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

^{*}, Sithara Mohan

^{*}Corresponding author. Email: johny.kakkattil@gmail.com

- 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 *n* from a continuous bivariate population with cumulative distribution function (cdf) *X*-variate in ascending order as *Y*-values paired with these order statistics are denoted by

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

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 *X* and *Y* with spearman's rho

## 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

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

When (2.4) is absolutely continuous the parameters satisfy the conditions

Comparing (2.3) with (2.4), we get

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

*x*and for all

*y*) is equivalent to

*X*and

*Y*are LRD.

The Morgenstern family belongs to

From Equation (2.7),

Hence

It is directly established that

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

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

By substituting

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

Let *i* = 1,2. The mean times to failure of

From [11],

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

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

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**and*

From (3.4), the

The

## 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,

We denote the EE distribution with parameters

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 *X* and *Y* measurements or concomitants. Then the cdf of

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

It follows from (4.6) and (4.7) that

*U*,

*V*,

*W*and

*Z*with

The

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

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

Relation 4.1

Relation 4.2

The *n, r, s*,

n | ||||||||
---|---|---|---|---|---|---|---|---|

(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.

n | ||||||||
---|---|---|---|---|---|---|---|---|

(1, 2) | (1, 3) | (2, 3) | ||||||

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.

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

Series System, |
|||||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |

0.5000 | 0.7500 | 0.9167 | 1.0417 | 1.1417 | |

Parallel System, |
|||||

1 | 2 | 3 | 4 | 5 | |

1.5000 | 2.2500 | 2.7500 | 3.1250 | 3.4250 |

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 *r* = 1, and *s* = 2. On the otherhand, if *r* = *n* − 1, and *s* = *n*. It is also observed that the MTTF of series system increases if *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 *n* = 10, 20 and 30.

Series System |
|||||||
---|---|---|---|---|---|---|---|

n |
λ |
α_{2} |
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.

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 *r* = 1, and *s* = 2. On the otherhand, if *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 *n* = 10, 20 and 30 and

n |
λ |
α_{2} |
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.

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

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

## ACKNOWLEDGMENTS

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

## REFERENCES

### 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 -