2. DEFINITIONS AND BACKGROUND

Let X be a continuous positive random variable representing a survival time with an absolutely continuous distribution function F(t), survival function F(t) = 1 − F(t), and cumulative hazard rate Λ(t) = −ln F(t) with F(0) = 1. Wherever necessary, it will be assumed that r_F(t) = Λ′(t) is the hazard rate corresponding to F(t). A key role in this article will be played by the mean residual life function (MRLF) μ_F(t) defined as

It will be assumed that μ_F(0) = E(X) < ∞. When discussing the variance of the residual lifetime X − t|X > t, it will be assumed that E(X²) < ∞. The variance residual life function (VRLF) is defined as

We refer to Gupta and Kirmani [17,18], Launer [28], and Gupta, Kirmani, and Launer [19] for details about the MRLF and the VRLF.

The above-defined functions highlight different aspects of survival and residual life distributions.The hazard rate and the mean residual life function are related by

Further, the hazard rate, the MRLF, and the VRLF are tied together by the relation

see Gupta [16].

It is well known that r_F(t) determines the distribution function uniquely and, hence, μ_F(t) also characterizes the distribution. Additionally, F(t) and μ_F(t) are connected by

Thus, r_F(t), μ_F(t), and F(t) are equivalent in the sense that given one of them, the other two can be determined. Hence, in the analysis of survival data, one sometimes estimates r_F(t) or μ_F(t) instead of F(t), according to the convenience of the procedure available.

In addition to the above functions, the residual coefficient of variation is given by γ_F(t) = σ_F(t)/μ_F(t). Further, the MRLF, the VRLF, and the residual coefficient of variation are connected by the relation

see Gupta [16].

We now describe briefly some aging classes of life distributions and their relationships

2.1. Some Criteria of Aging for the RLF

In this section we review some of the aging criteria and their relationships. We also describe how the aging properties of the original distribution are transformed into the aging properties of the residual life.

Let X be a continuous positive random variable representing the life of a component. Let F be the cumulative distribution function of X and F(x) = 1 − F(x) be the reliability function or the survival function of X. Then

is the survival function of a unit of age t. Evidently, any study of the phenomenon of aging should be based on

and functions related to this. Thus, the following hold:

1. F is said to be PF₂ if ln f (x) is concave, where f (·) is the density corresponding to F(·).
2. F is said to have increasing (decreasing) failure rate [IFR (DFR)] if
   
   is decreasing (increasing) in t. If F is absolutely continuous with density f, then F is in the IFR (DFR) class if r_F(t) = f (t)/F(t) is increasing (decreasing) in t.
3. F is said to have increasing (decreasing) failure rate average [IFRA (DFRA)] if
   
   is increasing (decreasing).
4. F is said to have new better (worse) than used [NBU (NWU)] if F_t(x) ≤ (≥) F(x) for x ≥ 0 and t ≥ 0.
5. F is said to have decreasing (increasing) mean residual life [DMRL (IMRL)] if the mean residual life
   
   is decreasing (increasing) assuming that the mean μ_F(0) exists.
6. F is said to have new better (worse) than used in expectation [NBUE (NWUE)] if μ_F(t) ≤ (≥) μ_F(0) for all t ≥ 0.
7. F is said to have decreasing variance residual life (increasing variance residual life) [DVRL (IVRL)] if σ_F²(t) is decreasing (increasing).

The chain of implications among these classes of distributions is

The reverse implications are not true; for counterexamples, see Bryson and Siddiqui [11]. Some extensions of these classes of distributions are contained in Klefsjo [26,27], Shaked [32], Singh and Deshpande [35], Deshpande, Kochar, and Singh [12], Basu and Ebrahimi [5,6], Abouammoh [1], Abouammoh and Ahmad [2], and Loh [30].

