankful to the referee for his/her valuable comments and suggestions which help us improve the quality of the paper. We thank the referee for giving the idea of the hazard ratio comparison with the AI order. The first author would like to thank the Department of Science and Technology, Government of India for giving the financial assistance for this research work (vide no. SR/S4/ MS:260/04 dated 18.04.2006). References Barlow, R.E., Proschan, F., 1975. Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York. Boland, P.J., El-Neweihi, E., 1995. Component redundancy vs system redundancy in the hazard rate ordering. IEEE Trans. Reliability 44 (4), 614–619.
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Bryson, M.C., Siddiqui, M.M., 1969. Some criteria for aging. J. Amer. Statist. Assoc. 64, 1472–1483. Fagiuoli, E., Pellerey, F., 1993. New partial orderings and applications. Naval Res. Logistics 40, 829–842. Hu, T., Kundu, A., Nanda, A.K., 2001. On generalized orderings and ageing properties with their implications. In: Hayakawa, Y., Irony, T., Xie, M. (Eds.), System and Bayesian Reliability: Essays in Honor of Prof. R.E. Barlow. World Scientific, New Jersey, pp. 199–228. Jiang, R., Ji, P., Xiao, X., 2003. Aging property of univariate failure rate models. Reliability Eng. System Safety 79, 113–116. Pocock, S.J., Gore, S.M., Kerr, G.R., 1982. Long-term survival analysis: the curability of breast cancer. Statist. Med. 1, 93–104. Rowell, G., Siegrist, K., 1998. Relative aging of distribution. Probab. Eng. Inform. Sci. 12, 469–478. Sengupta, D., Deshpande, J.V., 1994. Some results on the relative ageing of two life distributions. J. Appl. Probab. 31, 991–1003. Shaked, M., Shanthikumar, J.G., 1994. Stochastic Orders and their Applications. Academic Press, New York. Singh, H., Singh, R.S., 1997. On allocation of spares at component level versus system level. J. Appl. Probab. 34, 283–287.
Statistics & Probability Letters 77 (2007) 365–373 www.elsevier.com/locate/stapro
Properties of aging intensity function Asok K. Nanda, Subarna Bhattacharjee, S.S. Alam Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, West Bengal, India Received 6 January 2006; received in revised form 4 May 2006; accepted 7 August 2006 Available online 5 September 2006
Abstract Recently, the concept of aging intensity (AI) function has been introduced in the literature for evaluating the aging property of a unit (that may be a system or a living organism) quantitatively. In this paper, we discuss the properties of AI function and study its nature for various distributions. The closure properties of the aging classes defined in terms of AI function are also presented. We define an ordering, called aging intensity ordering, and study its closure properties under different reliability operations, viz., formation of k-out-of-n system, and increasing transformations. r 2006 Elsevier B.V. All rights reserved. Keywords: Aging property; k-out-of-n system; Spare allocation; Stochastic orders
1. Introduction An important phenomenon in reliability theory is ‘aging’ which is an inherent property of a unit that may be a living organism or a system of components. By aging we generally mean positive aging. Intuitively, aging means an increase of failure risk as a function of time. To be more specific, by aging (respectively, antiaging) we mean a mathematical specification of degradation (respectively, upgradation) of a unit over time. Antiaging is also known as negative aging. Failure rate, defined as the ratio of the density to its survival function, is one such measure of aging. It plays an important role in analyzing the failure pattern of a unit. Jiang et al. (2003) have pointed out that a unimodal failure rate can be effectively viewed as either approximately decreasing or approximately increasing or approximately constant. Clearly, such representation is qualitative. In their paper, a quantitative measure, called aging intensity (AI) function, defined as the ratio of the failure rate rðtÞ to a baseline failure rate, has been studied. One choice of the baseline failure rate Rt could be the average failure rate HðtÞ ¼ ð 0 rðuÞ duÞ=t. Thus, the AI function becomes LðtÞ ¼
rðtÞ ; HðtÞ
t40.
Here, in the present study, we explore the different properties of the AI function. Throughout this paper, the words ‘increasing (decreasing)’ and ‘nondecreasing (nonincreasing)’ are used interchangeably. We write Rþ ¼ ½0; 1Þ. The variables considered in this paper are all nonnegative. For a Corresponding author. Tel.: +91 3222 283686; fax: +91 3222 255303.
