On Some Generalized Orderings: In the Spirit of Relative Ageing
aa r X i v : . [ s t a t . A P ] O c t On Some Generalized Orderings:In the Spirit of Relative Ageing
Nil Kamal Hazra and Asok K. Nanda ∗ Department of Mathematics and StatisticsIISER Kolkata, Mohanpur CampusMohanpur 741252, IndiaOctober, 2014
Abstract
We introduce some new generalized stochastic orderings (in the spirit of rela-tive ageing) which compare probability distributions with the exponential distribu-tion. These orderings are useful to understand the phenomenon of positive ageingclasses and also helpful to guide the practitioners when there are crossing hazardrates and/or crossing mean residual lives. We study some characterizations of theseorderings. Inter-relations among these orderings have also been discussed.
Key Words:
Generalized ageing, hazard rate function, mean residual life function.
Lehmann [17] proposed proportional hazard (PH) rate model (commonly known asCox’s PH model, see Cox [6]) which is very useful to analyze the failure time data in re-liability and survival analysis. Later, Zahedi [24] introduced proportional mean residuallife model which is a parallel concept to Cox’s PH model. In many real life situations,the comparison of two crossing hazard rates and/or crossing mean residual lives has beenobserved, see, for instance, Pocock et al. [21], Champlin et al. [5], Begg et al. [2], Manteland Stablein [19], Gupta and Gupta [10], and Bekker and Mi [3]. Some methods underthe Cox proportional hazards framework have been developed to deal with the crossing ∗ e-mail: [email protected], [email protected], corresponding author. et al. [18]). Sengupta and Deshpande [22] discussed areasonable alternative approach based on the concept of relative ageing to handle thecrossing hazard rates problem. They have defined some stochastic orderings (some ofwhich are originally defined by Kalashnikov and Rachev [11]) based on the concept ofrelative ageing.Stochastic orderings and different ageing classes are extensively studied in reliabilitytheory. Stochastic orderings are used to compare two life distributions from differentaspects. Many different types of stochastic orderings have been developed (see Shakedand Shanthikumar [23]). On the other hand, positive ageing means an older system hasshorter remaining lifetime than a younger one in some stochastic sense. Many differenttypes of life distributions are characterized by their ageing properties. It has been ob-served that some stochastic orderings which compare probability distributions with theexponential distribution are found to be very useful to understand the phenomenon ofageing (see Barlow and Proschan [1], Deshpande et al. ([7], [8]), Kochar and Wiens [13],Lai and Xie [14], and the references therein). We introduce some new generalized stochas-tic orderings (in the spirit of relative ageing) which compare probability distributions withthe exponential distribution. These orderings may be useful to realize the phenomenonof positive ageing classes from different angles and also may be helpful to get guidancefor the crossing hazard rates and/or crossing mean residual lives problem.For any absolutely continuous nonnegative random variable X , let the probabilitydensity function be denoted by f X ( · ), the cumulative distribution function by F X ( · ) andthe survival function by F X ( · ) = 1 − F X ( · ). Let us write T X, ( x ) = f X ( x ) , and T X,s ( x ) = R ∞ x T X,s − ( t ) dt ˜ µ X,s − , (1.1)for s = 1 , , . . . , where ˜ µ X,s = Z ∞ T X,s ( t ) dt,s = 0 , , , . . . . We assume ˜ µ X,s to be finite for all s = 0 , , , . . . . We denote the randomvariable corresponding to the survival function T X,s ( · ) by X s . Clearly, X ≡ X . Wefurther define, for s = 1 , , . . . ,Λ X,s ( · ) = − log T X,s ( · ) , (1.2)2 X,s ( x ) = T X,s − ( x ) R ∞ x T X,s − ( t ) dt = T X,s − ( x )˜ µ X,s − T X,s ( x ) , and µ X,s ( x ) = R ∞ x T X,s ( t ) dtT X,s ( x ) , where Λ X,s ( · ), r X s ( · ) and µ X,s ( · ), respectively, represent the cumulative hazard function,the failure rate function and the mean residual life function corresponding to X s . Notethat, for s = 1 , , . . . , µ X,s (0) = ˜ µ X,s , and, for s = 2 , , . . . , r X,s ( x ) = 1 µ X,s − ( x ) . (1.3)Throughout the paper, increasing and decreasing properties are not used in strictsense. For any differentiable function k ( · ), we write k ′ ( t ) to denote the first derivative of k ( t ) with respect to t .The following well known definitions may be obtained in Fagiuoli and Pellerey [9]. Definition 1.1
For s = 1 , , . . . , X is said to be( i ) s-IFR if r X,s ( x ) is increasing in x ≥ ;( ii ) s-IFRA if x R x r X,s ( t ) dt is increasing in x > ;( iii ) s-NBU if T X,s ( x + t ) ≤ T X,s ( x ) .T X,s ( t ) for all x, t ≥ ;( iv ) s-NBUFR if r X,s (0) ≤ r X,s ( x ) for all x ≥ ( v ) s-NBAFR if r X,s (0) ≤ x R x r X,s ( x ) for all x > . It is easy to verify that each of the following equivalence relations holds:1-IFR ⇔ IFR, 2-IFR ⇔ DMRL, 3-IFR ⇔ DVRL,1-IFRA ⇔ IFRA, 2-IFRA ⇔ DMRLHA, 1-NBU ⇔ NBU,1-NBUFR ⇔ NBUFR, 2-NBUFR ⇔ NBUE, 3-NBUFR ⇔ NDVRL,1-NBAFR ⇔ NBAFR, 2-NBAFR ⇔ HNBUE.3or the definitions of IFR (Increasing in Failure Rate), IFRA (Increasing in FailureRate Average), NBU (New Better than Used), DMRL (Decreasing in Mean ResidualLife) and NBUE (New Better than Used in Expectation) classes one may refer to Brysonand Siddiqui [4], and Barlow and Proschan [1]; DVRL (Decreasing in Variance ResidualLife) and NDVRL (Net DVRL) classes are discussed in Launer [15]; DMRLHA (Decreas-ing Mean Residual Life in Harmonic Average) and NBUFR (New Better than Used inFailure Rate) classes are studied by Deshpande et al. [7]; NBAFR (New Better ThanUsed in Failure Rate Average) is due to Loh [16], whereas HNBUE (Harmonically NewBetter than Used in Expectation) is discussed in Klefsj¨o [12]. Similarly, the negativeageing notions, namely, DFR, DFRA, NWU, NWUFR, NWAFR, etc. are also found inthe literature.Let F be the class of distribution functions F : [0 , ∞ ) −→ [0 ,
