Sankhy¯ a : The Indian Journal of Statistics 2000, Volume 62, Series A, Pt. 2, pp. 258–266
GENERALIZED AGEING CLASSES IN TERMS OF LAPLACE TRANSFORMS By ASOK K. NANDA∗ Panjab University, Chandigarh SUMMARY. Using Laplace transform, various reliability classes have been characterized by different researchers. In this paper, we have characterized the general ageing classes (viz. s-IFR, s-IFRA, s-NBU and their duals), using Laplace transform, by means of necessary and sufficient conditions so that most of the existing Laplace transform characterization of ageing classes follow as particular cases.
1.
Introduction
Kebir (1994) (see also Shaked and Shanthikumar, 1994) has characterized likelihood ratio (LR) ordering, failure rate (FR) ordering, reversed hazard rate ordering and stochastic (ST) ordering with the help of Laplace transform. Using Laplace transform, Nanda (1995) has characterized, by means of necessary and sufficient conditions, the property that two life distributions are ordered in the s-FR and s-ST sense so that the characterization of likelihood ratio, failure rate, mean residual life (MR), variance residual life (VR), stochastic and harmonic average mean residual life (HAMR) orderings follow as particular cases. Vinogradov (1973) has characterized the IFR (Increasing Failure Rate) life distributions in terms of Laplace transform. IFRA (Increasing Failure Rate in Average), DMRL (Decreasing in Mean Residual Life), NBU (New Better than Used), NBUE (New Better than Used in Expectation) and their duals have been characterized by Block and Savits (1980) using the same transform. In Klefsj¨ o (1982), Laplace transform is used in order to give equivalent definition of HNBUE (Harmonically New Better than Used in Expectation) and its dual class. Thus, various reliability classes have been characterized by different researchers by means of Laplace transform. For various definitions of the above ageing classes, one may refer to Bryson and Siddiqui (1969), Deshpande et al. (1986) and the references there in. Paper received. October 1998; revised October 1999. AMS (1991) subject classification. Primary 60K10; Secondary 62N05. Key words and phrases. Positive ageing and negative ageing properties; Laplace transform; Variation diminishing property; Log-concavity. ∗ Presently visiting the University of Arizona, Tucson, USA.
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Fagiuoli and Pellerey (1993) have defined different ageing properties in a general way depending on the generalized orderings (s-FR, s-ST etc.). In Section 3, we give necessary and sufficient condition, in terms of Laplace transforms, for a distribution to have one of the generalized ageing criteria viz. s-IFR, s-IFRA, s-NBU and their duals (definitions follow). As a particular case, the characterization of IFR, IFRA, DMRL, DVRL (Decreasing in Variance Residual Life), NBU and their dual classes follows. This generalizes the results of Vinogradov (1973) and Block and Savits (1980). 2.
Notations, Definitions and Preliminaries
For any nonnegative absolutely continuous random variable X with density function f (x), survival function F (x) and mean µF , write T 0 (X, x) = f (x) and
R∞ x
T s (X, x) = for s = 1, 2, . . ., where
Z
(1)
T s−1 (X, t)dt , µs−1 (X)
(2)
T s (X, t)dt,
(3)
∞
µs (X) = 0
s = 0, 1, 2, . . . · Note that T 2 (X, x) is the survival function of the first derived distribution (also called equilibrium distribution) of X, which plays an important role in the aging concepts (cf. Deshpande et al., 1986). We call the first derived distribution of the first derived distribution as second derived distribution of F (or equivalently, of X) and so on. Thus, T s (X, x) is the survival function of the (s−1)th derived distribution of F . Also define T s−1 (X, x) rs (X, x) = R ∞ , T s−1 (X, t)dt x
(4)