E-mail address: [email protected] (A.K. Nanda). 0167-7152/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.08.002
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random variable Z, we write its density as f Z ðÞ, survival function as F¯ Z ðÞ, failure rate as rZ ðÞ, distribution function as F Z ðÞ, and AI function LZ ðÞ. We conclude this section by giving a few well known aging properties from Bryson and Siddiqui (1969) (see also Barlow and Proschan (1975)) followed by a brief discussion on partial orders available in Shaked and Shanthikumar (1994) and Fagiuoli and Pellerey (1993). Definition 1.1. A random variable X is said to be (i) increasing in failure rate (IFR) if rX ðtÞ is increasing in tX0; Rt (ii) increasing in failure rate average (IFRA) if 0 rX ðuÞ du=t is increasing in t40; (iii) decreasing in mean residual life (DMRL) if mX ðtÞ ¼ E½X tjX 4t is decreasing in tX0. If, in the above definition, the word ‘increasing’ is replaced by ‘decreasing’ and conversely, then we get the respective dual classes known as decreasing in failure rate (DFR), decreasing in failure rate average (DFRA), and increasing in mean residual life (IMRL), respectively. Let us recall a few well-known stochastic orders. Definition 1.2. A nonnegative random variable X is said to be smaller than another nonnegative random variable Y in (i) (ii) (iii) (iv) (v) (vi)
likelihood ratio order (denoted by X plr Y ) if f Y ðtÞ=f X ðtÞ is increasing in tX0; failure rate order (denoted by X pfr Y ) if rX ðtÞXrY ðtÞ, for all tX0; starting failure rate order (denoted by X psfr Y ) if rX ð0ÞXrY ð0Þ; stochastic order (denoted by X pst Y ) if F¯ X ðtÞpF¯RY ðtÞ, for all tX0; R1 1 increasing convex order (denoted by X picx Y ) if t F¯ X ðuÞ dup t F¯ Y ðuÞ du, for all tX0; mean residual life order (denoted by X pmrl Y ) if Z
1
F¯ X ðuÞ du
Z F¯ X ðtÞ p
t
1
F¯ Y ðuÞ du
F¯ Y ðtÞ
for all tX0;
t
(vii) variance residual life order (denoted by X pvrl Y ) if Z t
1
Z
1
F¯ X ðvÞ dv du=
u
Z
1 t
Z
1
F¯ Y ðvÞ dv du
is decreasing in tX0.
u
The present paper is arranged as follows. In Section 2, we present the AI function of various distributions, introduce the aging class based on AI function and study the closure properties of the aging class. In Section 3 of this paper, we define an ordering called AI ordering and give some of its properties. The closure of the AI order is studied here under different reliability operations, viz., formation of k-out-of-n system, and increasing transformations. 2. Properties of AI function Jiang et al. (2003) have developed a new notion, called AI, to quantitatively evaluate the aging property of a unit which may be a system or a component or a living being. AI function of a random variable X, denoted by LX ðtÞ, is defined as the ratio of the instantaneous failure rate to the failure rate average, i.e., rX ðtÞ H X ðtÞ tf X ðtÞ ; ¼ F¯ X ðtÞ ln F¯ X ðtÞ
LX ðtÞ ¼
t40.