2.2. Stochastic Order Relations

Let X and Y be nonnegative absolutely continuous random variables with density functions f (x) and g(x) and survival functions F(x) and G(x), respectively. Then, we have the following:

1. X is said to be smaller than Y in the likelihood ratio ordering, written as X ≥^LR Y, if f (x)/g(x) is nonincreasing in x.
2. X is said to be smaller than Y in the failure (hazard) rate ordering, written as X ≤^FR Y, if r_F(x) ≥ r_G(x) for all x.
3. X is said to be smaller than Y in the stochastic ordering, written as X ≤^st Y, if F(x) ≤ G(x) for all x.
4. X is said to be smaller than Y in the mean residual life ordering, written as X ≤^MRL Y, if μ_F(x) ≤ μ_G(x) for all x. Deshpande, Singh, Bagai, and Jain [13] show that X ≤^MRL Y if and only if
   
   is decreasing in x.
5. X is said to be smaller than Y in the increasing convex order, written as X ≤^icx Y, if
   
   for all x. Note that in the literature the increasing convex order has also been called stop loss ordering (Hesselager [22]) and ST2 ordering; see Belzunce, Candel, and Ruiz [7] and Shaked and Shanthikumar [33]. For some generalized variability ordering, see Zarek [39], Li and Zhu [29], and Bhattacharjee and Sethuraman [8].
6. X is said to be smaller than Y in the variance residual life ordering, written as X ≤^VRL Y, if σ_F²(x) ≥ σ_G²(x) for all x.
7. X is said to be smaller than Y in the Laplace transform ordering, written as X ≤^LT Y, if E(e^−sX) ≥ E(e^−sY). Shaked and Shanthikumar [33] showed that this is equivalent to
   
   .
8. X is said to be smaller than Y in the moment generating function ordering, written as X ≤^MGF Y, if E(e^sX) ≤ E(e^sY).

It is well known that

see Shaked and Shanthikumar [33].

3. HIGHER-ORDER EQUILIBRIUM DISTRIBUTIONS AND STOP LOSS MOMENTS

We define the sequence of induced distributions as follows. Let F(x) be the survival function of a nonnegative random variable X with MRLF given by μ_F(t). We now define a sequence {F₁,F₂,…} of survival functions induced by F as

where μ_n is the mean of the distribution F_n. Assume that E(X_n) < ∞ for the largest n used above so that μ₁,μ₂,…, μ_n will all be finite. In what follows, we will denote by X^[0] = X, corresponding to the survival function F₀ = F, the original random variable and X^[1],X^[2],… corresponding to F₁,F₂,…....

We now present the following result.

Theorem 3.1: Let φ be a convex function. Then

Proof:

where f₁(x) = F(x)/μ is the p.d.f. of the first induced distribution and

is its survival function. Thus,

Particular Cases:

and, in general,

where μ₀(t) = μ(t) = MRLF of F and μ_j(t) is the MRLF of the jth induced distribution; see also Stein and Dattero [36]. From this, it is clear that

where μ_j is the mean of the jth induced distribution.

3.1. Stop Loss Transform

Definition 3.1: The function π_X(t) = E [(X − t)₊] is called the stop loss transform of X, where (X − t)₊ = Max(X − t,0) represents the amount by which X exceeds the threshold t.

The kth stop loss moment is given by

see Willmot, Drekic, and Cai [38] and Hesselager, Wang, and Willmot [23]. Note that R_k(t) has also been called the kth partial moment in the literature; see Gupta and Gupta [15]. We now present the following result.

Theorem 3.2: The survival function of the kth equilibrium distribution is given by

Proof:

Proceeding in this way, we get

We now show that R_k(t) determines the distribution function uniquely.

Theorem 3.3: Under the above-stated conditions,

where R_k^(k)(t) denotes the kth derivative of R_k(t).

Proof: Applying Taylor's theorem to F(x), we get

This gives the desired result. █

Remark 3.1: For other proofs, see Gupta and Gupta [15] and Navarro, Franco, and Ruiz [31].

4. AGING PROPERTIES OF EQUILIBRIUM DISTRIBUTIONS AND SOME ORDER RELATIONS

It is well known that the failure rate of F₁ is the reciprocal of the MRLF of F and, in general, the failure rate of F_i (i = 1,2,…) is the reciprocal of the MRLF of F_i−1 (i = 1,2,…). This means that F_i is IFR is equivalent to F_i−1 is DMRL. In the following, we show that F_i−1 is DMRL is equivalent to F_i−2 is DVRL.