which represents the failure rate function corresponding to T s (X, x). The function rs (X, x) is, in fact, the failure rate function of the (s − 1)th derived distribution of F . For s = 1, r1 (X, x) is the failure rate function of X, defined as the ratio of the density to the survival function, whereas, for s = 2, r2 (X, x) is the reciprocal of the mean residual life function. Throughout in this paper, the set of nonnegative integers will be denoted by N and the set of positive integers will be denoted by N+ . The following definition is due to Fagiuoli and Pellerey (1993). Definition 2.1. For any s ∈ N, a continuous nonnegative random variable X is said to be (a) s-IFR (s-DFR) if rs (X, x) is nondecreasing (nonincreasing) in x > 0;
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asok k. nanda (b) s-IFRA (s-DFRA) if
0;
1 x
Rx 0
rs (X, t)dt is nondecreasing (nonincreasing) in x >
(c) s-NBU (s-NWU) if T s (X, x+t)T s (X, 0) ≤ (≥)T s (X, x).T s (X, t) for all x, t ≥ 0. For s ∈ N+ , the definition 2.1(c) above can be written as T s (X, x + t) ≤ (≥)T s (X, x).T s (X, t) for all x, t ≥ 0. It can be noted that the following equivalences hold: 0-IFR⇔ ILR; 1-IFR⇔ IFR; 2-IFR⇔ DMRL; 3-IFR⇔ DVRL; 1-IFRA⇔ IFRA; 1-NBU⇔ NBU. Note that X is s-IFR if and only if rs (X, t) is nondecreasing in t ≥ 0. This holds if and only if, for all x > 0, T s (X, x + t) is nonincreasing in t, T s (X, t)
(5)
which can be shown to hold if and only if T s (X, x) is nonincreasing in x. T s−1 (X, x)
(6)
³ ´ (s) n /n!, n = Let X(s) have survival function T s (X, x), s ∈ N+ . We write, µn (X) = E X(s) 1, 2, . . . · Let X be a nonnegative random variable with distribution function F and survival function F ≡ 1 − F having finite moments of all orders. Define the Laplace transform of X as Z ∞ φX (λ) = e−λx dF (x), 0
λ > 0 and define (as in Block and Savits, 1980) · ¸ (−1)n dn 1 − φX (λ) X aλ (n) = . , n! dλn λ n ∈ N, λ > 0, with aX λ (0) = (1 − φX (λ)) /λ and n X αX λ (n) = λ .aλ (n − 1),
n ∈ N+ , λ > 0 with αX λ (0) = 1 for λ > 0. Similarly, for a nonnegative random variable Y having survival function G(x), define αYλ (n) accordingly. Write, Γλ (n, x) = λe−λx
(λx)n−1 , (n − 1)!
generalized ageing classes in terms of laplace transforms n ∈ N+ , x ≥ 0 and
Z
261
∞
Γλ g(n) =
Γλ (n, x)g(x)dx 0
for any function g defined on [0, ∞), provided the integral is finite. Take Γλ g(0) = 1. The following result is given in Kebir (1994). R∞ + Lemma 2.1. αX λ (n) = 0 Γλ (n, x)F (x)dx for n ∈ N . For completeness, we reproduce the following result from Skaked and Shanthikumar (1994). Y Lemma 2.2. αX λ (n) and αλ (n) are discrete survival functions.
Let NλX and NλY be two discrete random variables having respective survival Y function αX λ (n) and αλ (n). Using the technique used by Shaked and Shanthikumar (1994) in proving Lemma 2.2, we can see that Z ∞ Γλ T s (X, n) = Γλ (n, x)T s (X, x)dx 0
and
Z Γλ T s (Y, n) =
∞
Γλ (n, x)T s (Y, x)dx 0
are discrete survival functions. The following lemma will be used in sequel. Lemma 2.3. For any s ∈ N, P∞ (i) k=n+1 Γλ T s (X, k) = λ.µs (X)Γλ T s+1 (X, n); P∞ (ii) k=0 Γλ T s (X, k) = 1 + λ.µs (X), where µs (X) is given in (3). Proof. ∞ X
Z Γλ T s (X, k) =
∞
λ 0
k=n+1
Z =
λ 0
∞
∞ X (λx)k−1 T s (X, x)dx (k − 1)! k=n+1 "Z # λx 1 e−t tn−1 dt T s (X, x)dx. Γ(n) 0
e−λx
Interchanging the order of integration and using (2), we have ∞ X
Γλ T s (X, k) = λ.µs (X).Γλ T s+1 (X, n).
k=n+1
This proves (i). To prove (ii), write ∞ X k=0
and use (i).
Γλ T s (X, k) = Γλ T s (X, 0) +
∞ X k=1
Γλ T s (X, k)
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Taking n = 0 and s = 1 in Lemma 2.3(i), we have the following corollary. ¢ ¢ ¡ ¡ Corollary 2.1. E NλX = λ.E(X) and E NλY = λ.E(Y ). The following lemma may be found in Block and Savits (1980) (see also Nanda, 1995). Lemma 2.4. Let x > 0 be a continuity point of the distribution function of X(s) . Let λ = λ(n, x) satisfy limn→∞ (n/λ) = x and limn→∞ (1/λ) = 0. Then, for any nonnegative integer s, lim Γλ T s (X, n) = T s (X, x).
n→∞
3.