1] with F (0) = 0.We assume that all F ( ∈ F ) have their finite generalized means e µ X,s , and are strictlyincreasing on their support. If F is not strictly increasing, we take the inverse as F − ( y ) = inf { x : F ( x ) ≥ y } . A function f ( · ) is called star-shaped (resp. antistar-shaped) if f ( x ) /x is increasing(resp. decreasing) in x >
0. On the other hand, it is called super-additive (resp. sub-additive) if, for all x, y , f ( x + y ) ≥ ( resp. ≤ ) f ( x ) + f ( y ).Let Y be another absolutely continuous nonnegative random variable with respectivegeneralized functions (analogous to the one defined above for X ) T Y,s ( · ) , e µ Y,s , Λ
Y,s ( · ), r Y,s ( · ) and µ Y,s ( · ). The random variable corresponding to the survival function T Y,s ( · ) isdenoted by Y s . For the sake of simplicity we write, for s = 1 , , . . . , Φ sX,Y ≡ Λ Y,s ( X s ) . In this note, we define some new generalized stochastic orderings, and some of theirproperties are also studied. In Sections 2, 3, 4, 5 and 6, we discuss s-IFR(R), s-IFRA(R),s-NBU(R), s-NBUFR(R) and s-NBAFR(R) orderings, respectively. We write ‘R’ withinparenthesis to mean that this ordering has been generated based on the concept of R elative ageing. Some characterizations of these orderings are discussed. We showthat these orderings are scale and base invariant. Inter-relations among these orderingshave also been discussed. We build a bridge by which these orderings could connect tothe generalized ageing classes, and vice versa.4 s-IFR(R) Ordering In this section we define and study s-IFR(R) ordering. This ordering interprets thatratio of the hazard rates of X s and Y s is increasing. This means that X s ages faster than Y s . We begin with the following definition. Definition 2.1
For any positive integer s , X (or its distribution F X ) is said to be more s -IFR(R) than Y (or its distribution F Y ) (written as X ≤ s − IF R ( R ) Y ) if the randomvariable Φ sX,Y has an IFR distribution. ✷ Remark 2.1
For s = 1 , Definition 2.1 gives X ≤ c Y , as discussed in Sengupta andDeshpande [22]. ✷ The following lemma may be obtained in Marshall and Olkin ([20], Section 21( f ), pp.699-700). Lemma 2.1
Let f ( · ) and g ( · ) be two real-valued continuous functions, and ζ ( · ) be astrictly increasing (resp. decreasing) and continuous function defined on the range of f and g . Then, for any real number c > , f ( x ) − cg ( x ) and ζ ( f ( x )) − ζ ( cg ( x )) have signchange property in the same (resp. reverse) order, as x traverses from left to right. ✷ In the following two propositions, we give some equivalent representations of the s-IFR(R) ordering. The second proposition can easily be verified by using Lemma 2.1 orProposition 2.C.8 of Marshall and Olkin [20], and Proposition 2.1(i).
Proposition 2.1
For s = 2 , , . . . , Definition 2.1 can equivalently be written in one ofthe following forms: ( i ) Λ X,s (cid:0) Λ − Y,s ( x ) (cid:1) is convex in x ≥ . ( ii ) r X,s ( x ) r Y,s ( x ) is increasing in x ≥ . ( iii ) µ X,s − ( x ) µ Y,s − ( x ) is decreasing in x ≥ . ( iv ) Λ X,s ( Y s ) has a DFR distribution. Proof:
We have Λ
X,s (cid:0) Λ − Y,s ( x ) (cid:1) = − log (cid:16) T X,s T − Y,s (cid:0) e − x (cid:1)(cid:17) = − log F Φ sX,Y ( x ) . (2.4)5hus, ( i ) follows from Definition 2.1, and conversely. Again, Definition 2.1 can equiva-lently be written as r Φ sX,Y ( x ) = (cid:18) ˜ µ Y,s − ˜ µ X,s − (cid:19) T X,s − T − Y,s ( e − x ) T X,s T − Y,s ( e − x ) ! e − x T Y,s − T − Y,s ( e − x ) ! (2.5)is increasing in x ≥
0, which holds if, and only if, (cid:18) ˜ µ Y,s − ˜ µ X,s − (cid:19) T X,s − T − Y,s ( u ) T X,s T − Y,s ( u ) ! uT Y,s − T − Y,s ( u ) ! is decreasing in u ∈ (0 , , or equivalently, r X,s ( x ) r Y,s ( x ) is increasing in x ≥ . This gives the equivalence of Definition 2.1 and ( ii ). The one-to-one connection between( ii ) and ( iii ) follows from (1.3). Note that ( iv ) holds if, and only if, r Y,s ( x ) r X,s ( x ) is decreasing in x ≥ , which is equivalent to ( ii ). ✷ Remark 2.2
The equivalences of ( i ) , ( ii ) and ( iv ) given in Proposition 2.1 are also truefor s = 1 . ✷ Proposition 2.2
Definition 2.1 can equivalently be written in one of the following forms: ( i ) For any real numbers a and b , Λ X,s Λ − Y,s ( x ) − ( ax + b ) changes sign at most twice,and if the change of signs occurs twice, they are in the order + , − , + , as x traversesfrom to ∞ . ( ii ) For any real numbers a and b , Λ − Y,s ( x ) − Λ − X,s ( ax + b ) changes sign at most twice,and if the change of signs occurs twice, they are in the order + , − , + , as x traversesfrom to ∞ . ( iii ) For any real numbers a and b , Λ − Y,s ( ax + b ) − Λ − X,s ( x ) changes sign at most twice,and if the change of signs occurs twice, they are in the order + , − , + , as x traversesfrom to ∞ . ( iv ) For any real numbers a and b , Λ − X,s ( x ) − Λ − Y,s ( ax + b ) changes sign at most twice,and if the change of signs occurs twice, they are in the order − , + , − , as x traversesfrom to ∞ . v ) For any real numbers a and b , Λ Y,s Λ − X,s ( x ) − ( ax + b ) changes sign at most twice,and if the change of signs occurs twice, they are in the order − , + , − , as x traversesfrom to ∞ . ( vi ) Λ Y,s Λ − X,s ( x ) is concave in x > . ✷ Below we state two lemmas which will be used in proving the upcoming theorem. Theproofs are omitted.
