MOMENTS PROPERTIES OF CONCOMITANTS OF GENERALIZED ORDER STATISTICS FROM FGMTBM EXPONENTIAL DISTRIBUTION Текст научной статьи по специальности «Биологические науки»

MOMENTS PROPERTIES OF CONCOMITANTS OF GENERALIZED ORDER STATISTICS FROM FGMTBM EXPONENTIAL

DISTRIBUTION

Mustafa Kamal1, Nayabuddin2, Intekhab Alam3, Ahmadur Rahman4, Abdul Salam5, Shazia Zarrin6

department of Basic Sciences, College of Science and Theoretical Studies, Saudi

Electronic University, KSA e-mail: m. kamal@seu.edu.sa 2Faculty of Public Health and Tropical Medicine, Jazan University, KSA e-mail: nhanif@jazanu.edu.sa 3Department of Management St. Andrews Institute of Technology &

Management Gurugram, Haryana, India "Department of Statistics and O.R., Aligarh Muslim University, India e-mail: ahmadur.st@gmail.com (Corresponding Author) 5Department of Epidemiology, Faculty of Public Health and Tropical Medicine,

Jazan University, KSA 6Uttaranchal Unani Medical College and Hospital, Haridwar, India. e-mail: shaziazarrin@gmail.com

Abstract

Concurrent or induced order statistics are produced when individuals in a random sample are ordered in compliance with the corresponding values of some other random sample. Concomitants are most helpful when k(n) individuals are to be chosen using a selection technique based on their X-values. The relevant Y-values are then used to reflect how well a characteristic has performed. In this paper, concomitants of generalized order statistics (GOS) from Farlie Gumbel Morgenstern type bivariate moment (FGMTBM) exponential distribution are obtained. Additionally, distribution function (df) and probaility density function (pdf) r-th generalized order statistics and a joint pdf of r-th and s-th GOS were also obtained. Furthermore, we provide the minimum variance linear unbiased estimator (MVLUE) of the position and scale parameters of the concomitants of the k-th upper record values and order statistics for the distribution under consideration. Finally, an implementation of the suggested methodology has been taken into account.

Keywords: Generalized order statistics, Concomitants, Record Values, FGMTBM exponential distribution, Single and Product Moments, Minimum variance linear unbiased estimator.

AMS Subject Classification: 62G30, 62E10

1. Introduction

In a wide range of statistical applications, order statistics (OS) and record values are often used in statistical modeling and inference. In both models, random variables are listed in descending order of magnitude. GOS offers an integrated approach to a wide variety of ordered random variable models with various interpretations. The theory of GOS is pioneered by [1]. Since then, a number of

writers have incorporated the idea of GOS into their works, including [2-3], and others.

Let n E N,n > 2,k > 0, m= (m1,m2, ...,mn-1) E Wn-1,Mr = Yj-r mj, such that yr = k + n — r + Mr > 0 for all r E {1,2, ...,n- 1}. Then X(r, n, m, k), r E {1,2, ...,n} are called gos if their joint pdf is given by

mu nm?-! [i - F(Xi)]mtf(Xi))[i - F(xn)]k-if(xn) (1.1)

on the cone F-1(0) < x1 < x2 < ••• <xn< F-1(1) of №.

Now, by selecting the proper values for the parameters, models such as ordinary OS (y¿ = n — i + 1;i = 1,2,...,n i.e.m1 = m2 = — = mn-1 = 0,k = 1,) , kth record values (yt = k i.e. m1 = m2... = mn-1 = —1,k EN,) , sequential OS (yt = (n — i + 1)a¿; ) (a1, a2,...,an > 0) , OS with non-integral sample size (y¿ = (a — i + 1);a > 0), Pfeifer's record values (y¿ = p¿; p1,p2,...,pn > 0) and progressive type-II censored OS (mi E N0,k E N) are obtained. The pdf of r — th GOS, X(r, n, m, k) is

fx(r,n,m,k) = ^ [F(x)Yr-1f(x)[gm(F(x))Y-1 (1.2)

and joint pdf of X(s,n,m,k) and X(r,n,m,k) , 1 < r < s < n, is

fx(r,s,n,m,k)(x,y) = {r_ ^^r-l)! [F(x)]mf(x) [gm(F(x))]r-1

where,

X [hm(F(y)) — hm(F(x))]s-r-1[F(y)]Ys-1f(y), a<x<y<p (1.3)

