Journal of Statistical Theory and Applications

Volume 20, Issue 2, June 2021, Pages 193 - 203

Dependence Concepts and Reliability Application of Concomitants of Order Statistics from the Morgenstern Family

Authors
Johny Scaria*, Sithara Mohan
Department of Statistics, Nirmala College Muvattupuzha, 686661, India
*Corresponding author. Email: johny.kakkattil@gmail.com
Corresponding Author
Johny Scaria
Received 6 June 2019, Accepted 1 July 2020, Available Online 30 March 2021.
DOI
https://doi.org/10.2991/jsta.d.210325.001How to use a DOI?
Keywords
Concomitants of order statistics; Dependence measures; Morgenstern type bivariate exponentiated exponential distribution; Reliability; Series and parallel systems
Abstract

The distribution theory and applications of concomitants from the Morgenstern family of bivariate distributions are discussed in Scaria and Nair, Biom. J. 41 (1999), 483–489. In the present study, some dependence concepts of concomitants of order statistics from the Morgenstern family are discussed. An application in reliability theory of designing a two component system using concomitants is also discussed.

Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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:nX2:nXn: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 [27]. 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

FX,Y(x,y)=FX(x)FY(y)[1+λ{1FX(x)}{1FY(y)}];1λ1.(1.1)

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 (2.12.3), respectively as

FY[r:n](y)=FY(y)[1+δλr{1FY(y)}];1λ1,(2.1)
fY[r:n](y)=fY(y)[1+δλr{12FY(y)}];1λ1,(2.2)
and
F[r,S.n](y1,y2)=FY(y1)FY(y2){1+δλr[1FY(y1)]+δλs[1FY(y2)]+δλ2[1FY(y1)][1FY(y2)]}.(2.3)
where δλr=λ(n2r+1n+1),δλs=λ(n2s+1n+1) and δλ2=λ2{n2s+1n+12r(n2s)(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

FX,Y(x,y)=FX(x)FY(y)[1+λ1{1FX(x)}+λ2{1FY(y)}+λ3{1FX(x)}{1FY(y)}],(2.4)
with corresponding marginal distributions and joint density function as
HX(x)=FX(x)[1+λ1{1FX(x)}],(2.5)
GY(y)=FY(y)[1+λ2{1FY(y)}],(2.6)
and
fX,Y(x,y)=fX(x)fY(y)[1+λ1{12FX(x)}+λ2{12FY(y)}+λ3{12FX(x)}{12FY(y)}],(2.7)

When (2.4) is absolutely continuous the parameters satisfy the conditions (1+λ1+λ2+λ3)>0, (1+λ1λ2λ3)>0, (1λ1+λ2λ3)>0 and (1λ1λ2+λ3)>0, where λ 's are real constants.

Comparing (2.3) with (2.4), we get λ1=δλr=λ(n2r+1n+1), λ2=δλs=λ(n2s+1n+1) and λ3=δλ2=λ2{n2s+1n+12r(n2s)(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

γ(x,y)=2logf(x,y)xy,
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 0λ1. That is the family is either Positive Quadrant Dependent (PQD) or Negative Quadrant Dependent (NQD) [11].

From Equation (2.7),

2logfX,Y(x,y)xy=(λ3λ1λ2)fX(x)fY(y)[1λ1FX(x)λ2FY(y)+λ3FX(x)FY(y)]2,(2.8)
showing that fX,Y(x,y) is TP2(RR2) if λ3()λ1λ2. So Cambanis family belongs to TP2 if λ3λ1λ2. And thus for concomitants of order statistics from the Morgenstern family,
λ3λ1λ2=λ2(4r(ns+1)(n+1)2(n+2))0.

Hence F[r,s:n](y1,y2) belongs to TP2 or Y[r:n] and Y[s:n] are LRD for 1λ1; 1r<sn.

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 TP2PQD and all other positive dependence concepts for 1λ1; 1r<sn.

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

τ=4FX,Y(x,y)fX,Y(x,y)dxdy1=29(λ3λ1λ2)(2.9)

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

τ=89λ2r(ns+1)(n+1)2(n+2)0;1r<sn.

