Concomitants of Order Statistics and Record Values from Generalization of FGM Bivariate-Generalized Exponential Distribution

H. M. Barakat; E. M. Nigm; M. A. Alawady; I. A. Husseiny

doi:10.2991/jsta.d.190822.001

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Volume 18, Issue 3, September 2019, Pages 309 - 322

Concomitants of Order Statistics and Record Values from Generalization of FGM Bivariate-Generalized Exponential Distribution

Authors

H. M. Barakat, E. M. Nigm^*, M. A. Alawady, I. A. Husseiny

Department of Mathematics, Faculty of Science Zagazig University, Zagazig, Egypt

^*Corresponding author. Email: s_nigm@yahoo.com

Corresponding Author

E. M. Nigm

Received 6 May 2018, Accepted 15 February 2019, Available Online 19 September 2019.

DOI: 10.2991/jsta.d.190822.001 How to use a DOI?
Keywords: Concomitants; Order statistics; Record values; Generalized exponential distribution; Generalization of FGM family
Abstract: We introduce the generalized Farlie–Gumbel–Morgenstern (FGM) type bivariate-generalized exponential distribution. Some distributional properties of concomitants of order statistics as well as record values for this family are studied. Recurrence relations between the moments of concomitants are obtained, some of these recurrence relations were not publishes before for Morgenstern type bivariate distributions. Moreover, most of the paper results are extended to arbitrary distributions.
Copyright: © 2019 The Authors. Published by Atlantis Press SARL.
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

Let (X,Y) be a bivariate absolutely continuous random variable (rv), formally defined by the distribution function (df)

(1)FX,Y(x,y)=FX(x)FY(y){1+λA(Fx(x))B(Fy(y))},

where, FX(x) and FY(y) are the marginals df’s of X and Y, respectively. Moreover, the two kernel A(x)→0 and B(y)→0, as x→1 and y→1, satisfy certain regularity conditions ensuring that FX,Y(x,y) is a df with absolutely continuous marginals FX(x) and FY(y). The model (1) was originally introduced by [1] for A(x)=1−x and B(y)=1−y and investigated by [2] for exponential marginals. Subsequent generalizations for this model is due to Farlie (1960) [3–6]. The successive generalizations of this model aims generally to enlarge the range of its correlation. If A(x)=1−FX(x) and B(y)=1−FY(y) in model (1) then we have the classical Farlie–Gumbel–Morgenstern (FGM) for arbitrary continuous marginals FX(x) and FY(y) with df

(2)FX,Y(x,y)=FX(x)FY(y){1+λ(1−FX(x))(1−FY(y))},−1≤λ≤1.

The extension of the model (2), due to [4] and denoted by HK-FGM, has the df and probability density function (pdf)

(3)FX,Y(x,y)=FX(x)FY(y){1+λ(1−F X p(x))(1−F Y p(y))},−1≤λ≤1,p≥1

and

(4)fX,Y(x,y)=fX(x)fY(y){1+λ((1+p)FX(x)−1)((1+p)FY(y)−1)},−1≤λ≤1,p≥1,

respectively, where fX(x) and fY(y) are the pdf’s of the rv’s X and Y respectively. The admissible range of the associated parameter λ is −max(1,p)−2≤λ≤p−1, and since p≥1, this admissible rang is p−2≤λ≤p−1.

Remark 1.1

It is worth mentioning that, under the conditions p>0 and −max(1,p)−2≤λ≤p−1, the model (4) is a bona fide (i.e., FX,Y(x,y) is a genuine bivariate df. However, when 0<p<1 the model (4) becomes very poor and is not allowing any improvement of the positive correlation compared to the classical FGM model (see, [6]). Therefore most of authors who tackle the model (4) postulate the condition p>1.

[7] studied some properties of the model (3) and (4)) for bivariate-generalized exponential (GE) distribution (denoted by MTBGED). Also, they studied some distributional properties of concomitants of order statistics as well as record values of this model. Moreover, they obtained some recurrence relations between moments of concomitants of order statistics. Recently, [8] extended all the results of [7] to HK-FGM family for bivariate-GE distribution (denoted by HK-FGMGE). Moreover, some new results, which were not be obtained by [7], for FGM family, were given. Finally, [8] studied the asymptotic behavior of the concomitants of order statistics and made some corrections of [7].