ð1Þ
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The larger the value of LX ðtÞ, the stronger the tendency of aging of the random variable X . The failure rate function uniquely determines the AI function but not conversely. In fact, two distributions having proportional failure rate functions do have the same AI function. It is to be noted that LX ðtÞ ¼ 1 for all t40 if and only if the failure rate function rX ðtÞ is constant (i.e., X follows exponential distribution). Thus, ‘LX ðtÞ 1 for t40,’ characterizes exponential distribution. Further, LX ðtÞ41 if rX ðtÞ is increasing in t (i.e., X is IFR); and LX ðtÞo1 if rX ðtÞ is decreasing in t (i.e., X is DFR). The next theorem shows that a constant AI function characterizes Weibull distribution. The proof is not difficult and hence is not included here. Theorem 2.1. For a random variable X, LX ðtÞ ¼ c, for t40, c being a constant if and only if X follows Weibull distribution with shape parameter c. On the basis of the monotonicity of the AI function, we define the following nonparametric family of distributions. Definition 2.1. A random variable X is said to be increasing in aging intensity (IAI) if the corresponding AI function LX ðtÞ is increasing in t40. We call the random variable X as decreasing in aging intensity (DAI) if LX ðtÞ is decreasing in t40. Further, it is seen that the monotonic behavior of the failure rate function is not, in general, transmitted to the monotonicity of the AI function as is evident from the following two counterexamples. Counterexample 2.1. Let X be a random variable having Erlang distribution with density function f X ðtÞ ¼ 4te2t ; tX0. Then, rX ðtÞ ¼ 4t=ð1 þ 2tÞ is increasing in tX0. So, X is IFR. Here, LX ðtÞ ¼ 4t2 =½ð1 þ 2tÞf2t lnð1 þ 2tÞg, which decreases in t40. Counterexample 2.2. Let X be a random variable having uniform distribution over ½a; b; 0paobo1. Then, the failure rate is given by rX ðtÞ ¼ ðb tÞ1 , aotob, which is increasing in t. So, X is IFR. It can be proved that LX ðtÞ ¼ t=½ðb tÞfln b lnðb tÞg increases with t, for aotob. From the foregoing two counterexamples it is observed that an IFR random variable can be IAI or DAI. For the random variable X having normal distribution with mean 4 and standard deviation unity, define Y ¼ ðX jX X4Þ, which is IFR. Then LY ðtÞ is nonmonotone in t. Thus, for an IFR random variable, the AI function could be nonmonotonic as well. Similar trend is also observed for a DFR random variable. For example, let a random variable X have the failure rate function ( 0:8 3t for 0ptp0:06; rX ðtÞ ¼ 0:62 for tX0:06: The corresponding AI function is given by 8 0:8 3t > > for 0otp0:06; < 0:8 1:5t LX ðtÞ ¼ 0:62t > > for tX0:06: : 0:0426 þ 0:62ðt 0:06Þ One can check that LX ðtÞ is decreasing for 0otp0:06, and increasing for tX0:06. 2.1. Closure properties of IAI and DAI classes In this section, we study whether IAI and DAI classes are closed under different reliability operations, viz., mixture of distributions, convolution of distributions and formation of k-out-of-n system. By k-out-of-n system we mean a system which operates as long as at least k out of the n components forming the system work. The following counterexamples show that IAI class is not closed under the aforesaid operations. It is to be mentioned here that a distribution with linear failure rate rX ðtÞ ¼ a þ bt with a40, bX0 is IAI.
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Counterexample 2.3. Let X 1 and X 2 be two random variables having respective failure rate functions rX 1 ðtÞ ¼ 1 þ 2t and rX 2 ðtÞ ¼ 2rX 1 ðtÞ; t40, and X be another random variable having survival function F¯ X ðtÞ ¼ 0:2F¯ X 1 ðtÞ þ 0:8F¯ X 2 ðtÞ. Then, for t40, 2
LX ðtÞ ¼
2tf4 þ 8t þ etþt ð0:5 þ tÞg . 2 tþt ð4 þ e Þf 0 ð8 þ 16u þ euþu2 ð1 þ 2uÞÞ=ð4 þ euþu2 Þ dug Rt
Although both X 1 and X 2 are IAI, the AI function of X is nonmonotone. Thus, IAI is not closed with respect to mixture. Next, we establish the fact that IAI class is not closed under convolution. Counterexample 2.4. Let X 1 and X 2 be X 2 and X 1 , respectively of Counterexample 2.3, then the AI function of the random variable X c ¼ X 1 þ X 2 is LX c ðtÞ ¼ f ðtÞ=gðtÞ where h n pffiffiffiffiffiffi 2 2 f ðtÞ ¼ 2t 3ðe1=6 ð1 þ 2tÞ þ e1=6þtþt ð1 þ 4tÞÞ þ e1=4þ2t=3þ4t =3 3pð1 þ 8t þ 4t2 Þ pffiffiffi pffiffiffiffiffiffi pffiffiffi oi 2 erfðð1 2tÞ=ð2 3ÞÞ e1=4þ2t=3þ4t =3 3pð1 þ 8t þ 4t2 Þ erfðð1 þ 4tÞ=2 3Þ , pffiffiffi 2 pffiffiffiffiffiffi 2 pffiffiffiffiffiffi 2 gðtÞ ¼ 3e1=6 ð3 þ 6etþt 2eð1=12Þð1þ4tÞ 3pð1 þ tÞ erfðð1 2tÞ=2 3 þ 2eð1=12Þð1þ4tÞ 3pð1 þ tÞ pffiffiffi pffiffiffi 2 pffiffiffiffiffiffi 1 2 erfðð1 þ 4tÞ=2 3ÞÞ ln e2tð1þtÞ ð3 þ 6etþt 2eð1=12Þð1þ4tÞ 3pð1 þ tÞ erfðð1 2tÞ=2 3Þ 9 pffiffiffi 2 pffiffiffiffiffiffi þ 2eð1=12Þð1þ4tÞ 3pð1 þ tÞ erfðð1 þ 4tÞ=2 3ÞÞ , and erfðzÞ is the ‘error function’ defined by erfðzÞ ¼ 2ð
Rz 0
pffiffiffi 2 et dtÞ= p. Here LX c ðtÞ, for t40, is nonmonotone.