Lemma 2.2
Let f ( · ) be a nonnegative, increasing, and convex function. Then f − ( · ) isconcave. ✷ Lemma 2.3
Let f ( · ) and g ( · ) be two nonnegative, increasing, and convex functions.Then f ( g ( · )) is convex. ✷ The following theorem shows some properties of the s-IFR(R) ordering.
Theorem 2.1
For any positive integer s , ( i ) X ≤ s − IF R ( R ) X . ( ii ) X ≤ s − IF R ( R ) Y and Y ≤ s − IF R ( R ) X hold simultaneously if, and only if, Λ X,s ( x ) = θ Λ Y,s ( x ) , for some θ > and for all x ≥ . ( iii ) If X ≤ s − IF R ( R ) Y and Y ≤ s − IF R ( R ) Z then X ≤ s − IF R ( R ) Z . Proof:
The proof of ( i ) is trivial. Now, X ≤ s − IF R ( R ) Y gives thatΛ X,s (cid:0) Λ − Y,s ( x ) (cid:1) is convex in x ≥ , which, by Lemma 2.2, reduces to the fact thatΛ Y,s (cid:0) Λ − X,s ( x ) (cid:1) is concave . (2.6)Further, Y ≤ s − IF R ( R ) X gives thatΛ Y,s (cid:0) Λ − X,s ( x ) (cid:1) is convex . (2.7)Combining (2.6) and (2.7), we get Λ Y,s (cid:0) Λ − X,s ( x ) (cid:1) = xθ , for some constant θ ( > X,s ( x ) = θ Λ Y,s ( x ), and hence ( ii ) is proved. On usingLemma 2.3, one can easily check that ( iii ) holds. ✷ The following lemma can be easily verified.7 emma 2.4
Let X ∼ F X ( x ) = e − λx . Then, for s = 1 , , . . . ,( i ) r X,s ( x ) = λ ;( ii ) T X,s ( x ) = e − λx . ✷ The following theorem shows that a random variable X has an s-IFR distribution if,and only if, X is smaller than exponential distribution in s-IFR(R) ordering. The prooffollows from Lemma 2.4. Theorem 2.2 If F Y ( x ) = e − λx , then X ≤ s − IF R ( R ) Y if, and only if, X is s-IFR. ✷ Below we give a lemma which will be used in proving the upcoming theorem, and canbe proved using Principle of Mathematical Induction.
Lemma 2.5
For any real numbers a ( > and b , and for s = 1 , , . . . , ( i ) T aX + b,s ( x ) = T X,s (cid:0) x − ba (cid:1) . ( ii ) ˜ µ aX + b,s = a ˜ µ X,s . ( iii ) r aX + b,s ( x ) = a r X,s (cid:0) x − ba (cid:1) . ✷ The following theorem shows that s-IFR(R) ordering is location and scale invariant.
Theorem 2.3 X ≤ s − IF R ( R ) Y if, and only if, ( aX + b ) ≤ s − IF R ( R ) ( aY + b ) , for any realnumbers a ( > and b . ✷ Proof:
Let Λ aX + b,s ( · ) and Λ aY + b,s ( · ) be the cumulative hazard rate functions of aX + b and aY + b , respectively. Then, on using Lemma 2.5, we have, for all x ≥ aX + b,s (Λ aY + b,s ) − ( x ) = − log (cid:16) T aX + b,s T − aY + b,s (cid:0) e − x (cid:1)(cid:17) = − log " T X,s T − aY + b,s ( e − x ) − ba ! = − log " T X,s b + aT − Y,s ( e − x ) − ba ! = Λ X,s Λ − Y,s ( x ) . (2.8)Thus, the result follows from Proposition 2.1(i). ✷ s-IFRA(R) Ordering In this section we discuss s-IFRA(R) ordering. The interior scenario of s-IFRA(R)ordering is that ratio of the cumulative hazard rates of X s and Y s is increasing. Definition 3.1
For any positive integer s , X (or its distribution F X ) is said to be more s -IFRA(R) than Y (or its distribution F Y ) (written as X ≤ s − IF RA ( R ) Y ) if the randomvariable Φ sX,Y has an IFRA distribution. ✷ Remark 3.1
For s = 1 , Definition 3.1 gives X ≤ ∗ Y , as discussed in Sengupta andDeshpande [22]. ✷ Some equivalent representations of s-IFRA(R) ordering are given in the following twopropositions. The second proposition can easily be proved by Lemma 2.1 and Proposi-tion 3.1(i).