Cr-1 = Wi=1 Yi ,Yi = k + (n — i)(m + 1), gm(x) = hm(x) — hm(0), x E (0,1)

(1 — x)m+1 , m^—1

and hm(x) = { m+i

L-log(l — x) , m = —1

Let F(x,y) be the df of some arbitrary bivariate population. Assume also that (X^Yi), i = 1,2, ...,n, are n pairs of independent random variables from the poulation having distrubution F(x,y). Let the ascending order of X variates is X(1,n,m,k) < X(2,n,m,k) <,...,< X(n,n,m,k), now, if we arrange the Y variates pairwise (necessarily not in increasing order) with these generalized ordered statistics, then Y variates are called the concomitants of GOS and are generally denoted by Y[hn^k],Y[2^mk],... ,Y[n^mk]. Now, the pdf of Y[r^mk], the rth concomitant of GOS is given as

k](y) = fvixiy1*) fx(r,n,m,k)(x) dx (1.4) and the joint pdf of Y[r}n}mM and Y[Sjnjmjk] is

9[r,s,n,m,k](yi,y2) = J_oo fYIX(yilx1) fYIX(y2lx2) fx(r,s,n,m,k)(x1'x2) dx2 (1.5)

The FGM family of bivariate distributions is frequently employed in practice. This family is

defined by the given marginal dfs Fx(x) and FY(y) of random variables X and Y and a parameter a. As a result, the FGM bivariate distribution function is given as follows:

FxAx-y) = Fx(x)FY(y)[1 + «(1 - Fx(x))(1 - FY(y))], (1.6)

together with associated pdf

fx,v(x,y) = fx(x)fy(y)[l + a(l - 2Fx(x))(l — 2FY(y))]. (1.7)

The marginals of fXjY(x,y) in this case are fx(x) and fY(y). The two random variables X and Y are independent when the parameter a, also known as the association parameter, is zero. For exponential marginals, such a model was first presented by [5] and examined by [6]. The general form in Eq. (1.6) is credited to [7] and [8]. The acceptable range of the association parameter a is —1 < a < 1, and the maximum value of the Pearson correlation coefficient p between X and Y must not exceed the value 1/3. Now, for given X, the conditional df and pdf of Y are:

Fnx(ylx) = FY(y)[1 + a(1 — Fx(x))(1 — FY(y))] (1.8)

and

fvix(ylx) = fy(y)[1 + «(1 — 2Fx(x))(1 — 2FY(y))] (1.9)

There have been various studies that discuss the concomitants of OS in the research. Weighted inverse Gaussian distribution used to conduct a comparative study on the concomitant of OS and record values by [9]. Shannon's entropy is calculated by [10] after taking into account the concomitants of distribution of FGM family. Readers might check [11, 12] for some outstanding reviews on the concomitants of GOS. In [13], parameter estimation is discussed. They estimated the parameters of the FGM-type bivariate exponential distribution using the concomitants of GOS. The concomitants of GOS for the bivariate Lomax distribution is investigated by [14]. In addition, [15] obtained the concomitants of GOS for bivariate Pareto distribution. Also, [16] explored the concomitants of m-GOS from the generalized FGM distribution family. For a few examples of recent studies based on the notion of concomitants of OS, readers may refer to [15, 17]. For some applications of OS in reliability and accelerated life testing, readers may refer to [18-22].

In this study, we examined the FGMTBM exponential distribution. For 6 > 0,0 < x,y < < a < 1, the pdf, df and the conditional pdf of Y given X of the FGMTBM exponential

distribution is described by the following equation [23].

f(x,y) = 84xy e-e(x+y){1 + a[1 — 2{1 — (1 + 8x)e-0x}] [1 — 2{1 — (1 + 8y)e-0y}]} (1.10)

F(x,y) = {1 — (1 + 8 x)e-ex}{1 — (1 + 8 y)e-ey}[1 + a(1 + 8x)(1 + 8y)e-0(x+y)] (1.11)

f(ylx) = 82y e-9y{1 + a[1 — 2{1 — (1 + 8x)e-0x}] [1 — 2{1 — (1 + 8y)e-0y}]} (1.12)

The marginal pdf and df of X are respectively

f(x) = 82x e-ex, 0<x <™, 8 > 0, (1.13)

F(x) = 1 — (1 + 8x)e-ex, 0<x <™, 8 > 0, (1.14)