Theorem 4.1: F_i−1 is DMRL is equivalent to F_i−2 is DVRL (i = 2,3,…).

Proof: It is enough to prove the result for i = 2. We have

Also,

This gives

This proves the result; see also Gupta et al. [19], Singh [34], Fagiuoli and Pellerey [14], and Willmot [37]. █

Corollary 4.1: Suppose F is a strictly increasing life distribution. Then F is DVRL (IVRL) if and only if the induced distribution corresponding to F is DMRL.

Remark 4.1: The restriction to strictly increasing F in Theorem 4.1 and Corollary 4.1 cannot be relaxed. This can be seen from Example 1 of Gupta et al. [19], where F is IVRL but the function E((X − t)²|X > t)/E((X − t)|X > t) is not increasing in the neighborhood of t = ½.

Remark 4.2: It is also clear from the above derivation that

From the above, we can state the following.

Theorem 4.2: Suppose F is a strictly increasing life distribution. Then F is DVRL (IVRL) if and only if μ_F1(t) ≤ (≥) μ_F(t).

Remark 4.3: The above result was also noticed by Deshpande et al. [13] and Abouammoh, Kanjo, and Khalique [3].

It is also clear from the above that μ_F1(t) = μ_F(t) if and only if F has an exponential distribution.

The following theorem addresses the stochastic comparisons of X^[1] and X.

Theorem 4.3: Suppose X is a nonnegative random variable with E(X) < ∞. Then X is NBUE (NWUE) if X^[1] ≤^st (≥^st) X.

Proof: The survival function of X^[1] is given by

This gives

Thus,

Similarly, F_X^[1](t) ≥ F_X(t) if and only if X is NWUE. █

We now present the following result dealing with the hazard rate comparison of X^[1] and X.

Theorem 4.4: Suppose X is a nonnegative random variable with E(X) < ∞. Then X is IMRL (DMRL) if and only if X ≤^FR (≥^FR) X^[1].

Proof:

The following result deals with the Laplace ordering of two random variables and their corresponding equilibrium distributions.

Theorem 4.5: Let X and Y be two nonnegative random variables with E(X) = E(Y). Then

Proof: We have

Now

The following two results deal with the stop loss ordering of two random variables and their corresponding equilibrium variables.

Theorem 4.6: Let X and Y be two nonnegative random variables with E(X) = E(Y). Then

Proof:

Theorem 4.7: Let X and Y be two nonnegative random variables with finite means. Then

Proof:

We now define the convex ordering as follows.

Definition 4.1: X ≤^icx Y if E(f (X)) ≤ E(f (Y)) for all increasing convex functions. This is equivalent to

Following the above concept, we define a similar ordering between the sth equilibrium as follows.

Definition 4.2: X ≤^s−icx Y if F_X^(s)(t) ≤ F_Y^(s)(t) for all t ≥ 0, where F_X^(s)(t) and F_Y^(s)(t) are the survival functions of their sth equilibrium distributions; see Klar and Muller [25].

Comparing the original variable and its equilibrium variable in the above ordering, we have

This is equivalent to

Denoting

the above inequality can be written as

see Stein and Dattero [36].

5. EQUILIBRIUM DISTRIBUTION OF A SERIES SYSTEM

Let X₁,X₂,…,X_n be independent random variables with survival functions F_X1,F_X2,…,F_Xn, respectively. When the components are arranged in a series system, we observe X₁ ∧ X₂ ∧ ··· ∧ X_n = Min(X₁,X₂,…,X_n). In this section, we will compare two systems whose components are the nth equilibrium distributions of X and Y and the nth equilibrium distribution of a system having components X and Y. More precisely, we have the following result due to Bon and Illayk [9].