[9] proposed a new generalization of FGM model (3), with marginals FU(u)=u and FV(v)=v, 0≤u,v≤1, by

(5)FU,V(u,v)=uv[1+λ(1−u p)(1−v p)]m,−1≤λ≤1,p>0,

where the admissible range of the parameter λ is −min(1,1mp2)≤λ≤1mp. The Spearman’s correlation coefficient of the model (5) is

(6)ρ★=12∑j=1nλ j(nj)Γ( j+1)Γ2ppΓj+1+2p2.

Clearly, if put m=p=1 in (6) then we get ρ⋆ for the model (2), while if put m=1 then we get ρ⋆ for the model (3).

A rv X is a two-parameter GE if it has the df

(7)FX(x)=(1−exp(−θx))α,x>0;θ>0,α>0,

and the pdf

(8)fX(x)=αθ(1−exp(−θx))α−1exp(−θx).

This distribution is a generalization of the exponential distribution and is more flexible, for being that, the hazard function of the exponential distribution is constant, but the hazard function of GE distribution can be constant, increasing or decreasing. [10] showed that the kth moment of GE(θ;α) is

μk(θ,α)=αk!θk ∑i=0ℵ(α−1)(−1)i(i+1)k+1α−1i,

where ℵ(x)=∞, if x is non-integer and ℵ(x)=x, if x is integer. Moreover, the mean, variance and moment generating function of GE(θ;α) are given, respectively, by

(9)μ1(θ,α)=E(X)=B(α)θ,Var(X)=C(α)θ2andMX(t)=αβ(α,1−tθ),

where B(α)=Ψ(α+1)−Ψ(1), C(α)=Ψ′(1)−Ψ′(α+1), β(a,b)=Γ(a)Γ(b)Γ(a+b) and Ψ(.) is the digamma function, while Ψ′(.) is its derivation (the trigamma function).

In this paper we studied some properties of the model (5) for bivariate-GE distribution (denoted by GFGM-GE). Also, we studied some distributional properties of concomitants of order statistics as well as record values of this model. Moreover, some recurrence relations between moments of concomitants of order statistics are obtained. It is more suitable for achievement our aim to put the model (5) in the following form (by using binomial expansion):

(10)FX,Y(x,y)=FX(x)FY(y)∑i=0mλimi(1−F Xp(x))i(1−F Yp(y))i.

In this case the pdf of the model (1.10) is given by

fX,Y(x,y)=fX(x)fY(y)1+∑i=1mλi(mi)(1−F Xp(x))i−1(1−(1+ip)F Xp(x)) .(1−F Yp(y))i−1(1−(1+ip)F Yp(y))1+∑i=1m(1−FXp(x))i−1.

2. SOME PROPERTIES OF GFGM-GE (θ1,α1;θ2,α2)

In this section we determined the correlation coefficient the model GFGM-GE (θ1,α1;θ2,α2). By using the Hoeffding formula (see [11]), we get

(11)COV(X,Y:λ;α1,α2;p;m)= ∫0∞∫0∞[FX,Y(x,y)−FX(x)FY(y)]dxdy =∑i=1mλimi ∫0∞∫0∞FX(x)FY(y)(1−F X p(x))i(1−F Y p(y))idxdy =1θ1θ2∑i=1mλimiI1I2,

where

It= ∫01ξαt(1−ξαtp)i11−ξdξ=∑j=0∞∫01ξαt+j(1−ξαtp)idξ =1αtp∑j=0∞∫01ξαt+j+1αtp−1(1−ξ)idξ=1αtp∑j=1∞β(αt+j+1αtp,i+1), t=1,2

and β(a,b) is the usual Beta function. Therefore, the correlation coefficient of the model GFGM-GE θ1,α1;θ2,α2 is given by

(12)ρ(X,Y:λ;α1,α2;p;m)=1p2∑i=1mλimi∏t=121C(αt)αt∑j=1∞β(αt+j+1αtp,i+1).