Using Equations (2.42.6), the Spearman's rho for the Cambanis family is calculated as

ρ=12[FX,Y(x,y)HX(x)GY(y)]dHX(x)dGY(y)=(λ3λ1λ2)3(2.10)

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

ρ=43λ2r(ns+1)(n+1)2(n+2)0;1r<sn.

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],

R(1)(t)=F¯(t,t)=Pr[Y[r:n]>t,Y[s:n]>t]=1FY[r,n](t)FYs,n](t)+F[r,s:n](t,t).(3.1)

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

R(1)(t)=12F(t)+[F(t)]22δλF(t)[1F(t)]2+δλ2[F(t)]2[1F(t)]2,(3.2)
where δλ=λ{nrs+1n+1}.

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

f(1)(t)=t(R(1)(t))=2f(t)2F(t)f(t)+2δλf(t){14F(t)+3[F(t)]2}2δλ2F(t)f(t){13F(t)+2[F(t)]2}.(3.3)

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

f(1)(t)=2f(t)f2:2(t)+2δλ[f(t)2f2:2(t)+f3:3(t)]δλ2[f2:2(t)2f3:3(t)+f4:4(t)],(3.4)
where fr:n(t) is the density of Yr:n.

From [11],

f(1)(t)+f(2)(t)=fY[r:n](t)+fY[s:n](t).(3.5)

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

f(2)(t)=2F(t)f(t)+2δλF(t)f(t)[23F(t)]+2δλ2F(t)f(t)[13F(t)+2[F(t)]2].(3.6)

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

f(2)(t)=f2:2(t)+2δλ[f2:2(t)f3:3(t)]+δλ2[f2:2(t)2f3:3(t)+f4:4(t)].(3.7)

Moments of T1 and T2

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

μ(1)(k)(t)=tkf(1)(t)dt=2μ(k)μ2:2(k)+2δλ[μ(k)2μ2:2(k)+μ3:3(k)]δλ2[μ2:2(k)2μ3:3(k)+μ4:4(k)].(3.8)

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

μ(2)(k)(t)=tkf(2)(t)dt=μ2:2(k)+2δλ[μ2:2(k)μ3:3(k)]+δλ2[μ2:2(k)2μ3:3(k)+μ4:4(k)],(3.9)
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,

FX(x)=(1eθx)α;x>0,θ>0,α>0,(4.1)
and
fX(x)=αθeθx(1eθx)α1;x>0,θ>0,α>0.(4.2)

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

E(X)=1θ[ψ(α+1)ψ(1)],(4.3)
Var(X)=1θ2[ψ(1)ψ(α+1)],
and
MX(t)=αBeta(α,1t/θ),
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,

FX,Y(x,y)=(1eθ1x)α1(1eθ2y)α2{1+λ[1(1eθ1x)α1][1(1eθ2y)α2]},x,y>0,θ1,θ2,α1,α2>0,1λ1,(4.4)

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,

F[r,s:n](y1,y2)=(1eθ2y1)α2(1eθ2y2)α2{1+δλr[1(1eθ2y1)α2]+δλs[1(1eθ2y2)α2]+δλ2[1(1eθ2y1)α2][1(1eθ2y2)α2]}.(4.5)

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

f(1)(t)=2(1+δλ)α2θ2eθ2t(1eθ2t)α212α2θ2eθ2t(1eθ2t)2α21(1+4δλ+δλ2)+6α2θ2eθ2t(1eθ2t)3α21(δλ+δλ2)4δλ2α2θ2eθ2t(1eθ2t)4α21,(4.6)
and
f(2)(t)=2(1+2δλ+δλ2)α2θ2eθ2t(1eθ2t)2α216(δλ+δλ2)α2θ2eθ2t(1eθ2t)3α21+4δλ2α2θ2eθ2t(1eθ2t)4α21.(4.7)