Proposition 3.1
Definition 3.1 can equivalently be written in one of the following forms: ( i ) Λ X,s (cid:0) Λ − Y,s ( x ) (cid:1) is star-shaped in x > . ( ii ) Λ X,s ( x )Λ Y,s ( x ) is increasing in x > . ( iii ) Λ X,s ( Y s ) has a DFRA distribution. Proof:
On using (2.4), the equivalence of Definition 3.1 and ( i ) follows. Note that ( i )can equivalently be written asΛ X,s (cid:0) Λ − Y,s ( x ) (cid:1) x is increasing in x > , or equivalently, Λ X,s ( x )Λ Y,s ( x ) is increasing in x > . Thus, the equivalence of ( i ) and ( ii ) is proved. The one-to-one connection between ( ii )and ( iii ) can be proved in the same line as is done in ( ii ) above. ✷ Proposition 3.2
Definition 3.1 can equivalently be written in one of the following forms: ( i ) For any real number a , Λ X,s Λ − Y,s ( x ) − ax changes sign at most once, and if thechange of sign does occur, it is in the order − , + , as x traverses from to ∞ . ( ii ) For any real number a , Λ − Y,s ( x ) − Λ − X,s ( ax ) changes sign at most once, and if thechange of sign does occur, it is in the order − , + , as x traverses from to ∞ . iii ) For any real number a , Λ − Y,s ( ax ) − Λ − X,s ( x ) changes sign at most once, and if thechange of sign does occur, it is in the order − , + , as x traverses from to ∞ . ( iv ) For any real number a , Λ − X,s ( x ) − Λ − Y,s ( ax ) changes sign at most once, and if thechange of sign does occur, it is in the order + , − , as x traverses from to ∞ . ( v ) For any real number a , Λ Y,s Λ − X,s ( x ) − ax changes sign at most once, and if thechange of sign does occur, it is in the order + , − , as x traverses from to ∞ . ( vi ) Λ Y,s Λ − X,s ( x ) is antistar-shaped in x > . ✷ Before going to the next theorem we give two lemmas without proof.
Lemma 3.1
Let f ( · ) be a nonnegative, increasing, and star-shaped function. Then f − ( · ) is antistar-shaped. ✷ Lemma 3.2
Let f ( · ) and g ( · ) be two nonnegative, increasing, and star-shaped functions.Then f ( g ( · )) is star-shaped. ✷ Some properties of the s-IFRA(R) ordering are discussed in the following theorem.
Theorem 3.1
For any positive integer s , ( i ) X ≤ s − IF RA ( R ) X . ( ii ) X ≤ s − IF RA ( R ) Y and Y ≤ s − IF RA ( R ) X hold simultaneously if, and only if, Λ X,s ( x ) = θ Λ Y,s ( x ) , for some θ > and for all x ≥ . ( iii ) If X ≤ s − IF RA ( R ) Y and Y ≤ s − IF RA ( R ) Z then X ≤ s − IF RA ( R ) Z . Proof:
The proof of ( i ) is trivial. To prove ( ii ) we proceed as follows. X ≤ s − IF RA ( R ) Y gives that Λ X,s (Λ − Y,s ( · )) is star-shaped , which, by Lemma 3.1, reduces to the fact thatΛ Y,s (Λ − X,s ( · )) is antistar-shaped . Further, Y ≤ s − IF RA ( R ) X gives thatΛ Y,s (Λ − X,s ( · )) is star-shaped . Y,s (Λ − X,s ( x )) = xθ , for some constant θ ( > X,s ( x ) = θ Λ Y,s ( x ), and hence ( ii ) is proved.Again, by Lemma 3.2, ( iii ) holds. ✷ The following theorem is a bridge between s-IFRA(R) ordering and s-IFRA ageingclass. The proof follows from Lemma 2.4.
Theorem 3.2 If F Y ( x ) = e − λx , then X ≤ s − IF RA ( R ) Y if, and only if, X is s-IFRA. ✷ Since, every IFR distribution is an IFRA distribution, we have the following theorem.
Theorem 3.3 If X ≤ s − IF R ( R ) Y then X ≤ s − IF RA ( R ) Y. ✷ The following theorem shows that s-IFRA(R) ordering is location and scale invariant.The proof follows from (2.8).
Theorem 3.4 X ≤ s − IF RA ( R ) Y if, and only if, ( aX + b ) ≤ s − IF RA ( R ) ( aY + b ) , for anyreal numbers a ( > and b . ✷ We start this section with the following definition of the s-NBU(R) ordering.
Definition 4.1
For any positive integer s , X (or its distribution F X ) is said to be more s -NBU(R) than Y (or its distribution F Y ) (written as X ≤ s − NBU ( R ) Y ) if the randomvariable Φ sX,Y has a NBU distribution. ✷ Remark 4.1
For s = 1 , Definition 4.1 gives X ≤ su Y , as discussed in Sengupta andDeshpande [22]. ✷ In the following proposition we give some equivalent representations of the s-NBU(R)ordering.
Proposition 4.1
Definition 4.1 can equivalently be written in one of the following forms: ( i ) Λ X,s (cid:0) Λ − Y,s ( x ) (cid:1) is super-additive in x ≥ . ( ii ) T − X,s (cid:16) T X,s ( x + t ) T X,s ( t ) (cid:17) ≥ T − Y,s (cid:16) T Y,s ( x + t ) T Y,s ( t ) (cid:17) , for all x, t > . ( iii ) Λ X,s ( Y s ) has a NWU distribution. roof: The equivalence of Definition 4.1 and ( i ) follows from (2.4). Again, ( i ) holds if,and only if, for all a, b > X,s (cid:0) Λ − Y,s ( a + b ) (cid:1) ≥ Λ X,s (cid:0) Λ − Y,s ( a ) (cid:1) + Λ X,s (cid:0) Λ − Y,s ( b ) (cid:1) , or equivalently, − log T X,s (cid:0) Λ − Y,s ( a + b ) (cid:1) T X,s (cid:0) Λ − Y,s ( a ) (cid:1) ! ≥ − log T X,s (cid:0) Λ − Y,s ( b ) (cid:1) . It can equivalently be written as T − X,s T X,s (cid:0) Λ − Y,s ( a + b ) (cid:1) T X,s (cid:0) Λ − Y,s ( a ) (cid:1) ! ≥ Λ − Y,s ( b ) . Writing a = Λ Y,s ( t ) , a + b = Λ Y,s ( x + t ) in the above inequality, we have, for x, t > T − X,s (cid:18) T X,s ( x + t ) T X,s ( t ) (cid:19) ≥ Λ − Y,s (cid:18) − log T Y,s ( x + t ) T Y,s ( t ) (cid:19) = T − Y,s (cid:18) T Y,s ( x + t ) T Y,s ( t ) (cid:19) . Thus, the equivalence of ( i ) and ( ii ) is proved. The proof of ( iii ) follows in the same lineas is done in ( ii ). ✷ To prove the next theorem we use two lemmas which are given below without proof.