Remark 2.1

For m=1, the correlation coefficient (12) reduces to

ρ(X,Y:λ;α1,α2;p;1)=λD(α1,p)D(α2,p)C(α1)C(α2),

where D(αi,p)=B(αi(1+p))−B(αi), i=1,2. The preceding formula was derived by [8].

The following theorem gives some interesting properties of the model GFGM-GE (θ1,α1;θ2,α2).

Theorem 2.1

Let −min1,1mp2≤λ≤1mp. Then,

(13)ρ(X,Y:λ;α1,α2;p;m)>ρ(X,Y:λ;α1,α2;p,1),∀m>1.

Moreover,

limα2→0α1→0ρ(X,Y:λ;α1,α2;p;m)=0

and

(14)limα1→8α2→8ρ(X,Y:λ;α1,α2;p;m)=ρ(X,Y:λ;p;m) =6π2∑i=1mλimi∫01(1−zp)2i1logzdz2.

Finally, a more accessible formula for ρ(X,Y:λ;p;m) is given by

(15)ρ(X,Y:λ;p;m)=6π2∑i=1mλimi∑j=1i(−1)i−jijlog(1+pj)2.

Proof of Theorem 2.1. The proof of the relation (13) follows immediately from the fact that the function β(x,y) is non-increasing in both x and y, and i≥1. In order to prove the relation (14), we start with the relation (11), where

(16)It=∫01ξαt1−ξαtpi11−ξdξ=1αt∫011−zpiz1αt1−z1αtdz

(by using the transformation z=ξαt). On the other hand, for any 0<z<1, we have

(17)limαt→∞z1αtαt(1−z1αt)=limθ→0θzθ1−zθ=−1logz

and

(18)limαt→∞C(αt)=π26.

Combining (16), (17) and (18), we get the relation (14). By using a result of [12] we get

∫01(zp−1)i1logzdz= ∑j=1i(−1)i−jijlog(1+pj),

which immediately proves (15).

Remark 2.2

From (15) with m=1, we have

ρ(X,Y:λ;p;1)=6π2log2(1+p),

which coincides with the result of [8].

In Table 1, we give some values of the correlation coefficient ρλmax=ρ(X,Y:λmax;p;m) defined in (15), where λmax=1mp. The result of this table shows that, when 0<p<1, the model becomes very poor and is not allowing any improvement of the positive correlation compared to the HK-FGM model or even the classical model FGM. Therefore, for the practical purposes, we always take p>1 (see also Remark 1.1). Moreover, Table 1 shows that the values of ρλmax are irregularly fluctuated with changing the values of the parameters p and m. However the maximum value of ρλmax, which is 0.4214, reaches at m=25 and p=4.5. This maximum value is a significant improvement comparing with the upper bound “0.2921”obtained by [7] for the model MTBGED and is also a satisfactory improvement comparing with the upper pound “0.3937”obtained by [8] for the model HK-FGMGE. This fact gives a satisfactory motivation to deal with GFGM-GE rather than MTBGED and HK-FGMGE.

m	p	λmax	ρλmax	m	p	λmax	ρλmax
1	0.01	100	0.006	1	3	0.33333	0.389
3	0.01	33.333	0.0062	50	3	0.0067	0.42
1	0.1	10	0.055	1	4	0.25	0.394
25	0.1	0.4	0.057	50	4	0.005	0.4210
1	1	1	0.29	25	4.5	0.0089	0.4214
50	1	0.02	0.32	1	5	0.2	0.399
100	1	0.01	0.33	25	5	0.008	0.411
3	1.5	0.2222	0.3513	50	5	0.004	0.417
1	2	0.5	0.366	1	6	0.16667	0.3836
3	2	0.16667	0.3849	50	6	0.0033	0.3983
50	2	0.01	0.399	500	6	0.00033	0.3988
100	2	0.005	0.3999	1000	6	0.000166	0.3998

Table 1

Some different values of the correlation coefficient for the family GFGM-GE.