It follows from (4.6) and (4.7) that

f(1)(t)=2(1+δλ)fU(t)fV(t)(1+4δλ+δλ2)+2(δλ+δλ2)fW(t)δλ2fZ(t),(4.8)
and
f(2)(t)=(1+2δλ+δλ2)fV(t)2(δλ+δλ2)fW(t)+δλ2fZ(t),(4.9)
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

μ(1)(k)(t)=2(1+δλ)E[Uk](1+4δλ+δλ2)E[Vk]+2(δλ+δλ2)E[Wk]δλ2E[Zk],(4.10)
and
μ(2)(k)(t)=(1+2δλ+δλ2)E[Vk]2(δλ+δλ2)E[Wk]+δλ2E[Zk].(4.11)

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

μ(1)(t)=1θ2{2ψ(α2+1)ψ(2α2+1)ψ(1)+2δλ[ψ(α2+1)2ψ(2α2+1)+ψ(3α2+1)]δλ2[ψ(2α2+1)2ψ(3α2+1)+ψ(4α2+1)]},(4.12)
and
μ(2)(t)=1θ2{ψ(2α2+1)ψ(1)+2δλ[ψ(2α2+1)ψ(3α2+1)]+δλ2[ψ(2α2+1)2ψ(3α2+1)+ψ(4α2+1)]}.(4.13)