2. PDF of Concomitants

For the FGMTBM exponential distribution as given in Eq. (1.10), using Eq. (1.12) and Eq. (1.2) in Eq. (1.4), the pdf of r — th concomitants Y[rn:mk]of GOS for —1 is given as:

9[r,n,m,k](y) = 'Z'r-1 d2ye~9y

x J" {1 + a[1 - 2{1 -(1 + вх)е-вх}] [1 - 2{1 - (1 + ву)е-вУ}]

x [(1 + вх)е-вху<--1-1[1-{(1 + вх)е-вх}т+1]г-1 в2хе-вхйх. (2.1)

Setting z = (1 + вх)е-вх in Eq. (2.1), we get

ir-m^r-i в2Уе-вУ il {1 + "(1 - 2^) [1 - 2{1 -(1 + ву)е-вУ}]} Zrr-1[1 - 2ш+1]г-1 dz.

(2.2)

Making transformation t = 1-zm+1, we get

1 Yr 1

^-- в2уе-ву J1 tr-1(1 - t{1 + a[1 - 2(1 - ty^+i] [1 - 2{1 - (1 + ву)е-вУ}]} dt

(r-1)!(m+1)r'

(2.3)

°r-1 r , в2уе-вУ{В(г,^~) + a[B(r,^~) - 2В(г,*+-Ш - 2{1 - (1 + ву)е-вУ}]}

(r—1)!(m+1)r-1 1 v m+1J L v m+1J v m+1JiL 1 v JJJ

(2.4)

Which after simplification yields

д[г,п,т,к](У) = в2уе-вУ[1 + a{1 - 2{1 -(1 + ву)е-вУ}}{1 - 2 RU (1 + ±)-1}]. (2.5)

n

It may be verified that J" g[r,n,m,k] (v)dv = 1

Remark 2.1: Set m = 0,k = 1 in Eq. (2.5), to get the pdf of r-th concomitants of OS from FGMTBM exponential distribution as

g[r-.n](y) = 82ye-9y[1 + «{1 - 2{1 - (1 + ey)e-ey}}{1 -2(^}].

Remark 2.2: At m = —1 in Eq. (2.5), to get the pdf of r-th concomitants of k - th upper record values from FGMTBM exponential distribution as

g[r,n,-i,k^(y) = e2ye-0y[1 + «{1 - 2{1 - (1 + ey)e-ey}}{1 - 2(±m 3. Single Moments

Now, by using the results from the previous section, we obtain the moments of Y[rnmik] for the FGMTBM exponential distribution in this section. As a result, in the light of Eq. (2.5), the moments of Y[r,n,m,k] are given as:

EK]n,mM] = 62 J" ykl+1e-9y[1 + a{1-2[1-(1 + ey)e-ey]}{1 - 2 UU (1 + 1)-1}]dy. (3.1)

Note that

n

J" xa-1e-@xdx = (3.2)

Using Eq. (3.2) in Eq. (3.1), we get after simplification

eK^]] = tj^2[1 + «{202- W - 2nu (1 +1)-1}]. (3.3)

Remark 3.1: Insert m = 0,k = 1 in Eq. (3.3) to retrieve the moments of OS concomitants from the FGMTBM exponential distribution as:

E[Y[r-.n]\ [1 + a{2M+2 — W

Remark 3.2: The moments of concomitants of the k-th upper record statistics from the FGMTBM exponential distribution can be obtained as follows by setting m = —1 in Eq. (3.3):

^-W =lii+2[1 + — m — 2(fc+i)r}\

Table 3.1: Mean of the concomitant of OS for FGMTBM exponential distribution with 6 = 0.4

n r a = -0.9 a = -0.5 a = -0.1 a = 0.1 a = 0.5 a = 0.9

1 1 5 5 5 5 5 5

2 1 4.437 4.687 4.937 5.062 5.312 5.562

2 5.562 5.312 5.062 4.937 4.687 4.437

3 1 4.156 4.531 4.906 5.093 5.468 5.843

2 5 5 5 5 5 5

3 5.843 5.468 5.093 4.906 4.531 4.156

4 1 3.987 4.437 4.887 5.112 5.562 6.012

2 4.662 4.812 4.962 5.037 5.187 5.337

3 5.337 5.187 5.037 4.962 4.812 4.662

4 6.012 5.562 5.112 4.887 4.437 3.987

Table 3.2: Mean of the concomitant of record value for FGMTBM exponential distribution with 6 = 0.4 and k = 2.