3. CONCOMITANTS OF ORDER STATISTICS BASED ON GFGM-GE

In the last two decades much attention has been paid to the concomitants of order statistics models, see, e.g. [7,8,13,14]. The importance of these models increased owing to the rigorous demand of natural science, social science and economics to study the problems which generally depend on two different dependent characteristics. Let (X1,Y1),(X2,Y2),...,(Xn,Yn) be a random sample from a bivariate df FX,Y(x,y). If we arrange the X−variate in ascending order as X1:n≤X2:n≤....≤Xn:n, then, the Y−variate paired with these order statistics are denoted by Y[1:n],Y[2:n],...,Y[n:n] and termed the concomitants of order statistics. The concept of concomitants of order statistics was first introduced by [15] and almost simultaneously under the name of induced order statistics by [16]. These concomitant order statistics are of interest in selection and prediction problems based on the ranks of the X’s. Another application of concomitants of order statistics is in ranked-set sampling. It is a sampling scheme for situations where measurement of the variable of primary interest for sampled items is expensive or time-consuming while ranking of a set of items related to the variable of interest can be easily done. A comprehensive review of ranked-set sampling can be found in [17]. For a recent comprehensive review of possible applications of the concomitants of order statistics, see [18].

Let X∼GE(θ1;α1) and Y∼GE(θ2;α2). Since the conditional pdf of Y[r:n] given X[r:n]=x is fY[r:n]|Xr:n(y|x)=fY|X(y|x), then the pdf of Y[r:n] is given by

(19)f[r:n](y)= ∫0∞fY|X(y|x)fr:n(x)dx,

where fr:n(x)=1β(r,n−r+1)F Xr−1(x)(1−FX(x))n−rfX(x) is the pdf of the rth order statistic Xr:n and fY|X(y|x) can be computed by using (7), (8) and (10). The following theorem gives the useful representation of the pdf f[r:n](y).

Theorem 3.1

Let Uj∼GE(θ2,α2(jp+1)) and Vi∼GE(θ2,α2((j+1)p+1)). Then

f[r:n](y)=fY(y)+∑i=1mλimiSr,n(t)(p,i)

×∑j=0ℵ(i−1)i−1j(−1)j1jp+1fUj(y)−1+pi(j+1)p+1fVj(y),t=1,2,

where

Sr,n(1)(p,i)=1p∑l=0n−rn−rl(−1)lΔl:r,n⋆(p,i),

Δl:r,n⋆(p,i)=β(r+lp,i)−(1+pi)β(r+lp+1,i)β(r,n−r+1),

Sr,n(2)(p,i)= ∑l=0ℵ(i−1)i−1l(−1)lΔl:r,n⋆⋆(p,i)

and

Δl:r,n⋆⋆(p,i)=β(lp+r,n−r+1)−(1+pi)β((l+1)p+r,n−r+1)β(r,n−r+1).

Proof.

Clearly, the relation (19), can be written in the form

f[r:n](y)=fY(y)1+∑i=1mλimi1−F Yp(y)i−11−(1+pi)F Yp(y)(J1−(1+pi)J2)

=fY(y)+ ∑i=1mλimi(J1−(1+pi)J2)∑j=0ℵ(i−1)i−1j(−1)jfUj(y)jp+1−(1+pi)fVj(y)(j+1)p+1,

where

(20)J1β(r,n−r+1)=∫0∞1−F Xp(x)i−1F Xr−1(x)(1−FX(x))n−rfX(x)dx =∑ℓ=0n−r(n−rℓ)(−1)ℓ∫0∞(1−F Xp(x))i−1F Xℓ+r−1(x)fX(x)dx =∑ℓ=0n−r(n−rℓ)(−1)ℓα1θ1∫0∞(1−(1−e−θ1x)α1p)i−1(1−e−θ1x)α1(ℓ+r)−1e−θ1xdx = ∑ℓ=0n−r(n−rℓ)(−1)ℓα1∫01(1−ξα1p)i−1ξα1(ℓ+r)−1dξ =1p∑ℓ=0n−r(n−rℓ)(−1)ℓ∫01tℓ+rp−1(1−t)i−1dt=1p∑ℓ=0n−r(n−rℓ)(−1)ℓβ(r+ℓp,i)