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

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

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

n α2 λ θ2μ(1)(t)
(r, s) = (1, 2) (r, s) = (1, 3) (r, s) = (2, 3) (r, s) = (n − 2, n − 1) (r, s) = (n − 2, n) (r, s) = (n − 1, n)
10 1 −1 0.7879 0.7449 0.7096 0.3460 0.3207 0.3031
−.6 0.6618 0.6391 0.6191 0.4009 0.3846 0.3709
−.2 0.5503 0.5437 0.5375 0.4647 0.4589 0.4533
.2 0.4533 0.4589 0.4647 0.5375 0.5437 0.5503
.6 0.3709 0.3846 0.4009 0.6191 0.6391 0.6618
1 0.3031 0.3207 0.3460 0.7096 0.7449 0.7879
2 −1 1.2857 1.2314 1.1860 0.7132 0.6798 0.6554
−.6 1.1252 1.0961 1.0703 0.7867 0.7652 0.7470
−.2 0.9819 0.9734 0.9653 0.8707 0.8631 0.8558
.2 0.8558 0.8631 0.8707 0.9653 0.9734 0.9819
.6 0.7470 0.7652 0.7867 1.0703 1.0961 1.1252
1 0.6554 0.6798 0.7132 1.1860 1.2314 1.2857
3 −1 1.6195 1.5605 1.5109 0.9923 0.9554 0.9281
−.6 1.4447 1.4130 1.3848 1.0737 1.0500 1.0298
−.2 1.2881 1.2788 1.2699 1.1662 1.1578 1.1498
.2 1.1498 1.1578 1.1662 1.2699 1.2788 1.2881
.6 1.0298 1.0500 1.0737 1.3848 1.4130 1.4447
1 0.9281 0.9554 0.9923 1.5109 1.5605 1.6195
4 −1 1.8701 1.8084 1.7566 1.2129 1.1742 1.1452
−.6 1.6874 1.6544 1.6248 1.2987 1.2738 1.2525
−.2 1.5237 1.5139 1.5046 1.3959 1.3871 1.3787
.2 1.3787 1.3871 1.3959 1.5046 1.5139 1.5237
.6 1.2525 1.2738 1.2987 1.6248 1.6544 1.6874
1 1.1452 1.1742 1.2129 1.7566 1.8084 1.8701
5 −1 2.0705 2.0072 1.9540 1.3946 1.3546 1.3247
−.6 1.8830 1.8490 1.8187 1.4831 1.4575 1.4355
−.2 1.7147 1.7047 1.6951 1.5832 1.5742 1.5655
.2 1.5655 1.5742 1.5832 1.6951 1.7047 1.7147
.6 1.4355 1.4575 1.4831 1.8187 1.8490 1.8830
1 1.3247 1.3546 1.3946 1.9540 2.0072 2.0705
20 1 −1 0.8474 0.8243 0.8034 0.2955 0.2847 0.2760
−.6 0.6936 0.6815 0.6702 0.3654 0.3577 0.3508
−.2 0.5596 0.5562 0.5528 0.4512 0.4482 0.4453
.2 0.4453 0.4482 0.4512 0.5528 0.5562 0.5596
.6 0.3508 0.3577 0.3654 0.6702 0.6815 0.6936
1 0.2760 0.2847 0.2955 0.8034 0.8243 0.8474
2 −1 1.3612 1.3321 1.3054 0.6451 0.6305 0.6184
−.6 1.1659 1.1504 1.1359 0.7397 0.7294 0.7201
−.2 0.9939 0.9894 0.9850 0.8530 0.8491 0.8453
.2 0.8453 0.8491 0.8530 0.9850 0.9894 0.9939
.6 0.7201 0.7294 0.7397 1.1359 1.1504 1.1659
1 0.6184 0.6305 0.6451 1.3054 1.3321 1.3612