Characterization of the Generalized Ageing Properties
Using Laplace transform, Vinogradov (1973) has characterized IFR distributions, while the necessary and sufficient conditions, for a distribution to be DMRL, IFRA, NBU, NBUE and their duals, in terms of Laplace transforms, have been given by Block and Savits (1980). Here, in this section, we use Laplace transform to give necessary and sufficient condition for a distribution to belong to the generalized ageing class (definition 2.1). This generalizes the results of Vinogradov (1973) and Block and Savits (1980). In the following theorem, a characterization of s-IFR distributions is given. Theorem 3.1. For s ∈ N+ , a random variable X is s-IFR if and only if Γλ T s (X, n) is nonincreasing in n ∈ N+ , Γλ T s−1 (X, n) for λ > 0. Proof. N ecessity: By (6), X is s-IFR if and only if T s (X, x) is nonincreasing in x. T s−1 (X, x) h i n−1 is T P2 in (n, x) for x ∈ (0, ∞) and n ∈ N+ , the required result Since, e−λx (λx) (n−1)! follows by using variation diminishing property. Suf f iciency: Given that Γλ T s (X, n) is nonincreasing in n, Γλ T s−1 (X, n) which gives, for k ∈ N+ , Γλ T s (X, n).Γλ T s−1 (X, n + k) ≥ Γλ T s (X, n + k).Γλ T s−1 (X, n).
(7)
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Let x(> 0) and x + y(> 0) be two points of continuity of both of T s−1 and T s . Write λ = (n/x) and k = [ny/x], the greatest integer contained in (ny/x). Then λn+1 −λt n t dt, one can easily see limn→∞ n+k λ = x + y. Now, by writing, dGn (t) = n! e that Gn+k−1 converges weekly to a distribution degenerate at x + y (see also Block and Savits (1980)). Hence, by Lemma 2.4, we have lim Γλ T s (X, n + k) = T s (X, x + y).
n→∞
. . . (8)
Since the discontinuity points of a distribution function (or equivalently, of a survival function) are at most countable, by taking limits as n approaches infinity, (7) gives the required result. Corollary 3.1. A random variable X, having survival function F , is (a) IFR if and only if Γλ F (n) is log concave in n for all λ > 0; P∞ (b) DMRL if and only if k=n Γλ F (k)/Γλ F (n) is nonincreasing in n ∈ N+ for all λ > 0; P∞ (c) DVRL if and only if k=n Γλ T 2 (X, k)/Γλ T 2 (X, n) is nonincreasing in n ∈ N+ , for all λ > 0. Proof. (a) Take s = 1 in the above theorem. Then, X is IFR if and only if, for all λ > 0, Γλ T 0 (X, n) is nondecreasing in n ∈ N+ . (9) Γλ T 1 (X, n) On using Lemma 2.3, (9) can equivalently be written as, for all λ > 0, Γλ T 1 (X, n − 1) − Γλ T 1 (X, n) is nondecreasing in n ∈ N+ , Γλ T 1 (X, n) which is equivalent to saying that, for all λ > 0, Γλ T 1 (X, n − 1) is nondecreasing in n ∈ N+ . Γλ T 1 (X, n) This means, for all λ > 0, Γλ F (n − 1) is nondecreasing in n ∈ N+ . Γλ F (n) That is, Γλ F (n) is log-concave in n ∈ N+ for all λ > 0. To prove (b), put s = 2 in the above theorem. Then, X is DMRL if and only if, for all λ > 0, Γλ T 2 (X, n) is nonincreasing in n ∈ N+ . Γλ F (n) On using Lemma 2.3, (10) can equivalently be written as, for all λ > 0, ∞ X Γλ F (k) is nonincreasing in n ∈ N+ , Γ F (n) λ k=n+1
(10)
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which, further, is equivalent to saying that, for all λ > 0, ∞ X Γλ F (k) is nonincreasing in n ∈ N+ . Γ F (n) λ k=n
This proves (b). Further, taking s = 3, in the above theorem, we get that X is DVRL if and only if, for all λ > 0, Γλ T 3 (X, n) is nonincreasing in n ∈ N+ . Γλ T 2 (X, n)
(11)
On using Lemma 2.3, (11) gives the required result. REMARK 3.1. Corollary 3.1(a) is given in Vinogradov (1973), and Corollary 3.1(b) is proved in Block and Savits (1980). Theorem 3.2. For any positive integer s, (a) X is s-IFRA if and only if, for all λ > 0, £ ¤1/n Γλ T s (X, n) is nonincreasing in n ∈ N+ ; (b) X is s-NBU if and only if, for all λ > 0, Γλ T s (X, m + n) ≤ Γλ T s (X, m).Γλ T s (X, n) for m, n > 0. Proof. Define, for n ∈ N+ , 1
(s)
aλ (n) =
λn+1
.Γλ T s (X, n + 1).