Order statistics have nice applications in the study of the successive times of failures of the components or of systems. The kth order statistic is the lifetime of the ðn k þ 1Þ-out-of-n system. Next, we cite a counterexample to show that IAI class is not closed under the formation of a k-out-of-n system. Counterexample 2.5. Let X be a random variable with failure rate rX ðtÞ ¼ 1 þ 2t, t40, and f X ð2:3Þ ðtÞ be the density function of the 2nd order statistic in a sample of size 3 from the distribution of X . Then 2 2 2 LX ð2:3Þ ðtÞ ¼ ½6tðetþt 1Þð1 þ 2tÞ=½ð3etþt 2Þ lnfe3tð1þtÞ ð2 þ 3etþt Þg, the AI function of X ð2:3Þ , is nonmonotone. Next we deal with DAI class of distributions to study its closure properties under mixture, and formation of k-out-of-n system. In fact, it is also observed that DAI class of distributions is not closed under the aforementioned reliability operations which will be clear from the following counterexamples. Counterexample 2.6. Let X 1 and X 2 be two random variables having gamma distributions with respective probability density functions given by f X 1 ðtÞ ¼ 12t2 et , t40, and f X 2 ðtÞ ¼ 16 t3 et , t40, each of which is DAI. Let the survival function of the mixture random variable X be F¯ X ðtÞ ¼ 0:2F¯ X 1 ðtÞ þ 0:8F¯ X 2 ðtÞ. Then it follows that DAI class is not closed under mixture of distributions as LX ðtÞ ¼ ½t3 ð0:75 þ tÞ=½7:5et þ ð1:88233 þ tÞð3:98443 þ 1:86767t þ t2 Þ, the AI function of X , is nonmonotone for t40. From the counterexample given below, we see that DAI class is not closed under the formation of a k-outof-n system. Counterexample 2.7. Let X be a random variable having gamma distribution with probability density function f X ðtÞ ¼ 16et t3 , tX0, and f X ð2:3Þ ðtÞ be the density function of the 2nd order statistic X ð2:3Þ in a sample of size 3
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from this distribution. Then, the corresponding AI function is LX ð2:3Þ ðtÞ ¼ hðtÞ=kðtÞ, where hðtÞ ¼ 3t4 ð6 6et þ 6t þ 3t2 þ t3 Þ, and kðtÞ ¼ ð6 þ 6t þ 3t2 þ t3 Þð6 9et þ 6t þ 3t2 þ t3 Þ 3t e ln ð6 þ 9et 6t 3t2 t3 Þð6 þ 6t þ 3t2 þ t3 Þ2 . 108 It can be verified that LX ð2:3Þ ðtÞ is nonmonotone, although the underlying distribution is DAI. 3. Some properties of AI order We define a probabilistic order based on the AI function LðtÞ as follows. Definition 3.1. A random variable X is said to be smaller than another random variable Y in the AI order (denoted by X pAI Y ) if LX ðtÞXLY ðtÞ, for all t40. We mention a family of parametric distributions where AI ordering between the random variables is present. Example 3.1. Let X i be a random variable having Weibull distribution with survival function F¯ X i ðtÞ ¼ expðtai =bi Þ, ai ; bi 40; tX0; i ¼ 1; 2. If a1 Xa2 , then X 1 pAI X 2 . Theorem 3.1. For two random variables X and Y, the following conditions are equivalent. (i) RX pAI Y . R t t (ii) 0 rX ðuÞ du= 0 rY ðuÞ du is increasing in t40. (iii) ln F¯ X ðtÞ= ln F¯ Y ðtÞ is increasing in t40. A probabilistic order based on the monotonicity of the ratio rX ðtÞ=rY ðtÞ is studied in Sengupta and Deshpande (1994) and Rowell and Siegrist (1998). The random variable X is said to be aging