Lemma 4.1
Let f ( · ) be a nonnegative, increasing, and super-additive function. Then f − ( · ) is sub-additive. ✷ Lemma 4.2
Let f ( · ) and g ( · ) be two nonnegative, increasing, and super-additive func-tions. Then f ( g ( · )) is super-additive. ✷ The following theorem discusses some properties of the s-NBU(R) ordering.
Theorem 4.1
For any positive integer s , ( i ) X ≤ s − NBU ( R ) X . ( ii ) X ≤ s − NBU ( R ) Y and Y ≤ s − NBU ( R ) X hold simultaneously if, and only if, Λ X,s ( x ) = θ Λ Y,s ( x ) , for some θ > and for all x ≥ . ( iii ) If X ≤ s − NBU ( R ) Y and Y ≤ s − NBU ( R ) Z then X ≤ s − NBU ( R ) Z . roof: It is easy to verify ( i ). Let X ≤ s − NBU ( R ) Y . ThenΛ X,s (cid:0) Λ − Y,s ( x ) (cid:1) is super-additive . By Lemma 4.1, the above statement can equivalently be written asΛ
Y,s (cid:0) Λ − X,s ( x ) (cid:1) is sub-additive . (4.9)Further, Y ≤ s − NBU ( R ) X gives thatΛ Y,s (cid:0) Λ − X,s ( x ) (cid:1) is super-additive . (4.10)Combining (4 .
9) and (4 . Y,s (cid:0) Λ − X,s ( x ) (cid:1) = xθ , for some constant θ ( > X,s ( x ) = θ Λ Y,s ( x ). The proof of ( iii ) follows fromLemma 4.2. ✷ In the following theorem we represent the relationship between s-NBU(R) orderingand s-NBU ageing. The proof follows from Lemma 2.4.
Theorem 4.2 If F Y ( x ) = e − λx , then X ≤ s − NBU ( R ) Y if, and only if, X is s-NBU. Since every star-shaped function is super-additive, we have the following theorem.
Theorem 4.3 If X ≤ s − IF RA ( R ) Y then X ≤ s − NBU ( R ) Y. ✷ The following theorem shows that s-NBU(R) ordering is scale and base invariant. Theproof follows from (2.8).
Theorem 4.4 X ≤ s − NBU ( R ) Y if, and only if, ( aX + b ) ≤ s − NBU ( R ) ( aY + b ) , for anyreal numbers a ( > and b . ✷ We discuss s-NBUFR(R) ordering in this section.
Definition 5.1
For any positive integer s , X (or its distribution F X ) is said to be more s -NBUFR(R) than Y (or its distribution F Y ) (written as X ≤ s − NBUF R ( R ) Y ) if therandom variable Φ sX,Y has a NBUFR distribution. ✷ In the following proposition we give some equivalent conditions of the s-NBUFR(R)ordering. 13 roposition 5.1
For s = 2 , . . . , Definition 5.1 can equivalently be written in one ofthe following forms: ( i ) r X,s ( x ) r Y,s ( x ) ≥ ˜ µ Y,s − ˜ µ X,s − , for all x ≥ . ( ii ) µ Y,s − ( x ) µ X,s − ( x ) ≥ ˜ µ Y,s − ˜ µ X,s − , for all x ≥ . ( iii ) Λ X,s ( Y s ) has a NWUFR distribution. Proof: Φ sX,Y is NBUFR if, and only if, for all x ≥ r Φ sX,Y ( x ) ≥ r Φ sX,Y (0) , or equivalently, T X,s − (cid:16) T − Y,s ( u ) (cid:17) T X,s (cid:16) T − Y,s ( u ) (cid:17) ≥ (cid:18) T X,s − (0) T Y,s − (0) (cid:19) T Y,s − (cid:16) T − Y,s ( u ) (cid:17) u , for all u ∈ (0 , , which holds if, and only if, T X,s − ( x ) T X,s ( x ) ≥ T Y,s − ( x ) T Y,s ( x ) , for all x ≥ . (5.11)This can equivalently be written as r X,s ( x ) r Y,s ( x ) ≥ ˜ µ Y,s − ˜ µ X,s − , for all x ≥ . Thus, the equivalence of Definition 5.1 and ( i ) is established. The equivalence of ( i ) and( ii ) follows from (1.3). The proof of the equivalence of ( i ) and ( iii ) is obvious. ✷ Remark 5.1
For s = 1 , Definition 5.1 can equivalently be written in one of the followingforms: ( i ) r X, ( x ) r Y, ( x ) ≥ f X (0) f Y (0) , for all x ≥ . ( ii ) Λ X, ( Y ) has a NWUFR distribution. ✷ The following theorem discusses some properties of the s-NBUFR(R) ordering.