r a = -0.9 a = -0.5 a = -0.1 a = 0.1 a = 0.5 a = 0.9

1 4.437 4.687 4.937 5.062 5.312 5.562

2 5.187 5.104 5.02 4.979 4.895 4.812

3 5.687 5.381 5.076 4.923 4.618 4.312

4 6.02 5.567 5.113 4.886 4.432 3.979

4. Joint PDF of Two Concomitants

Now, using Eqs. (1.3) and (1.12) in Eq. (1.5), the joint pdf of the r-th and s-th concomitants of GOS Y[r ^m k] and Y[s^mk\ for m ± —1 for the FGMTBM exponential distribution can be derived as:

g[r,s,n,m,k](yi,y2) = ^-¿-Ir-u. e'y^e-9^-^

x /o°° Jo2 {1 + «[1 — 2{1 — (1 + ex1)e-e^}] [1 — 2{1 — (1 + dyje-9^]]]

x {1 + a[1 - 2{1 -(1 + вх2)е-вХ2}] [1 - 2{1 - (1 + ву2)е-ву2}]} x (в4в е-вх1е-вх2){-^(1 - [(1 + вх1)е-вх1]т+1)}г-1{(1 + вх2)е-вх2}^-1 x {--— [(1 + вх2)е-вх2]т+1 + — [(1 + вх1)е-вх1]т+1}5-г-1

1 ш+1 2 ш+1

x [(1 + вх1)е-вх1]т dx1dx2 (4.1)

And utilizing the transformations U = 1- (1 + вх1)е-вХ1 and V = 1 - (1 + вх2)е-вХ2 in Eq. (4.1), we get:

в^е^е-^2 Л Jv=v {1 + «(1 - 2U) [1 - 2{1 -(1 + ву^1}]

(r-1)!(s-r-l)!(m+l)s~2

x {1 + a(1 — 2V) [1 — 2{1 — (1 + dy2)e-9y2}]}Um [1 — um+1]r-1 [Um+1 — V^+1]s-r-1 Vrs-1du dv

(4.2)

Subsequently, using the transformation V = Ut in the preceding equation, we obtained the following results:

,SyiP-ey2

- в4у1У2в-ву1е-ву2 J1 J01 {1 + a(1 - 2U)[1 - 2{1 -(1 + ву^е-^1}]}

(r-iy.(s-r-iy.(m+l)s-2 J1J2 J0 J0

x {1 + a(1 - 2Ut) [1 - 2{1 -(1 + 9y2)e-0y2}]}UYr-1 [1 - um+1]r-11Ys-1 [1 - tm+1]s-r:1 du dt

(4.3)

By setting p = um+1 and q = tm+1 in Eq. (4.3), we get after simplification

= e^*-**1*-**2 fi fi ^+a(1 - [1 --(1 + eyi)e-eyi)]}

1 1 Yr -,

x {1 + a(1 - 2pm+1 qm+1) [1 - 2{1 - (1 + dy2)e:9y2}]}p^+1-1 (1 - p)r-1 q^+1-1 (1 - q)s-r-1 dp dq

(4.4)

Upon further simplification of the preceding result, we obtain

g[r,s,n,m,k](yi.y2) = O^e^e-^2 [1 + a2 [1- 2{1 - (1 + eyi)e-ey1}] [1 - 2{1 - (1 + dy2)e-ey2}]

x{1-2 Wi=i (1 +1)-1 - 2 n=i (1 + f)-1 + 4 nri=i (1 + 2)-1}

ri Yi Yi

+«{[1 - 2{1 -(1 + eyi)e-eyi}] + [1- 2{1 - (1 + 9y2)e-8y2}]}{1 - 2 n?=i (1 + ^)-1}] (4.5)

Yi

It may be varified that f0°° f0°° g[r,s,n,m,k] (yi,y2)dyidy2 = 1

Remark 4.1: We can obtain the joint pdf of concomitants of OS by putting m = 0,k = 1 in Eq. (4.5), as well as the joint pdf of concomitants of the k-th upper record values for the FGMTBM exponential distribution by setting m = -1.