3 −1 1.7017 1.6699 1.6410 0.9166 0.9003 0.8867
−.6 1.4891 1.4722 1.4564 1.0217 1.0104 1.0000
−.2 1.3013 1.2964 1.2916 1.1467 1.1424 1.1383
.2 1.1383 1.1424 1.1467 1.2916 1.2964 1.3013
.6 1.0000 1.0104 1.0217 1.4564 1.4722 1.4891
1 0.8867 0.9003 0.9166 1.6410 1.6699 1.7017
4 −1 1.9558 1.9227 1.8925 1.1331 1.1159 1.1015
−.6 1.7338 1.7162 1.6997 1.2440 1.2321 1.2213
−.2 1.5374 1.5323 1.5273 1.3754 1.3709 1.3666
.2 1.3666 1.3709 1.3754 1.5273 1.5323 1.5374
.6 1.2213 1.2321 1.2440 1.6997 1.7162 1.7338
1 1.1015 1.1159 1.1331 1.8925 1.9227 1.9558
5 −1 2.1584 2.1245 2.0935 1.3121 1.2944 1.2794
−.6 1.9307 1.9126 1.8955 1.4267 1.4145 1.4033
−.2 1.7289 1.7236 1.7184 1.5622 1.5576 1.5531
.2 1.5531 1.5576 1.5622 1.7184 1.7236 1.7289
.6 1.4033 1.4145 1.4267 1.8955 1.9126 1.9307
1 1.2794 1.2944 1.3121 2.0935 2.1245 2.1584
30 1 −1 0.8693 0.8535 0.8387 0.2796 0.2729 0.2672
−.6 0.7052 0.6969 0.6890 0.3536 0.3486 0.3439
−.2 0.5629 0.5606 0.5583 0.4465 0.4445 0.4425
.2 0.4425 0.4445 0.4465 0.5583 0.5606 0.5629
.6 0.3439 0.3486 0.3536 0.6890 0.6969 0.7052
1 0.2672 0.2729 0.2796 0.8387 0.8535 0.8693
2 −1 1.3889 1.3690 1.3502 0.6234 0.6142 0.6062
−.6 1.1806 1.1701 1.1599 0.7238 0.7172 0.7109
−.2 0.9982 0.9951 0.9922 0.8468 0.8442 0.8416
.2 0.8416 0.8442 0.8468 0.9922 0.9951 0.9982
.6 0.7109 0.7172 0.7238 1.1599 1.1701 1.1806
1 0.6062 0.6142 0.6234 1.3502 1.3690 1.3889
3 −1 1.7318 1.7101 1.6897 0.8923 0.8820 0.8730
−.6 1.5052 1.4937 1.4827 1.0042 0.9968 0.9899
−.2 1.3060 1.3027 1.2994 1.1399 1.1370 1.1342
.2 1.1342 1.1370 1.1399 1.2994 1.3027 1.3060
.6 0.9899 0.9968 1.0042 1.4827 1.4937 1.5052
1 0.8730 0.8820 0.8923 1.6897 1.7101 1.7318
4 −1 1.9872 1.9650 1.9433 1.1074 1.0965 1.0870
−.6 1.7507 1.7387 1.7271 1.2256 1.2178 1.2105
−.2 1.5424 1.5389 1.5355 1.3683 1.3653 1.3623
.2 1.3623 1.3653 1.3683 1.5355 1.5389 1.5424
.6 1.2105 1.2178 1.2256 1.7271 1.7387 1.7507
1 1.0870 1.0965 1.1074 1.9433 1.9650 1.9872
5 −1 2.1907 2.1675 2.1456 1.2855 1.2743 1.2644
−.6 1.9479 1.9356 1.9238 1.4077 1.3997 1.3922
−.2 1.7340 1.7303 1.7268 1.5548 1.5517 1.5487
.2 1.5487 1.5517 1.5548 1.7268 1.7303 1.7340
.6 1.3922 1.3997 1.4077 1.9238 1.9356 1.9479
1 1.2644 1.2743 1.2855 2.1456 2.1675 2.1907

