1. INTRODUCTION

The concept of concomitants, when the bivariate data are ordered by one of its components, was first introduced by [1]. Let (X1,Y1),(X2,Y2),…,(Xn,Yn) be a random sample of size n from a continuous bivariate population with cumulative distribution function (cdf) FX,Y(x,y). If we arrange the X-variate in ascending order as X1:n≤X2:n≤…≤Xn:n, then the Y-values paired with these order statistics are denoted by Y[1:n],Y[2:n],…,Y[n:n] and are termed as concomitants of order statistics. Recent studies on concomitants of order statistics are in the works of [2–7]. A fine review of works on concomitants of order statistics is available in [8]. Concomitants have found a variety of applications in many applied fields such as selection procedure, ocean engineering, inference problems, prediction analysis and double sampling plans. In particular, ranked set sampling (RSS) proposed by [9] has found a remarkable application of concomitants of order statistics. Inspired from RSS, Scaria and Thomas [10] extended the theory of concomitants and introduced the concept of second order concomitants of order statistics. A noteworthy advantage of employing second order concomitants in sampling is that this technique will reduce the number of measurements required, and therefore reduce the researcher's costs.

Concepts of stochastic dependence is widely discussed in literature and it permeates throughout our daily life. Lai and Xie [11] discussed different dependence concepts, dependence orderings and measures of dependence in detail. The concepts of dependence in the bivariate and multivariate cases are presented in [12,13]. Esary and Proschan [14] gave definitions to the terms right-tail increasing (RTI) and left-tail decreasing (LTD). The term likelihood ratio dependent (LRD) was first defined by [15]. The concept of TP2 (totally positive of order 2) was introduced by [16]. For several works of positive dependence we refer to [15,17]. All these concepts are widely used in reliability theory.

Generally, the component life times of a system may be dependent on one another. Hence we need probability models that prescribe the dependence structures among the life time variables. In multi-component systems, measures of dependence between component lives are a major aspect to be considered in selecting the appropriate model. The Morgenstern system of distributions is a popular, and well-known family of bivariate dependent variables, and its numerous generalizations are scattered in the literature. Dependence properties of this family are closely associated with the correlation coefficient although a priori the pivotal parameter of the family is not associated with this concept. It is already established that the Morgenstern family is suitable in reliability modelling [18]. The Morgenstern system of bivariate distributions discussed in [19] includes all cdfs of the form

Here the parameter λ measures the association between X and Y with spearman's rho λ/3 and Kendall's tau 2λ/9. Lai and Xie [11] discussed the dependence structure of the Morgenstern family. No previous work has been done regarding the dependence structure of the concomitants from the Morgenstern family. The present paper is organized as follows. In Section 2, we discuss the dependence concepts of the concomitants from the Morgenstern family. The distribution theory of lifetimes of two component systems using concomitants is discussed in Section 3. In Section 4, we discuss an application of concomitants in reliability modelling of designing a two component system from the Morgenstern type bivariate exponentiated exponential distribution (MTBEED).

2. CONCOMITANTS OF ORDER STATISTICS AND SOME OF THE DEPENDENCE MEASURES

The general distribution theory of concomitants from the Morgenstern family is discussed in [20]. The cdf, the probability density function (pdf) of the rth concomitant Y[r:n] and the joint distribution function of Y[r:n] and Y[s:n] are given in Equations (2.1–2.3), respectively as

and where δλr=λ(n−2r+1n+1),δλs=λ(n−2s+1n+1) and δλ2=λ2{n−2s+1n+1−2r(n−2s)(n+1)(n+2)}

Equation (2.3) reveals that the joint distribution function of concomitants is a special case of the bivariate Cambanis family, introduced by [21] specified by

with corresponding marginal distributions and joint density function as and

When (2.4) is absolutely continuous the parameters satisfy the conditions (1+λ1+λ2+λ3)>0, (1+λ1−λ2−λ3)>0, (1−λ1+λ2−λ3)>0 and (1−λ1−λ2+λ3)>0, where λ 's are real constants.