faster than another random variable Y (written as X pAF Y ) if rX ðtÞ=rY ðtÞ is increasing in t. Based on AF order, a generalized stochastic order has been proved in Hu et al. (2001). The crossing hazards phenomenon has been studied in connection with prognostic studies in the treatment of breast cancer by Pocock et al. (1982). An increasing hazards ratio is a reasonable alternative to the proportional hazards model in this case, see Sengupta and Deshpande (1994), for instance. By using variation diminishing property, it can easily be verified that AF order defined above is stronger than AI order. The reflexive, commutative, and antisymmetric properties of the AI order are given below. The proof is omitted. Proposition 3.1. (i) X pAI X . (ii) If X pAI Y and Y pAI Z, then X pAI Z. (iii) If X pAI Y and Y pAI X , then X and Y have proportional failure rates. Before we study the different properties of the AI order, a natural question arises—what is the relationship between AI order and other existing stochastic orders? Below we take an attempt to answer such a question. The following counterexample shows that AI ordering does not imply increasing convex ordering. Counterexample 3.1. Let X be a random variable having exponential distribution with survival function F¯ X ðtÞ ¼ e0:5t , t40; and Y be another random variable having Weibull distribution with survival function
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¯ Y ðtÞ ¼ et0:6 , t40. Then, we have LX ðtÞ ¼ 1 and LY ðtÞ ¼ 0:6, for all t40. Clearly, X pAI Y . Let gðtÞ ¼ F R1 R1 ¯ ¯ t F X ðuÞ du and hðtÞ ¼ t F Y ðuÞ du. If X picx Y , then gðtÞphðtÞ, for all tX0. But from the values gð1:5Þ ¼ 0:944733, hð1:5Þ ¼ 0:7912, gð4:5Þ ¼ 0:210798, hð4:5Þ ¼ 0:321707, we can conclude that gðtÞ4 / hðtÞ for all tX0, i.e., X 4 / icx Y . This shows that X pAI Y RX picx Y . Since AI ordering does not imply increasing convex ordering, it is obvious that AI ordering implies none of likelihood ratio ordering, failure rate ordering, and stochastic ordering. The following counterexample shows that AI ordering does not imply variance residual life ordering. Counterexample 3.2. Let X be a random variable having exponential distribution with mean 2, and Y be 0:8 t40. Then, we another random variable having Weibull distribution with survival function F¯ Y ðtÞ ¼ et R, for 1 R1 ¯ have L ðtÞ ¼ 1 and L ðtÞ ¼ 0:8, for all t40. Here, X p Y . Let bðtÞ ¼ ð Y AI t u F X ðvÞ dv duÞ= R1 R1 X ð t u F¯ Y ðvÞ dv duÞ. If X pvrl Y , then bðtÞ must be a decreasing function of t, but the values bð2Þ ¼ 3:05293, bð5Þ ¼ 3:51248, bð10Þ ¼ 3:40778 reveal that bðtÞ is not a decreasing function of t, proving that X 4 / vrl Y . Thus, X pAI Y RX pvrl Y . Since AI ordering does not imply variance residual life ordering, it is obvious that AI ordering does not imply mean residual life ordering. The following counterexample shows that likelihood ratio ordering does not imply AI ordering. Counterexample 3.3. Let X and Y be two independent random variables having lognormal distribution with density functions 2 1 f X ðtÞ ¼ pffiffiffiffiffiffi eð1=2Þðln tþ4Þ ; t 2p
t40
2 1 f Y ðtÞ ¼ pffiffiffiffiffiffi eð1=2Þðln t4Þ ; t 2p
t40.