MTTF, mean times to failure.

Table 1

θ2 times MTTF of two component series system using concomitants.

n α2 λ θ2μ(2)(t)
(r,s)
(1, 2) (1, 3) (2, 3) (n2,n1) (n2,n) (n1,n)
10 1 −1 1.9394 1.8914 1.8359 1.1086 1.0429 0.9697
−.6 1.7746 1.7427 1.7082 1.2718 1.2336 1.1927
−.2 1.5952 1.5835 1.5716 1.4262 1.4138 1.4012
.2 1.4012 1.4138 1.4262 1.5716 1.5835 1.5952
.6 1.1927 1.2336 1.2718 1.7082 1.7427 1.7746
1 0.9697 1.0429 1.1086 1.8359 1.8914 1.9394
2 −1 2.5628 2.5111 2.4504 1.6504 1.5777 1.4961
−.6 2.3840 2.3493 2.3115 1.8315 1.7893 1.7439
−.2 2.1878 2.1751 2.1620 2.0020 1.9884 1.9745
.2 1.9745 1.9884 2.0020 2.1620 2.1751 2.1878
.6 1.7439 1.7893 1.8315 2.3115 2.3493 2.3840
1 1.4961 1.5777 1.6504 2.4504 2.5111 2.5628
3 −1 2.9441 2.8911 2.8285 2.0017 1.9264 1.8417
−.6 2.7602 2.7245 2.6855 2.1894 2.1458 2.0987
−.2 2.5580 2.5448 2.5313 2.3659 2.3519 2.3375
.2 2.3375 2.3519 2.3659 2.5313 2.5448 2.5580
.6 2.0987 2.1458 2.1894 2.6855 2.7245 2.7602
1 1.8417 1.9264 2.0017 2.8285 2.8911 2.9441
4 −1 3.2196 3.1658 3.1023 2.2615 2.1849 2.0985
−.6 3.0330 2.9968 2.9572 2.4527 2.4083 2.3604
−.2 2.8276 2.8143 2.8005 2.6323 2.6181 2.6034
.2 2.6034 2.6181 2.6323 2.8005 2.8143 2.8276
.6 2.3604 2.4083 2.4527 2.9572 2.9968 3.0330
1 2.0985 2.1849 2.2615 3.1023 3.1658 3.2196
5 −1 3.4353 3.3812 3.3170 2.4677 2.3903 2.3029
−.6 3.2471 3.2107 3.1706 2.6610 2.6162 2.5677
−.2 3.0398 3.0263 3.0124 2.8426 2.8282 2.8133
.2 2.8133 2.8282 2.8426 3.0124 3.0263 3.0398
.6 2.5677 2.6162 2.6610 3.1706 3.2107 3.2471
1 2.3029 2.3903 2.4677 3.3170 3.3812 3.4353
20 1 −1 2.0097 1.9852 1.9585 0.9426 0.9058 0.8669
−.6 1.8207 1.8042 1.7870 1.1775 1.1566 1.1349
−.2 1.6118 1.6058 1.5996 1.3964 1.3899 1.3833
.2 1.3833 1.3899 1.3964 1.5996 1.6058 1.6118
.6 1.1349 1.1566 1.1775 1.7870 1.8042 1.8207
1 0.8669 0.9058 0.9426 1.9585 1.9852 2.0097
2 −1 2.6388 2.6124 2.5835 1.4660 1.4251 1.3816
−.6 2.4341 2.4163 2.3975 1.7270 1.7039 1.6799
−.2 2.2061 2.1995 2.1927 1.9692 1.9620 1.9547
.2 1.9547 1.9620 1.9692 2.1927 2.1995 2.2061
.6 1.6799 1.7039 1.7270 2.3975 2.4163 2.4341
1 1.3816 1.4251 1.4660 2.5835 2.6124 2.6388
3 −1 3.0221 2.9951 2.9654 1.8104 1.7680 1.7228
−.6 2.8119 2.7935 2.7741 2.0812 2.0572 2.0323
−.2 2.5768 2.5699 2.5630 2.3320 2.3246 2.3170
.2 2.3170 2.3246 2.3320 2.5630 2.5699 2.5768
.6 2.0323 2.0572 2.0812 2.7741 2.7935 2.8119
1 1.7228 1.7680 1.8104 2.9654 2.9951 3.0221
4 −1 3.2986 3.2713 3.2411 2.0667 2.0235 1.9774
−.6 3.0855 3.0668 3.0472 2.3425 2.3181 2.2928
−.2 2.8468 2.8398 2.8327 2.5979 2.5903 2.5825
.2 2.5825 2.5903 2.5979 2.8327 2.8398 2.8468
.6 2.2928 2.3181 2.3425 3.0472 3.0668 3.0855
1 1.9774 2.0235 2.0667 3.2411 3.2713 3.2986
5 −1 3.5150 3.4875 3.4570 2.2707 2.2270 2.1804
−.6 3.3001 3.2813 3.2614 2.5496 2.5250 2.4993
−.2 3.0592 3.0521 3.0450 2.8077 2.8001 2.7923
.2 2.7923 2.8001 2.8077 3.0450 3.0521 3.0592
.6 2.4993 2.5250 2.5496 3.2614 3.2813 3.3001
1 2.1804 2.2270 2.2707 3.4570 3.4875 3.5150
30 1 −1 2.0339 2.0175 2.0000 0.8817 0.8562 0.8296
−.6 1.8367 1.8257 1.8142 1.1432 1.1289 1.1142
−.2 1.6177 1.6136 1.6118 1.3858 1.3813 1.37684
.2 1.3768 1.3813 1.3858 1.6118 1.6136 1.6177
.6 1.1142 1.1289 1.1432 1.8142 1.8257 1.8367
1 0.8296 0.8562 0.8817 2.0000 2.0175 2.0339
2 −1 2.6648 2.6471 2.6283 1.3982 1.3697 1.3401
−.6 2.4516 2.4396 2.4271 1.6891 1.6731 1.6568
−.2 2.2126 2.2081 2.2035 1.9575 1.9526 1.9476
.2 1.9476 1.9526 1.9575 2.2035 2.2081 2.2126
.6 1.6568 1.6731 1.6891 2.4271 2.4396 2.4516
1 1.3401 1.3697 1.3982 2.6283 2.6471 2.6648
3 −1 3.0489 3.0308 3.0114 1.7399 1.7105 1.6797
−.6 2.8299 2.8175 2.8047 2.0418 2.0253 2.0084
−.2 2.5835 2.5789 2.5742 2.3199 2.3148 2.3097
.2 2.3097 2.3148 2.3199 2.5742 2.5789 2.5835
.6 2.0084 2.0253 2.0418 2.8047 2.8175 2.8299
1 1.6797 1.7105 1.7399 3.0114 3.0308 3.0489
4 −1 3.3257 3.3074 3.2877 1.9949 1.9649 1.9335
−.6 3.1038 3.0912 3.0782 2.3025 2.2857 2.2684
−.2 2.8536 2.8488 2.8441 2.5855 2.5803 2.5751
.2 2.5751 2.5803 2.5855 2.8441 2.8488 2.8536
.6 2.2684 2.2857 2.3025 3.0782 3.0912 3.1038
1 1.9335 1.9649 1.9949 3.2877 3.3074 3.3257
5 −1 3.5423 3.5238 3.5040 2.1981 2.1677 2.1360
−.6 3.3185 3.3059 3.2927 2.5092 2.4922 2.4747
−.2 3.0660 3.0613 3.0564 2.7953 2.7900 2.7847
.2 2.7847 2.7900 2.7953 3.0564 3.0613 3.0660
.6 2.4747 2.4922 2.5092 3.2927 3.3059 3.3185
1 2.1360 2.1677 2.1981 3.5040 3.5238 3.5423