Theorem 5.1
For any positive integer s , ( i ) X ≤ s − NBUF R ( R ) X . ( ii ) X ≤ s − NBUF R ( R ) Y and Y ≤ s − NBUF R ( R ) X hold simultaneously if, and only if, Λ X,s ( x ) = θ Λ Y,s ( x ) , for some θ > and for all x ≥ . iii ) If X ≤ s − NBUF R ( R ) Y and Y ≤ s − NBUF R ( R ) Z then X ≤ s − NBUF R ( R ) Z . Proof:
The proof of ( i ) is obvious. To prove ( ii ) we proceed as follows. On using (5.11), X ≤ s − NBUF R ( R ) Y reduces to the fact that, for all x ≥ T X,s − ( x ) T X,s ( x ) ≥ (cid:18) T X,s − (0) T Y,s − (0) (cid:19) (cid:18) T Y,s − ( x ) T Y,s ( x ) (cid:19) , (5.12)and Y ≤ NBUF R ( R ) X gives T Y,s − ( x ) T Y,s ( x ) ≥ (cid:18) T Y,s − (0) T X,s − (0) (cid:19) (cid:18) T X,s − ( x ) T X,s ( x ) (cid:19) , or equivalently, T X,s − ( x ) T X,s ( x ) ≤ (cid:18) T X,s − (0) T Y,s − (0) (cid:19) (cid:18) T Y,s − ( x ) T Y,s ( x ) (cid:19) . (5.13)Combining (5.12) and (5.13), we have T X,s − ( x ) T X,s ( x ) = (cid:18) T X,s − (0) T Y,s − (0) (cid:19) (cid:18) T Y,s − ( x ) T Y,s ( x ) (cid:19) . (5.14)Again, from (2.4) and (2.5) we have ddx (cid:0) Λ X,s Λ − Y,s ( x ) (cid:1) = (cid:18) ˜ µ Y,s − ˜ µ X,s − (cid:19) T X,s − T − Y,s ( e − x ) T X,s T − Y,s ( e − x ) ! e − x T Y,s − T − Y,s ( e − x ) ! = 1 θ , (5.15)where θ = (cid:18) ˜ µ X,s − ˜ µ Y,s − (cid:19) (cid:18) T Y,s − (0) T X,s − (0) (cid:19) , and the second equality follows from (5.14). Hence, from (5.15) we haveΛ X,s (cid:0) Λ − Y,s ( x ) (cid:1) = xθ , or equivalently, Λ X,s ( x ) = θ Λ Y,s ( x ) . Thus, ( ii ) is proved. Again, X ≤ s − NBUF R ( R ) Y gives that, for all x ≥ T X,s − ( x ) T X,s ( x ) ≥ (cid:18) T X,s − (0) T Y,s − (0) (cid:19) (cid:18) T Y,s − ( x ) T Y,s ( x ) (cid:19) , (5.16)15nd Y ≤ s − NBUF R ( R ) Z gives T Y,s − ( x ) T Y,s ( x ) ≥ (cid:18) T Y,s − (0) T Z,s − (0) (cid:19) (cid:18) T Z,s − ( x ) T Z,s ( x ) (cid:19) , or equivalently, (cid:18) T X,s − (0) T Y,s − (0) (cid:19) (cid:18) T Y,s − ( x ) T Y,s ( x ) (cid:19) ≥ (cid:18) T X,s − (0) T Z,s − (0) (cid:19) (cid:18) T Z,s − ( x ) T Z,s ( x ) (cid:19) . (5.17)Thus, from (5.16) and (5.17) we have T X,s − ( x ) T X,s ( x ) ≥ (cid:18) T X,s − (0) T Z,s − (0) (cid:19) (cid:18) T Z,s − ( x ) T Z,s ( x ) (cid:19) . Thus, X ≤ s − NBUF R ( R ) Z . ✷ In the following theorem we give a relationship between s-NBUFR(R) ordering ands-NBUFR ageing.
Theorem 5.2 If F Y ( x ) = e − λx , then X ≤ s − NBUF R ( R ) Y if, and only if, X is s-NBUFR. ✷ Since, every NBU distribution is a NBUFR distribution, we have the following theo-rem.
Theorem 5.3 If X ≤ s − NBU ( R ) Y then X ≤ s − NBUF R ( R ) Y. ✷ The following theorem shows that s-NBUFR(R) ordering is scale and base invariant.
Theorem 5.4 X ≤ s − NBUF R ( R ) Y if, and only if, ( aX + b ) ≤ s − NBUF R ( R ) ( aY + b ) , forany real numbers a ( > and b . Proof:
For all x ≥
0, and for any real numbers a ( >
0) and b , the hazard rate functionof the random variable Φ saX + b,aY + b is given by r Φ saX + b,aY + b ( x ) = ddx (cid:0) Λ aX + b,s (Λ aY + b,s ) − ( x ) (cid:1) = ddx (cid:0) Λ X,s Λ − Y,s ( x ) (cid:1) = r Φ sX,Y ( x ) , (5.18)where the second equality holds from (2.8). Thus, the result follows from Definition 5.1.16 s-NBAFR(R) Ordering We begin this section with the following definition.
Definition 6.1
For any positive integer s , X (or its distribution F X ) is said to be more s -NBAFR(R) than Y (or its distribution F Y ) (written as X ≤ s − NBAF R ( R ) Y ) if therandom variable Φ sX,Y has a NBAFR distribution. ✷ Some equivalent representations of the s-NBAFR(R) ordering are discussed in thefollowing theorem.
Proposition 6.1
For s = 2 , , . . . , Definition 6.1 can equivalently be written in one ofthe following forms: ( i ) Λ − X,s ( x ˜ µ Y,s − ) ≤ Λ − Y,s ( x ˜ µ X,s − ) , for all x > . ( ii ) Λ X,s ( x )Λ Y,s ( x ) ≥ ˜ µ Y,s − ˜ µ X,s − , for all x > . ( iii ) Λ X,s ( Y s ) has a NWAFR distribution. Proof: Φ sX,Y has a NBAFR distribution if, and only if, for all x > − x log F Φ sX,Y ( x ) ≥ r Φ sX,Y (0) , or equivalently, Λ X,s Λ − Y,s ( x ) x ≥ (cid:18) ˜ µ Y,s − ˜ µ X,s − (cid:19) (cid:18) T X,s − (0) T Y,s − (0) (cid:19) . (6.19)This can equivalently written asΛ − Y,s ( x ) ≥ Λ − X,s (cid:18) x ˜ µ Y,s − ˜ µ X,s − (cid:19) . (6.20)Replacing x by x ˜ µ X,s − in (6.20), we getΛ − X,s ( x ˜ µ Y,s − ) ≤ Λ − Y,s ( x ˜ µ X,s − ) . Thus, the equivalence of Definition 6.1 and ( i ) is proved. The one-to-one connectionbetween ( i ) and ( ii ) follows from (6.19). The equivalence of ( i ) and ( iii ) can be provedin the same line as is done in ( i ). ✷ Remark 6.1
For s = 1 , Definition 6.1 can equivalently be written in one of the followingforms: i ) Λ − X, ( xf X (0)) ≤ Λ − Y, ( xf Y (0)) , for all x > . ( ii ) Λ X, ( x )Λ Y, ( x ) ≥ f X (0)˜ f Y (0) , for all x > . ( iii ) Λ X, ( Y ) has a NWAFR distribution. ✷ The following theorem gives some properties of the s-NBAFR(R) ordering.