(by using the substitute ξ=1−e−θ1x and then use the substitution t=ξα1p) and xs

(21)J2β(r,n−r+1)=∫0∞1−F Yp(x)i−1F Xp+r−1(x)(1−FX(x))n−rfX(x)dx =∑ℓ=0n−r(n−rℓ)(−1)ℓ∫0∞(1−F Xp(x))i−1F Xℓ+p+r−1(x)fX(x)dx =∑ℓ=0n−r(n−rℓ)(−1)ℓα1θ1∫0∞(1−(1−e−θ1x)α1p)i−1(1−e−θ1x)α1(ℓ+p+r)−1e−θ1xdx =1p∑ℓ=0n−r(n−rℓ)(−1)ℓα1∫01(1−ξα1p)i−1ξα1(ℓ+p+r)−1dξ=1p∑ℓ=0n−r(n−rℓ)(−1)ℓβ(r+ℓp+1,i)

(by using the substitute ξ=1−e−θ1x and then use the substitution t=ξα1p). Therefore, by combining (20) and (21), we get

Sr,n(1)(p,i)=J1−(1+pi)J2=1p∑ℓ=0n−rn−rℓ(−1)ℓΔℓ:r,n⋆(p,i),

where

Δℓ:r,n⋆(p,i)=β(r+ℓp,i)−(1+pi)β(r+ℓp+1,i)β(r,n−r+1).

On the other hand,

J1β(r,n−r+1)= ∫0∞1−F Xp(x)i−1F Xr−1(x)(1−FX(x))n−rfX(x)dx =α1θ1 ∫0∞(1−(1−e−θ1x)α1p)i−1(1−e−θ1x)α1r−1(1−(1−e−θ1x)α1)n−re−θ1xdx.

Upon substituting ξ=1−e−θ1x, we get

J1β(r,n−r+1)=α1 ∫01(1−ξα1p)i−1ξα1r−1(1−ξα1)n−rdξ.

Moreover, by using the substitution u=ξα1, we get

J1β(r,n−r+1)=1p∫01ur−1(1−u)n−r(1−up)i−1du,

after simple calculations, we get

(22)J1β(r,n−r+1)=∑ℓ=0ℵ(i−1)(ℓi−1)(−1)ℓβ(ℓp+r,n−r+1).

Similarly

(23)J2β(r,n−r+1)=∑ℓ=0ℵ(i−1)(ℓi−1)(−1)ℓβ((ℓ+1)p+r,n−r+1).

Therefore, by combining (22) and (23), we get

Sr,n(2)(p,i)=(J1−(1+pi)J2)= ∑j=0ℵ(i−1)i−1ℓ(−1)ℓΔℓ:r,n⋆⋆(p,i),

where

Δℓ:r,n⋆⋆(p,i)=β(ℓp+r,n−r+1)−(1+pi)β((ℓ+1)p+r,n−r+1)β(r,n−r+1).

This completes the proof of the theorem.

The following corollary is a direct consequence of Theorem 3.1.

Corollary 3.1

Let μ[r:n](k)=E(Y[r:n]k), k=1,2,…. Then,

(24)μ[r:n](k)=μk(θ2,α2)+∑i=1mλimiSr,n(t)(p,i)D(k;p,i) =μk(θ2,α2)+∑i=1mλimiSr,n(t)(p,i)∑j=0ℵ(i−1)i−1j(−1)jE(Ujk)jp+1−(1+pi)E(Vjk)(j+1)p+1,t=1,2,

where, E(Ujk) and E(Vjk) can be easily computed by using the relation (9). Therefore, the mean μ[r:n]=E(Y[r:n]) is given by

(25)μ[r:n]=B(α2)θ2+∑i=1mλimiSr,n(t)(p,i)D(1,p,i),t=1,2,

where

D(1,p,i)=1θ2 ∑j=0ℵ(i−1)i−1j(−1)jB(α2(jp+1))jp+1−(1+pi)B(α2((j+1)p+1))(j+1)p+1.

Corollary 3.2

When m=1, we get μ[r:n] for the family HK-FGM-GE, (see [8])

(26)μ[r:n]=1θ2(1+λΔ0:r,n⋆⋆(p,1))B(α2)−λΔ0:r,n⋆⋆(p,1)B(α2(p+1)).