MTTF, mean times to failure.

Table 2

θ2 times MTTF of two component parallel system using concomitants.

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

μ1(t)=μxμyμx+μy,
and
μ2(t)=μx+μyμxμyμx+μy,
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.

Series System, θ2μ1(t)
α2 1 2 3 4 5
0.5000 0.7500 0.9167 1.0417 1.1417

Parallel System, θ2μ2(t)

α2 1 2 3 4 5
1.5000 2.2500 2.7500 3.1250 3.4250
Table 3

Independent case.

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.

Series System
n λ α2 1 2 3 4 5
10 −.2 10.06 30.92 40.51 46.27 50.19
−.6 32.36 50.03 57.60 61.99 64.93
−1 57.58 71.43 76.67 79.52 81.35
20 −.2 11.92 32.52 41.95 47.59 51.43
−.6 38.72 55.45 62.44 66.44 69.11
−1 69.48 81.49 85.63 87.75 89.05
30 −.2 12.58 33.09 42.47 48.07 51.88
−.6 41.04 57.41 64.20 68.06 70.61
−1 73.86 85.19 88.92 90.77 91.88

MTTF, mean times to failure.

Table 4

The percentage relative gain in MTTF of a series system.

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.

n λ α2 Parallel System
1 2 3 4 5
10 −1 29.29 13.90 7.06 3.03 0.30
20 −1 33.98 17.28 9.90 5.56 2.63
30 −1 35.59 18.44 10.87 6.42 3.42

MTTF, mean times to failure.

Table 5

The percentage relative gain in MTTF of a parallel system.

Using relations (4.1) and (4.2), we deduce that the percentage relative gain in the MTTF of series and parallel systems using the (n1)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, α25 and n30, the percentage relative increase in the MTTF of a series system is greater than ninety one. Similarly, if λ=±1, α2=1 and n30, 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.

REFERENCES

1.H.A. David, Bull. Inst. Int. Stat., Vol. 45, 1973, pp. 295-300.
2.M. Ahsanullah, Bull. Malays. Math. Sci. Soc., Vol. 32, No. 2, 2009, pp. 101-117.
5.J. Scaria and N.U. Nair, J. Appl. Stat. Sci., Vol. 14, No. 5, 2005, pp. 251-262.
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19.N.L. Johnson and S. Kotz, Distributions in Statistics, Continuous Multivariate Distributions, John Wiley and Sons, New York and London, 1972.
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Journal
Journal of Statistical Theory and Applications
Volume-Issue
20 - 2
Pages
193 - 203
Publication Date
2021/03/30
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
https://doi.org/10.2991/jsta.d.210325.001How to use a DOI?
Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Johny Scaria
AU  - Sithara Mohan
PY  - 2021
DA  - 2021/03/30
TI  - Dependence Concepts and Reliability Application of Concomitants of Order Statistics from the Morgenstern Family
JO  - Journal of Statistical Theory and Applications
SP  - 193
EP  - 203
VL  - 20
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.210325.001
DO  - https://doi.org/10.2991/jsta.d.210325.001
ID  - Scaria2021
ER  -