Theorem 6.1
For any positive integer s , ( i ) X ≤ s − NBAF R ( R ) X . ( ii ) X ≤ s − NBAF R ( R ) Y and Y ≤ s − NBAF R ( R ) X hold simultaneously if, and only if, Λ X,s ( x ) = θ Λ Y,s ( x ) , for some θ > and for all x > . ( iii ) If X ≤ s − NBAF R ( R ) Y and Y ≤ s − NBAF R ( R ) Z then X ≤ s − NBAF R ( R ) Z . Proof:
The proof of ( i ) is obvious. Note that X ≤ s − NBAF R ( R ) Y holds if, and only, if,for all x >
0, Λ
X,s (Λ − Y,s ( x )) ≥ (cid:18) x ˜ µ Y,s − ˜ µ X,s − (cid:19) (cid:18) T X,s − (0) T Y,s − (0) (cid:19) . (6.21)Again, Y ≤ s − NBAF R ( R ) X holds if, and only if, for all x > Y,s (Λ − X,s ( x )) ≥ (cid:18) x ˜ µ X,s − ˜ µ Y,s − (cid:19) (cid:18) T Y,s − (0) T X,s − (0) (cid:19) . (6.22)Replacing x by Λ X,s (Λ − Y,s ( x )) in (6.22), we haveΛ X,s (Λ − Y,s ( x )) ≤ (cid:18) x ˜ µ Y,s − ˜ µ X,s − (cid:19) (cid:18) T X,s − (0) T Y,s − (0) (cid:19) . (6.23)Combining (6.21) and (6.23), we haveΛ X,s (Λ − Y,s ( x )) = θx, where θ = (cid:18) ˜ µ Y,s − ˜ µ X,s − (cid:19) (cid:18) T X,s − (0) T Y,s − (0) (cid:19) . Thus, Λ
X,s ( x ) = θ Λ Y,s ( x ), and hence ( ii ) is proved. Again, X ≤ s − NBAF R Y givesΛ X,s (Λ − Y,s ( x )) ≥ (cid:18) x ˜ µ Y,s − ˜ µ X,s − (cid:19) (cid:18) T X,s − (0) T Y,s − (0) (cid:19) , (6.24)18nd Y ≤ s − NBAF R Z givesΛ Y,s (Λ − Z,s ( x )) ≥ (cid:18) x ˜ µ Z,s − ˜ µ Y,s − (cid:19) (cid:18) T Y,s − (0) T Z,s − (0) (cid:19) . (6.25)Now, Λ X,s (Λ − Z,s ( x )) = Λ X,s Λ − Y,s (cid:0) Λ Y,s Λ − Z,s ( x ) (cid:1) ≥ Λ X,s Λ − Y,s (cid:18) x ˜ µ Z,s − ˜ µ Y,s − T Y,s − (0) T Z,s − (0) (cid:19) ≥ (cid:18) x ˜ µ Z,s − ˜ µ Y,s − (cid:19) (cid:18) T Y,s − (0) T Z,s − (0) (cid:19) (cid:18) ˜ µ Y,s − ˜ µ X,s − (cid:19) (cid:18) T X,s − (0) T Y,s − (0) (cid:19) = (cid:18) x ˜ µ Z,s − ˜ µ X,s − (cid:19) (cid:18) T X,s − (0) T Z,s − (0) (cid:19) , where the first inequality holds from (6.25) and using the fact that Λ X,s Λ − Y,s ( · ) is anincreasing function. The second inequality follows from (6.24). Thus, X ≤ s − NBAF R ( R ) Z . ✷ The following theorem shows that X is smaller than exponential random variable ins-NBAFR(R) ordering if, and only if, X has a NBAFR distribution. The proof followsfrom Lemma 2.4. Theorem 6.2 If F Y ( x ) = e − λx , then X ≤ s − NBAF R ( R ) Y if, and only if, X is s-NBAFR. ✷ Since, every NBUFR distribution is a NBAFR distribution, we have the followingtheorem.
Theorem 6.3 If X ≤ s − NBUF R ( R ) Y then X ≤ s − NBAF R ( R ) Y. ✷ That the s-NBAFR(R) ordering is scale and base invariant, is shown in the followingtheorem. On using (5.18), the proof follows from Definition 6.1.
Theorem 6.4 X ≤ s − NBAF R ( R ) Y if, and only if, ( aX + b ) ≤ s − NBAF R ( R ) ( aY + b ) , forany real numbers a ( > and b . ✷ In this paper we give some new generalized stochastic orderings, and study some oftheir properties. These orderings may be helpful to visualize the positive ageing classesfrom different aspects. To handle the crossing hazard rates and/or crossing mean residual19ives problem, we may find out a new direction with the help of these orderings. Thisunified study is meaningful because it gives a complete scenario of the existing results(available in the literature) together with the new results. The usefulness of relative age-ing is well explained in Sengupta and Deshpande [22], and Kalashnikov and Rachev [11].Keeping the importance of the relative ageing in mind, we have taken an attempt to dis-cuss different kinds of relative ageing in a unified way so that different kinds of relativeageing properties come under a single umbrella. Further, the different characterizationsof these relative ageing properties are important because of their theoretical insight inone hand, and the systems belonging to this ageing classes help the practitioners (viz.reliability and design engineers) manipulate it for its nice mathematical properties onthe other. To make the usefulness of these kind of orderings more appealing, let us takea particular example as discussed below.