Comparing (2.3) with (2.4), we get λ1=δλr=λ(n−2r+1n+1), λ2=δλs=λ(n−2s+1n+1) and λ3=δλ2=λ2{n−2s+1n+1−2r(n−2s)(n+1)(n+2)}.

Concepts of stochastic dependence are widely applicable in statistics. Among the dependence concepts, correlation is still the most widely used concept in applications. Some of the positive dependence measures are discussed in [11]. In this work we deal with the association measures (Kendall's tau and Spearman's rho) of concomitants of order statistics from the bivariate Morgenstern family.

Holland and Wang [22,23], defined a local dependence function

assuming the partial derivative of the second order exists and f(x,y) is the pdf of the bivariate distribution. Holland and Wang [23] also showed that γ(x,y)≥0 (for all x and for all y) is equivalent to f(x,y) belonging to TP2, or X and Y are LRD.

The Morgenstern family belongs to TP2 if 0≤λ≤1. That is the family is either Positive Quadrant Dependent (PQD) or Negative Quadrant Dependent (NQD) [11].

From Equation (2.7),

showing that fX,Y(x,y) is TP2 (RR2) if λ3≥(≤)λ1λ2. So Cambanis family belongs to TP2 if λ3≥λ1λ2. And thus for concomitants of order statistics from the Morgenstern family,

Hence F[r,s:n](y1,y2) belongs to TP2 or Y[r:n] and Y[s:n] are LRD for −1≤λ≤1; 1≤r<s≤n.

It is directly established that TP2 implies other dependency concepts such as stochastic increase (SI), right corner set increasing (RCSI), right (left) tail increasing (decreasing), association, PQD, weakly positive quadrant dependence and positive correlation (see [13,24]). Thus concomitants of order statistics from Morgenstern family belongs to TP2⇒PQD and all other positive dependence concepts for −1≤λ≤1; 1≤r<s≤n.

There are a variety of ways to measure dependence. The most widely known scale-invariant measures of association are Kendall's tau and Spearman's rho. The dependence measurement for the Cambanis family is computed by Kendall's tau and is calculated by using Equations (2.4) and (2.7), as

By substituting λ1,λ2 and λ3 in Equation (2.9), the Kendall's tau for the rth and sth concomitants from the Morgenstern family is

Using Equations (2.4–2.6), the Spearman's rho for the Cambanis family is calculated as

By substituting λ1,λ2 and λ3 in Equation (2.10), the Spearman's rho for the rth and sth concomitants from the Morgenstern family is

3. DISTRIBUTION THEORY OF LIFETIMES OF TWO COMPONENT SYSTEM USING CONCOMITANTS OF ORDER STATISTICS

Let FY[r:n] and FY[s:n] denote the distribution function of the component lifetimes of the concomitants of order statistics Xr:n and Xs:n respectively. Let T1 = min (Y[r:n],Y[s:n]), and T2 = max (Y[r:n],Y[s:n]) denote the lifetimes of the series and parallel systems of two components, respectively. Let F(i)(t),R(i)(t) and f(i)(t) denote the distribution function, survival function and density functions of Ti, i = 1,2. The mean times to failure of T1 and T2 are denoted by μ(1) and μ(2), respectively.

From [11],

Using the Equations (2.1), (2.3) and (3.1), we find

where δλ=λ{n−r−s+1n+1}.

The corresponding density function can be obtained from Equation (3.2) as,

Using the formula for the density of order statistics in (3.3), we find

where fr:n(t) is the density of Yr:n.

From [11],

Using Equations (2.2), (3.3) and (3.5), we get

Using the formula for the density of order statistics in (3.6), we find

Moments of T1 and T2

From (3.4), the kth moment of T1 can be derived as

The kth moment of T2 follows from (3.7) as,

where μ(k)=E[Yk] and μr:n(k)=E[Yr:nk].

4. APPLICATION IN RELIABILITY

Gupta and Kundu [25] introduced the exponentiated exponential (EE) distribution as a generalization of the standard exponential distribution with corresponding cdf and pdf are respectively,

and

We denote the EE distribution with parameters θ and α as EE(θ,α). Gupta and Kundu [25] showed that the mean, variance and moment generating function of the random variable X with EE(θ,α) can be given as

and where ψ(.) is the digamma function, ψ′(.) is its derivative, and Beta(a,b)=Γ(a)Γ(b)Γ(a+b).