5. Product Moments of Two Concomitants The product moments of two concomitants Y[rnmik] and Y[Sinmk] are given as:

E(Y[r,n,m,k]Y[Ps,n,m,k]) = Ç yl yl 9[r,s,n,rn,k](yi,y2) dy± dy2 (5.1)

In view of Eq. (4.5) and Eq. (5.1), we have

HYUrn,k]YL,rn,k]) = e4 C y2P+1e-ey2 (C y1+1 e-eyi [1 + «2 [1- 2{1 - (1 + Oyje-^1]]

x [1 - 2(1 -(1 + ey2)e-ey2}]{1 - 2 nU (1 +1)-1 - 2 n?=i (1 +1)-1 + 4Uri=i (1 + 2)-1]

+«{[1 - 2(1 -(1 + dy^e-^]] + [1 - 2(1 -(1 + 8y2)e-ey2}]}(1 - 2 RU (1 + 1)-1]]dy1]dy2

Yi

(5.2)

E(Y[r,n,m,k]Y[s,n,m,k]) = ^e^H + a2 [1 - 2(1 - (1 + e^e-»*]]

■ f)-1 - 2 nu (1+1)-1 + 4 nu (1+f)-1] + *[(2l+2-22+M) Yi Yi Yi 2

+ [1 - 2(1 -(1 + ey2)e-ey2]]][1 - 2 nu (1 + 1)-1]]dy2 (5.3)

Yi

Solving Eq. (5.2), we get after simplification

pry I yP N _ T(l+2) T(P+2)C[ 2 f2l+2-2l+3 + l+4. (2p+2-2p+3+P+4., C(I[r,n,m,k]'[s,n,m,k]) = gl+p ([1 + a ( 2l+2 )( 2p+2 )]

x[1 -2 Wi=1 (1 + 7)-1 - 2 nu (1 +1)-1 + 4 nu (1 + 7)-1]

Yi Yi Yi

+«[(-~2-) + (-Ï+T1—)] [1 - 2 nU (1 + ") 1]] (5.4)

Remark 5.1: The product moments of the concomitants of OS from the FGMTBM exponential distribution can be obtained by setting m = 0,k = 1 in Eq. (5.4) as follows:

pryl yP \ - r(l + 2)T(p+2) fr 2 2l+2-2l+3 + l+4. (2P+2-2P+3 + p+4., b(Y[r-.n]Y[S-.n]) = —g+ï-{[1 + a (-2+2-J(-2p+2-)]

x (n-s+i)(n-s+2).. _ 2(n-Hj _ 2(n-r+l)]

L v (n+1)(n+2) J v n+i J v n+i yJ

2l+2-2l+3 + l+4. 2+2-2V+3+P+4^vA n,n-r+1.„ + «[(-—2-) + (-—2-)][1 _

Remark 5.2: The product moments of the concomitants of k _ th upper record value from the FGMTBM exponential distribution can be obtained by setting m = _1 in Eq. (5.4) as follows:

pryl yi-P \ - r(l+2)T(p + 2) fr 2 f2l+2-2l+3 + l+4. 2?+2-2V+3+p+4„

£ (Y[r,n,-1Jk]Y[s,n,-1,k]) = gï+ï {[1 + a ( 2+2 J ( 2p+2 )]

X[1 + 4(—)r - 2(—)r - 2(—)s]

L Vk+2J Vk+V Vk+1J J

+ «[(-—2-) + (-—2-)][1 - 2(ft+T)S]}

Table 5.1: Covariance of the concomitant of OS for FGMTBM exponential distribution with 6 = 0.4

n s r a = -0.9 a = -0.5 a = -0.1 a = 0.1 a = 0.5 a = 0.9

4 1 1 30.981 19.839 12.493 10.243 8.589 10.731

2 1 13.102 13.488 11.939 10.439 5.988 -0.397

2 11.567 6.996 4.179 3.429 3.246 4.817

3 1 1.297 9.011 11.46 10.71 5.261 -5.452

2 -0.692 2.378 3.695 3.695 2.378 -0.692

3 -2.682 -4.253 -4.07 -3.32 -0.503 4.067

4 1 -4.431 6.41 11.056 11.056 6.41 -4.431

2 -6.877 -0.363 3.285 4.035 3.386 -0.127

3 -9.322 -7.136 -4.485 -2.985 0.363 4.177

4 -11.768 -13.91 -12.256 -10.006 -2.66 8.481

Table 5.2: Covariance of the concomitant of record value for FGMTBM exponential distribution with 6=0.4, and k = 2.