Example 7.1
Let X be random variable having µ X, ( t ) = 1 / (4 + 11 t ) , t ≥ , and Y beanother random variable having µ Y, ( t ) = 1 / (4 + 5 t ) , t ≥ . Then r X, ( t ) = 4 + 11 t − t t , and r Y, ( t ) = 4 + 5 t − t t . By drawing the figures of r X, ( t ) and r Y, ( t ) , it can be shown that X and Y have crossinghazard rates. Note that r X, ( t ) r Y, ( t ) = (cid:18) (4 + 11 t ) − t (4 + 5 t ) − t (cid:19) (cid:18) t t (cid:19) , which can be shown to non-monotone, and hence X (cid:2) − IF R ( R ) Y . Further r X, ( t ) r Y, ( t ) = 4 + 11 t t is increasing in t, and hence X ≤ − IF R ( R ) Y follows from Proposition 2.1. ✷ In the above example we see that hazard rates of X and Y have crossed each other. So,none of the two dominates the other in terms of their failure rates. In order to decide onthe better system, i.e., to see which one is ageing slower, we take s = 1, i.e., we comparethem in terms of 1-IFR(R) order. It is noted that none of the two dominates the other asfar as 1-IFR(R) order is concerned. This means that if we concentrate our study basedon 1-IFR(R) order only, we cannot conclude which of the two is better. To overcomethis difficulty we take s = 2, which gives a comparison, known as 2-IFR(R) order. Here20e see that the ratio of r X, ( t ) and r Y, ( t ) is monotone − clearly showing the dominanceof one over the other. Thus, we are now in a position to say that X is ageing fastercompared to Y . So the system with life distribution Y is better. The above examplegives the importance of s -IFR(R) ( s ≥
2) order. In a similar spirit, the other generalizedorders are defined and studied to help the reliability practitioners to decide on how tochoose the better one. We conclude our discussion by mentioning the following chain ofimplications of the generalized stochastic orderings. X ≤ s − IF R ( R ) Y ⇒ X ≤ s − IF RA ( R ) Y ⇓ X ≤ s − NBU ( R ) Y ⇓ X ≤ s − NBUF R ( R ) Y ⇒ X ≤ s − NBAF R ( R ) Y. Acknowledgements
The authors are thankful to the Editor, and anonymous Reviewers for their valu-able constructive comments and suggestions which lead to an improved version of themanuscript. Financial support from Council of Scientific and Industrial Research, NewDelhi (CSIR Grant No. 09/921(0060)2011-EMR-I) is sincerely acknowledged by Nil Ka-mal Hazra.
References [1] Barlow, R.E. and Proschan, F. (1975).
Statistical Theory of Reliability and LifeTesting . Holt, Rinehart and Winston, New York.[2] Begg, C.B., Mcglave, P.B., Bennet, J.M., Cassileth, P.A. and Oken, M.M. (1984).A critical comparison of allogeneic bone marrow transplantation and conventionalchemotherapy as treatment for acute non-lymphomytic leukemia.
Journal of ClinicalOncology , , 369-378.[3] Bekker, L. and Mi, J. (2003). Shape and crossing properties of mean residual lifefunctions. Statistics and Probability Letters , , 225-234.[4] Bryson, M.C. and Siddiqui, M.M. (1969). Some criteria for ageing. Journal of theAmerican Statistical Association , , 1472-1483.215] Champlin, R., Mitsuyasu, R., Elashoff, R. and Gale, R.P. (1983). Recent advancesin bone marrow transplantation. In: UCLA Symposia on Molecular and CellularBiology, ed. R.P. Gale, New York , , 141-158.[6] Cox, D.R. (1972). Regression models and life-tables. Journal of the Royal StatisticalSociety, Series B , , 187-220.[7] Deshpande, J.V., Kochar, S.C. and Singh, H. (1986). Aspects of positive ageing. Journal of Applied Probability , , 748-758.[8] Deshpande, J.V., Singh, H., Bagai, I. and Jain, K. (1990). Some partial ordersdescribing positive ageing. Communications in Statistics. Stochastic Models , ,471-481.[9] Fagiuoli, E. and Pellerey, F. (1993). New partial orderings and applications. NavalResearch Logistics , , 829-842.[10] Gupta, R.C. and Gupta, P.L. (2000). On the crossing of reliability measures. Statis-tics and Probability Letters , , 301-305.[11] Kalashnikov, V. V. and Rachev, S.T. (1986). Characterization of queueing modelsand their stability. In: Probability Theory and Mathematical Statistics, eds. Yu. K.Prohorov et al., VNU Science Press, Amsterdam , , 37-53.[12] Klefsj¨o, B. (1982). The HNBUE and HNWUE classes of life distributions. NavalResearch Logistics Quarterly , , 331-344.[13] Kochar, S.C. and Wiens, D.P. (1987). Partial orderings of life distributions withrespect to their ageing properties. Naval Research Logistics , , 823-829.[14] Lai, C.D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability .Springer, New York.[15] Launer, R.L. (1984). Inequalities for NBUE and NWUE life distributions.
OperationsResearch , , 660-667.[16] Loh, W.Y. (1984). A new generalization of the class of NBU distributions. IEEETransactions on Reliability , R-33 , 419-422.[17] Lehmann, E.L. (1953). The power of rank test.
Annals of Mathematical Statistics , , 23-43. 2218] Liu, K., Qiu, P. and Sheng, J. (2007). Comparing two crossing hazard rates by Coxproportional hazards modelling. Statistics in Medicine , , 375-391.[19] Mantel, N. and Stablein, D.M. (1988). The crossing hazard function problem. Jour-nal of the Royal Statistical Society, Series D , , 59-64.[20] Marshall, A.W. and Olkin, I. (2007). Life Distributions . Springer, New York.[21] Pocock, S.J., Gore, S.M. and Keer, G.R. (1982). Long-term survival analysis: thecurability of breast cancer.
Statistics in Medicine , , 93-104.[22] Sengupta, D. and Deshpande, J.V. (1994): Some results on the relative ageing oftwo life distributions. Journal of Applied Probability , , 991-1003.[23] Shaked, M. and Shanthikumar, J.G. (2007). Stochastic Orders . Springer, New York.[24] Zahedi, H. (1992). Proportional mean remaining life model.
Journal of StatisticalPlanning and Inference ,29