Theorem 5.1: Let X and Y be independent nonnegative random variables. Let X^[k] and Y^[k] be respectively the independent equilibrium variables of X and Y of order k. If X and Y have the DMRL property and if the nth moments of X and Y exist, then

where (X ∧ y)^[n] is the equilibrium variable of X ∧ Y of order n.

Proof: For n = 1,X^[1] and Y^[1] are independent and the p.d.f. of X^[1] ∧ Y^[1] is given by

Also, the p.d.f. of (X ∧ Y)^[1] is

These give

Since X and Y have the DMRL property, the right-hand side of the above equation is decreasing and, hence, X^[1] ∧ Y^[1] ≤^LR (X ∧ Y)^[1].

For the general case, we proceed by induction and assume that the result is true of k = n. It can be verified that

where

and likewise for B(t). We now show that A(t) and B(t) are decreasing functions of t.

Since Y has the DMRL property, all of its equilibrium variables have the IFR property and also the DMRL property. This means that Y^[n] ≤^HR Y^[n−1]. Thus,

Also, the induction hypothesis implies that

Therefore, X^[n−1] ∧ Y^[n] ≤^HR (X ∧ Y)^[n−1]. This implies that A(t) is decreasing. Similary, B(t) is decreasing and the proof is complete. █

Remark 5.1: For the lower bound on the nth moment of a series system, see the above remark.

6. BIVARIATE EQUILIBRIUM DISTRIBUTION

In order to define bivariate equilibrium distribution, we first review the bivariate hazard rates and bivariate MRLF as follows. Let (X₁,X₂) be a nonnegative bivariate random variable with survival function F(x₁,x₂).The hazard components of (X₁,X₂) are given by h₁(x₁,x₂) and h₂(x₁,x₂), where

and the MRLF of (X₁,X₂) is given by (μ₁(x₁,x₂),μ₂(x₁,x₂)), where

and likewise for μ₂(x₁,x₂). The hazard components and the MRLF components are related by

Keeping these properties in mind, Gupta and Sankaran [21] defined the bivariate equilibrium random variable (Y₁,Y₂) by means of the conditional distribution as follows.

The p.d.f. of Y₁ given Y₂ > y₂ is

Likewise, the p.d.f. of Y₂ given Y₁ > y₁ is

It can be easily seen that the marginal distributions of Y₁ and Y₂ coincide with the equilibrium distributions of X₁ and X₂, respectively. Following the above scheme, one can easily define equilibrium distributions in higher dimensions.

We now present two examples.

Example 6.1: Gumbel's bivariate exponential distribution with survival function

It is well known that for this distribution, the conditional distribution of X₁ given X₂ > x₂ and the conditional distribution of X₂ given X₁ > x₁ are exponential. Also, the marginal distributions of X₁ and X₂ are exponentials. However, the conditional distribution of X₁ given X₂ = x₂ and that of X₂ given X₁ = x₁ are not exponential. For other properties of this model, see Arnold [4].

It can be easily verified that in this case

This shows that the distribution of Y₁ given Y₂ > y₂ is exponential with parameter a + cy₂. Similarly, the distribution of Y₂ given Y₁ > y₁ is exponential with parameter b + cy₁.

We will now obtain the equilibrium distributions in the case of series and parallel systems.

Example 6.2: Generalized bivariate Pareto distribution with survival function

The above survival function was obtained as a mixture of independent exponential random variables; see Hutchinson and Lee [24, p.118].

It can be verified that, in this case,

where

We will now obtain the equilibrium distributions in the case of series and parralel systems.

7. SOME CONCLUSIONS AND COMMENTS

In this article we have presented the equilibrium distribution including its higher derivates from a reliability point of view. Since the equilibrium distribution is intimately connected with the original distribution, it provides a useful tool in studying several important properties of the original distribution. Aging properties of equilibrium distributions and their stochastic orderings with the original distributions are explored. The relation between the equilibrium distribution of a series system and a series system of equilibrium distributions, consisting of two components, is investigated. Extension to bivariate equilibrium distributions is provided along with some examples. It is hoped that this article will be useful to theoreticians and practitioners in studying the applications of equilibrium distribution in reliability.