s r a = -0.9 a = -0.5 a = -0.1 a = 0.1 a = 0.5 a = 0.9

1 1 7.352 2.287 0.098 0.081 2.204 7.202

2 1 21.549 7.57 0.633 -0.193 3.431 14.099

2 5.002 1.537 0.059 0.064 1.565 5.052

3 1 31.013 11.092 0.991 -0.377 4.25 18.697

2 14.091 4.944 0.411 -0.123 2.268 9.275

3 6.607 2.016 0.072 0.092 2.118 6.79

4 1 37.323 13.44 1.229 -0.499 4.795 21.762

2 20.151 7.214 0.646 -0.248 4 12.09

3 12.5 4.236 0.305 -0.034 2.535 9.439

4 9.298 2.836 0.101 0.131 2.987 9.57

6. Application

Finding the MVLUE of the location and scale parameters is an intriguing application of this work. Although the [24] methodology could be utilized to obtain MVLUE but it is quite challenging to acquire the inverse of the variance-covariance matrix in closed form for this methodology.

Therefore, these estimates are computed using numerical techniques.

Assume that the distribution of the random variables have ^ and a as the location and scale parameters. Now, based on the [24] technique, the MVLUE of 8 is given as 8 =

(A'V-1A)-1(A'V-1y), where, V = (Vij) represents the variance of the i — th and j — th concomitants, V-1 represents the inverse of the matrix V, and y' is the observed value of the vector Y' = Yintmk],Y[2^mk],... ,Y[n^m,k]. For n = 0 and a = 1, A and 8 can be obtained as:

A =

1,

1

1

,n,m, k] Vd[2 ,n,m,k] ••• №-d[n,n,m,k]

P=[M a]

Table 6.1: Coefficients of MVLUE of i and a for FGMTBM exponential distribution based on OS with n = 4,

9 = 0.4

a Estimate Coefficients

-0.9 P- -0.0828 -0.1638 1.26766 -0.5523

ô -0.5017 -0.9922 7.67812 -3.3452

-0.5 P- 5.14576 3.93097 -8.4672 0.39049

ô -0.9503 -0.825 1.83408 -0.0588

-0.1 P 14.4297 28.8015 -32.732 -9.4992

ô -2.7511 -5.8872 6.69428 1.944

0.1 P -9.1516 -35.805 33.9053 12.0516

ô 1.98121 7.01331 -6.6369 -2.3576

0.5 p -5.0134 -3.5421 10.2922 -0.7368

ô 1.07724 0.74098 -2.047 0.2288

0.9 p -1.9202 -0.6639 1.18162 2.40244

ô 0.40411 0.19224 -0.1153 -0.481

Table 6.2: Coefficients of MVLUE of i and a for FGMTBM exponential distribution based on record values

with n = 4, 9 = 0.4 and k = 2

a Estimate Coefficients

-0.9 P -9.0704 37.5315 -32.689 5.22765

ô 2.27034 -8.4608 7.36915 -1.1787

-0.5 P -9.6243 57.7496 -68.289 21.1632

ô 2.26517 -12.314 14.5604 -4.5111

-0.1 P 26.7814 -5.3193 24.1669 -44.629

ô -5.2229 1.07976 -4.8968 9.04001

0.1 p -32.758 4.67707 12.0107 17.0699

ô 6.66864 -0.9234 -2.3731 -3.3721

0.5 p -19.659 26.1102 4.18588 -9.6375

ô 3.88893 -4.915 -0.7883 1.81434

0.9 p -7.5351 10.0486 -1.2653 -0.2481

ô 1.53447 -1.8065 0.22748 0.04455

7. Conclusions

In this study, FGMTBM exponential distributions was considered and concomitants of GOS were obtained. Additionally, pdf pdf r-th GOS and a joint pdf of r-th and s-th GOS were also obtained. Furthermore, the MVLUE of the location and scale parameters of the concomitants of the k-th upper record values, as well as OS, were obtained for the distribution under consideration. Finally, an example was considered for the purpose of putting the suggested methodology into exercise. When k(<n) individuals are to be picked using a selection method based on their X-values, concomitants are most useful. The performance of a characteristic is then depicted using the pertinent Y-values. When individuals in a random sample are arranged in accordance with the corresponding values of another random sample, concurrent or induced OS are created.

References

[1] Kamps, U. (1995). A concept of generalized order statistics, B.G. Teubner Stuttgart, Germany (Ph. D. Thesis).

[2] Khan, A.H., Khan R.U. and Yaqub, M. (2006). Characterization of continuous distributions through conditional expectation of generalized order statistics, J. Appl. Prob. Statist 1, 15-131.