N ecessity: (a) Note that Z (s) aλ (n)
∞
= Z0 ∞ = =
0 (s)
un −λu T s (X, u)du e n! un T s (Y, u)du n!
µn+1 (Y ),
where T s (Y, u) = e−λu T s (X, u). Note that T s (X, ·) is IFRA (NBU) if and only if T s (Y, ·) is IFRA (NBU). Further, since s-IFRA property of X is equivalent to IFRA property of X(s) , by using Corollary 6.5 of Barlow and Proschan (1975), X is s-IFRA implies h i1/(n+1) (s) µn+1 (X) is nonincreasing in n. This further gives
£ ¤1/n Γλ T s (x, n) is nonincreasing in n.
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This proves the necessity of (a). Again, X is s-NBU if and only if X(s) is NBU, which implies (s) (s) µm+n (X) ≤ µ(s) m (X).µn (X) for m, n ≥ 0. This gives Γλ T s (X, m + n) ≤ Γλ T s (X, m).Γλ T s (X, n), giving necessary part of (b). Sufficiency: (a) Given that, for all n, k ∈ N+ and for all λ > 0, £ ¤1/n £ ¤1/(n+k) Γλ T s (X, n) ≥ Γλ T s (X, n + k) , or equivalently, for all n, k ∈ N+ and for all λ > 0, £ ¤n/(n+k) Γλ T s (X, n) ≥ Γλ T s (X, n + k) . If we take x(> 0) and x + y(> 0) as the continuity points of T s (X, ·), then on using the same type of argument as is used in proving the sufficiency of Theorem 3.1, we have, as n approaches infinity, the above expression reduces to £ ¤x/(x+y) T s (X, x) ≥ T s (X, x + y) . Since this is true for all continuity points x of T s (X, x) and the continuity points are dense, the above expression reduces to ¤1/x £ T s (X, x) is nonincreasing in x. This means X is s-IFRA. To prove (b), given that, for n, k > 0, and for all λ > 0, Γλ T s (X, n + k) ≤ Γλ T s (X, n).Γλ T s (X, k). Now, taking limit as n approaches infinity on both sides of the above expression and using an argument similar to one used in proving the sufficiency part of Theorem 3.1, we have T s (X, x + y) ≤ T s (X, x).T s (X, y) for x, y > 0. Hence the result. Corollary 3.2. A random variable X is £ ¤1/n (a) IFRA if and only if Γλ F (n) is nonincreasing in n ∈ N+ , for all λ > 0; (b) NBU if and only if Γλ F (m + n) ≤ Γλ F (m).Γλ F (n) for all m, n > 0 and for all λ > 0. Proof. Taking s = 1 in (a) and (b) of the above theorem, the result follows. Remark 3.2. All the above results follow for the respective dual classes with an obvious modification. Remark 3.3. Corollary 3.2 is given in Block and Savits (1980).
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Acknowledgements. This paper is based on part of the author’s dissertation written at the Panjab University, Chandigarh under the guidance of Professor Harshinder Singh and Dr. Kanchan Jain. The author thanks the referee for comments which has improved the exposition of the earlier draft. References
Barlow, R.E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, INC. Block, H.W. and Savits, T.H. (1980). Laplace transforms for classes of life distributions. Annals of probability, 8(3), 465-474. Bryson, M.C. and Siddiqui, M.M. (1969). Some criteria for ageing. Jour. Amer. Statist. Assoc., 64, 1472-1483. Deshpande,J.V., Kochar,S.C. and Singh, H. (1986). Aspects of positive ageing. Jour. Applied Probability, 23, 748-758. Fagiuoli, E. and Pellerey, F. (1993). New partial orderings and applications. Naval Research Logistics, 40, 829-842. Kebir, Y. (1994).Laplace transform characterization of probabilistic orderings. Probability in the Engineering and Informational Sciences, 8, 69-77. Klefsj¨ o, B. (1982). The HNBUE and HNWUE classes of life distributions. Naval Research Logistics Quarterly, 29, 331-344. Nanda, A. K. (1995). Stochastic orders in terms of Laplace transforms. Calcutta Statistical Association Bulletin, 45(179-180), 195-201. Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and their Applications. Academic Press, New York. Vinogradov, O. P. (1973). The definitions of distribution functions with increasing hazard rate in terms of Laplace transforms. Theory of Probability and its Applications, 18, 811-814.
Asok K. Nanda Department of Statistics Panjab University Chandigarh-160014, INDIA. e-mail:
[email protected]