Let Y be the life time of a very expensive component of a two component system and X be an inexpensive variable (directly measurable or observable) which is correlated with Y. Suppose (X, Y) follows MTBEED. Then from (1.1), the cdf of (X, Y) is,

Let (Xr:n,Xs:n), (r<s) be the pair of rth and sth order statistics of the inexpensive variable X and (Y[r:n],Y[s:n]) be the associated Y measurements or concomitants. Then the cdf of (Y[r:n],Y[s:n]) follows from Equation (2.3) as,

It follows from (3.3) and (3.6) that the density function of the lifetime of a series, and parallel systems using the expensive components Y[r:n] and Y[s:n] are, respectively

and

It follows from (4.6) and (4.7) that

and where fU(t), fV(t), fW(t) and fZ(t) are pdf's of random variables U, V, W and Z with EE(θ2,α2), EE(θ2,2α2), EE(θ2,3α2) and EE(θ2,4α2) respectively.

The kth moment of T1 and T2 follows from (4.8) and (4.9), respectively as

and

The mean times to failure (MTTF) of the corresponding systems are obtained by setting k = 1 in (4.10) and (4.11), and using Equation (4.3). They are, respectively

and

The following relations are directly from (4.12) and (4.13).

Relation 4.1 μ(1)[r,s:n](−λ)=μ(1)[n−s+1,n−r+1:n](λ).

Relation 4.2 μ(2)[r,s:n](−λ)=μ(2)[n−s+1,n−r+1:n](λ).

The θ2 times MTTF of series and parallel systems of a two component systems using concomitants are tabulated in Tables 1 and 2 for specific values of n, r, s, α2 and λ.

The MTTF of a series and parallel system based on independent components are respectively,

and see [11]. The θ2 times MTTF of a series and parallel systems based on independent components for α2 = 1, 2, 3, 4 and 5 are tabulated in Table 3.

The following conclusions are evident from the above Tables.

Comparing Tables 1 and 3, we can say that the selection of components in the series system based on concomitants will substantially increase the MTTF. It follows from Table 1 that if λ<0, the MTTF is optimal for the pair r = 1, and s = 2. On the otherhand, if λ>0, the MTTF is optimal for the pair r = n − 1, and s = n. It is also observed that the MTTF of series system increases if λ, and n increases numerically. The percentage relative gains in MTTF of a series system using the first and second concomitants over independent cases are tabulated in Table 4 for λ = −0.2, −0.6 and −1; α2 = 1, 2, 3, 4 and 5; and n = 10, 20 and 30.

Similarly, when comparing Tables 2 and 3, we can say that the selection of components in the parallel system based on concomitants will substantially increase the MTTF. It follows from Table 2 that if λ=−1, α2 = 1, 2, 3, 4 and 5, the MTTF is optimal for the pair r = 1, and s = 2. On the otherhand, if λ=1, α2 = 1, 2, 3, 4 and 5, the MTTF is optimal for the pair r = n − 1, and s = n. The percentage relative gains in MTTF of a parallel system using the first and second concomitants over independent cases are tabulated in Table 5 for λ = −1; n = 10, 20 and 30 and α2 = 1, 2, 3, 4 and 5.

Using relations (4.1) and (4.2), we deduce that the percentage relative gain in the MTTF of series and parallel systems using the (n−1)th and nth concomitants over independent case are the same as that of the above table by replacing λ with −λ.

Thus we conclude that the design of two component series or parallel systems using concomitants substantially increases the MTTF of both the systems. Moreover, the selection of components based on concomitants is more effective in series systems than parallel systems. If λ=±1, α2≥5 and n≥30, the percentage relative increase in the MTTF of a series system is greater than ninety one. Similarly, if λ=±1, α2=1 and n≥30, the percentage relative increase in the MTTF of a parallel system is greater than thirty five.

ACKNOWLEDGMENTS

The authors are grateful to the editor and reviewers for their suggestions in improving this paper.