and
It is easy to verify that f Y ðtÞ=f X ðtÞ increases with tX0. But cðtÞ ¼ ln F¯ Y ðtÞ= ln F¯ X ðtÞ is not monotone as evident from the values cð0:2Þ ¼ 0:000113422, cð0:5Þ ¼ 0:0000709368, cð6Þ ¼ 0:000731632. Hence, by Theorem 3.1, we have X 4 / AI Y . This means X plr Y , but X 4 / AI Y . Since likelihood ratio ordering does not imply AI ordering, it can be concluded that none of the other stochastic orderings mentioned in Section 1 implies AI ordering. Given two random variables, we state a few conditions under which there will be AI ordering between them. The proof is omitted. Theorem 3.2. For two nonnegative random variables X and Y, if X is IFRA and Y is DFRA, then X pAI Y . Below we take a counterexample where Y is not DFRA. but X pAI Y . This shows that the conditions that X is IFRA and Y is DFRA are sufficient conditions for X pAI Y to hold. Counterexample 3.4. Let the random variables X and Y have the failure rate functions rX ðtÞ ¼ 2t, t40, and rY ðtÞ ¼ 1:5t0:5 , t40. Then LX ðtÞ ¼ 2 and LY ðtÞ ¼ 1:5, for all t40. Here, X and Y both are IFRA, but X pAI Y . Remark 3.1. If X is DMRL and Y is DFR, then X and Y may be not ordered in AI order, which is justified by taking the distributions of X and Y as the following Counterexample shows. Counterexample 3.5. Consider the survival function of the random variable X as F¯ X ðtÞ ¼ et
for 0ptp1,
¼e
1
¼e
1:22t
for 1ptp1:1, for tX1:1.
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pffi Also take the survival function of the random variable Y as F¯ Y ðtÞ ¼ e t , tX0. Here, X is DMRL (cf. Bryson and Siddiqui, 1969) and Y is IMRL. But
8 pffiffi > > t ln F¯ X ðtÞ < p1ffi ¼ t pffiffi ln F¯ Y ðtÞ > > : 2 t 1:2 pffi t
if 0otp1; if 1ptp1:1; if tX1:1;
which is not monotone. Hence, by Theorem 3.1, we have X 4 / AI Y . We study the closure properties of the AI order under different reliability operations as follows. Here, we give the conditions under which the series systems formed by two different sets of components are ordered in AI order. Theorem 3.3. Let fX i ; i ¼ 1; 2; . . . ; mg, fY j ; j ¼ 1; 2; . . . ; ng, be two sets of independent random variables such that X i pAI Y j for all i; j. Then min1pipm X i pAI min1pjpn Y j . P Pn Proof. Write X ¼ min1pipm X i and Y ¼ min1pjpn Y j . Here, rX ðtÞ ¼ m i¼1 rX i ðtÞ and rY ðtÞ ¼ j¼1 rX j ðtÞ. Since, X i pAI Y j , for i ¼ 1; 2; . . . ; m and j ¼ 1; 2; . . . ; n, we have, for t40, Z t Z t rX i ðtÞ rY j ðuÞ duXrY j ðtÞ rX i ðuÞ du, 0
0
for i ¼ 1; 2; . . . ; m and j ¼ 1; 2; . . . ; n. This gives m n Z t n m Z t X X X X rX i ðtÞ rY j ðuÞ du rY j ðtÞ rX i ðuÞ duX0. i¼1
j¼1
0
j¼1
i¼1
(2)
0
Now, "
# Pn Pm r ðtÞ r ðtÞ Y j X j¼1 LX ðtÞ LY ðtÞ ¼ t Pm i¼1 Pn R t . Rt i i¼1 0 rX i ðuÞ du j¼1 0 rY j ðuÞ du X0. The second equality follows from the fact that the failure rate function of a series system is the sum of the component failure rates, whereas the inequality follows from (2). Hence, the result follows. & If fX i ; i ¼ 1; 2; . . . ; mg and fY j ; j ¼ 1; 2; . . . ; ng, be two sets of independent random variables such that X i pAI Y j for all i; j 2 f1; 2; . . . ; minðm; nÞg, then Theorem 3.3 may not hold as the following counterexample shows. Counterexample 3.6. Let X i be a random variable having Rayleigh distribution with failure rate function rX i ðtÞ ¼ t, i ¼ 1; 2. Further, let Y j be a random variable having failure rate rY j ðtÞ ¼ 1, for j ¼ 1; 2, and rY 3 ðtÞ ¼ 3t2 . Then LX i ðtÞ ¼ 2; i ¼ 1; 2, LY j ðtÞ ¼ 1, for j ¼ 1; 2, and LY 3 ðtÞ ¼ 3. Here, LX i ðtÞXLY j ðtÞ, for all t40 and i; j 2 f1; 2g. Thus, X i pAI Y j , i; j 2 f1; 2g. But, one can easily verify that LX ðtÞ LY ðtÞ ¼
2 t2 X0 2 þ t2 p0
pffiffiffi for 0otp 2 pffiffiffi for tX 2,