[3] Tavangar M. and Asadi M. (2008) On a characterization of generalized Pareto distribution based on generalized order statistics, Commun. Statist. Theory Meth. 37, 1347-1352.

[4] Athar H. and Nayabuddin (2013). Recurrence relations for single and product moments of generalized order statistics from Marshall- Olkin extended general class of distributions, J. Stat. Appl. Prob. 2(2), 63-72.

[5] Morgenstern, D. (1956). Einfache Beispiele Zweidimensionaler Verteilunngen, Mitteilungsblatt fur Mathemtische Statistik, 8, 234-235.

[6] Gumbel, E.J. (1960). Bivariate exponential distributions, J. Amer. Statist. Assoc. 55, 698-707.

[7] Farlie, D.J.G. (1960). The performance of some correlation coefficients for a general bivariate distribution, Biometrika , 47, 307-323.

[8] Johnson, N.L. and Kotz, S. (1975). On some generalized Farlie-Gumbel-Morgenstern distributions, Commun. statist. Theor. Meth. 4, 415-427.

[9] Das, K. K., Das, B. and Baruah, B. K. (2012). A comparative study on concomitant of order statistics and record values for weighted inverse Gaussian distribution. International Journal of Scientific and Research Publication 2, 1-7.

[10] Tahmasebi, S. and Behboodian, J. (2012). Shannon information for concomitants of generalized order statistics in Farlie-Gumbel-Morgenstern (FGM) family. Bulletin of the Malaysian Mathematical Science Society 34, 975-981.

[11] Ahsanullah, M. and Beg, M. I. (2006). Concomitant of generalized order statistics in Gumbel's bivariate exponential distribution. Journal of Statistical Theory and Applications 6, 118-132.

[12] Beg, M. I. and Ahsanullah, M. (2007). Concomitants of generalized order statistics from Farlie Gumbel Morgenstern type bivariate Gumbel distribution, Statistical Methodology. 1-20.

[13] Chacko, M. and Thomas, P.Y. (2011) . Estimation of a parameter of Morgenstern type bivariate exponential distributionusing concomitants of order statistics, Statistical Methodology, 8, 363-376.

[14] Nayabuddin, (2013). Concomitants of generalized order statistics from bivariate Lomax distribution, ProbStat Forum 6, 73-88.

[15] Nayabuddin, Athar H. and Al- Helan, M. (2018). Concomitants of generalized order statistics from bivariate Burr XII distribution, Aligarh Journal of statistics, 18, 1-18.

[16] Domma, F. and Giordano, S. (2016). Concomitants of m-generalized order statistics from generalized Farlie-Gumbel-Morgenstern distribution family. Journal of Computational and Applied Mathematics, 294, 413-435

[17] Alawady, Barakat, H. M., Shengwu Xiong and Abd Elgawad, M. A.(2020). Concomitants of generalized order statistics from iterated Farlie-Gumbel-Morgenstern type bivariate distribution, Communications in Statistics - Theory and Methods, DOI: 10.1080/03610926.2020.1842452

[18] Saxena, S., Zarrin, S., & Kamal, M. (2012). Computation of reliability and Bayesian analysis of system reliability for Mukherjee Islam failure model. American Journal of Mathematics and Statistics, 2(2), 1-4.

[19] Kamal, M. (2013). Application of geometric process in accelerated life testing analysis with type-I censored Weibull failure data. Reliability: Theory & Applications, 8(3 (30)), 87-96.

[20] Kamal, M. (2021). Parameter estimation for progressive censored data under accelerated life test with k levels of constant stress. Reliability: Theory & Applications, 16(3 (63)), 149-159.

[21] Kamal, M., Rahman, A., Zarrin, S., & Kausar, H. (2021). Statistical inference under step stress partially accelerated life testing for adaptive type-II progressive hybrid censored data. Journal of Reliability and Statistical Studies, 585-614.

[22] Kamal, M. (2022). Parameter estimation based on censored data under partially accelerated life testing for hybrid systems due to unknown failure causes. CMES-Computer Modeling in Engineering & Sciences, 130(3), 1239-1269.

[23] Hasnain, S.A. (2013). Exponentiated moment Exponential Distribution, National College of Business Administration and Economics, Lahore, Pakistan. (Ph.D. Thesis).

[24] Llyod, E.H.(1952) Least squares estimation of location and scale parameter using order statistics, Biometrika, 39, 88-95.