/ AI Y . where X ¼ min1pip2 X i and Y ¼ min1pjp3 Y j . Thus, X 4 Remark 3.2. The AI ordering is not closed under the formation of parallel system. To show this, let X 1 and X 2 be two independent and identically distributed (iid) random variables having exponential distribution with survival function F¯ X 1 ðtÞ ¼ e2t , tX0. Also, let Y 1 and Y 2 be two iid random variables having Weibull 0:9 distribution with survival function F¯ Y 1 ðtÞ ¼ et , tX0. Here, it is easy to see that LX i ðtÞ ¼ 1 and LY i ðtÞ ¼ 0:9, for t40 and i; j ¼ 1; 2. Thus, X i pAI Y j for all i; j ¼ 1; 2. If we write X ¼ max1pip2 X i and Y ¼ max1pjp2 Y j ,
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then it can be shown that LX ðtÞ ¼
ð2e2t
tð4e2t 4e4t Þ , e4t Þ lnð2e2t e4t Þ
and 0:9
LY ðtÞ ¼
0:9
1:8t0:9 et ð1 et Þ . f1 ð1 et0:9 Þ2 g lnf1 ð1 et0:9 Þ2 g
Write LX ðtÞ LY ðtÞ ¼ LðtÞ. Then we obtain Lð0:2Þ ¼ 0:0907007, Lð3Þ ¼ 0:0211412, and Lð7Þ ¼ 0:03095, showing that X 4 / AI Y . The following theorem shows that, under very mild condition, the AI order is closed under increasing transformation. The proof is easy and therefore omitted. Theorem 3.4. Let X and Y be two continuous random variables. Then X pAI Y if and only if fðX ÞpAI fðY Þ, for any strictly increasing continuous function f : Rþ !Rþ , with fð0Þ ¼ 0. Design engineers are well aware of the fact that under appropriate assumptions, the lifetime of a system where active spare allocation is made at the component level is stochastically larger than that of a system where active spare allocation is made at the system level. Boland and El-Neweihi (1995) have shown that this result cannot be extended to the failure rate ordering even for a series system unless the spares match with the original components. This result has been further strengthened by Singh and Singh (1997) who have shown that if the components and the spares have independent and identically distributed lifetimes, then spare allocation at the component level is superior to that at the system level in likelihood ratio ordering. The question now arises—whether componentwise spare allocation is better than systemwise spare allocation in AI ordering. Here, we cite a counterexample in order to answer the question in negative. Counterexample 3.7. Let X 1 ; X 2 , and X 3 represent the lifetimes of three iid components forming a system with lifetime Z ¼ min½X 1 ; maxðX 2 ; X 3 Þ. Let X 1 , X 2 and X 3 be the lifetimes of independent spare components having identical distributions as that of the original components. Let componentwise and systemwise allocations of the spare components to the original components result in two systems having lifetimes Z1 and Z 2 , respectively. Then, the survival functions of Z 1 and Z 2 are given, respectively, by F¯ Z1 ðtÞ ¼ f1 ð1 F¯ X ðtÞÞ2 gf1 ð1 F¯ X ðtÞÞ4 g and F¯ Z2 ðtÞ ¼ 1 ½1 F¯ X ðtÞf1 ð1 F¯ X ðtÞÞ2 g2 . Now, Z 1 XAI Z 2 , if and only if hðpÞ ¼ ln F¯ Z1 ðtÞ= ln F¯ Z2 ðtÞ is increasing in p ¼ F¯ X ðtÞ 2 ð0; 1. But, hð0:2Þ ¼ 0:784261, hð0:4Þ ¼ 0:725561, hð0:95Þ ¼ 0:913528, which show that hðpÞ is not increasing in p. Hence, Z 1 Z / AI Z 2 . Acknowledgments The authors are very much thankful to the referee for his/her valuable comments and suggestions which help us improve the quality of the paper. We thank the referee for giving the idea of the hazard ratio comparison with the AI order. The first author would like to thank the Department of Science and Technology, Government of India for giving the financial assistance for this research work (vide no. SR/S4/ MS:260/04 dated 18.04.2006). References Barlow, R.E., Proschan, F., 1975. Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York. Boland, P.J., El-Neweihi, E., 1995. Component redundancy vs system redundancy in the hazard rate ordering. IEEE Trans. Reliability 44 (4), 614–619.
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