Remark 3.1

It is worth mentioning that, if we replace μk(θ2,α2) by E(Yk). Moreover, Uj and Vj in D(k;p,i) are taken to be such that Uj∼F Yjp+1(y) and Vj∼F Y(j+1)p+1(y), then the representation (24) holds for any two arbitrary distributions FX(x) and FY(y).

Now, by using the two representations in relation (24), as well as (25), at t=1 and t=2, we can derive some useful recurrence relations satisfied by the moments μ[r:n](k),k=1,2,…. The following theorem give a new recurrence relation by using the representation at t=1. It is worth mentioning that this recurrence relation was not proved even for the model FGM-GE. Moreover, in view of Remark 3.1, all the next recurrence relations are satisfied for arbitrary distributions FX(x) and FY(y), if only we would consider the obvious changes illustrated in Remark 3.1.

Theorem 3.2.

Let p be an integer and k=1,2,…., then

1Qr,n(2)(p)μ[r+2p:n+2p](k)+1Qr,n(1)(p)μ[r+p:n+p](k)−2μ[r:n](k)=1Qr,n(2)(p)+1Qr,n(1)(p)−2μk(θ2,α2)−1p∑i=1mλimiD(k,p,i)∑l=0n−rn−rl(−1)lηl:r,n(p,i),

where

Qr,n(j)(p)=Γ(r)Γ(n+jp+1)Γ(r+jp)Γ(n+1),j=1,2,

and

ηl:r,n(p,i)=3−p(1+i)r+p(1+i)+lβ(r+lp,i+1)β(r,n−r+1) −(1+pi)3−p(1+i)r+p(2+i)+lβ(r+lp+1,i+1)β(r,n−r+1).

Proof.

Starting with Δl:r,n⋆(p,i), after simple calculations, we can show that

Δl:r+p,n+p⋆(p,i)=β(r+lp+1,i)β(r+p,n−r+1)−(1+pi)β(r+lp+2,i)β(r+p,n−r+1) =(n+p)!(r−1)!(r+p−1)!(n!)[r+lpr+lp+iβ(r+lp,i)β(r,n−r+1)−(1+pi)r+lp+1r+lp+i+1β(r+lp+1,i)β(r,n−r+1)].

On the other hand, since Qr,n(1)(p)=(n+p)!(r−1)!(r+p−1)!(n!), we get Δl:r+p,n+p⋆(p,i)=Qr,n(1)(p)[Δl:r,n⋆(p,i)−ξ1], where

ξ1=β(r+lp,i+1)−(1+pi)β(r+lp+1,i+1)β(r,n−r+1).

Therefore,

(27)1Qr,n(1)(p)Δl:r+p,n+p⋆(p,i)−Δl:r,n⋆(p,i)=−ξ1.

Similarly, after some calculations, we get

Δl:r+2p,n+2p⋆(p,i)=β(r+lp+2,i)β(r+2p,n−r+1)−(1+pi)β(r+lp+3,i)β(r+2p,n−r+1) =(n+2p)!(r−1)!(r+2p−1)!(n!)r+lp(r+lp+1)r+lp+i(r+lp+i+1)β(r+lp,i)β(r,n−r+1) −(1+pi)(r+lp+1)(r+lp+2)(r+lp+i+1)(r+lp+i+2)β(r+lp+1,i)β(r,n−r+1).

Thus, we get

(28)Δl:r+2p,n+2p⋆(p,i)=Qr,n(2)(p)[Δl:r,n⋆(p,i)−ξ2],

where

ξ2=ξ1+l+rpβ(r+lp,i+1)(r+lp+i+1)β(r,n−r+1)−(1+pi)(1+l+rp)β(r+lp+1,i+1)(l+rp+i+2)β(r,n−r+1).

Therefore,

(29)1Qr,n(2)(p)Δj:r+2p,n+2p⋆(p,i)−Δl:r,n⋆(p,i)=−ξ2.

Put x=r+lp, we can write

ξ2=i(2x+i+1)(x+i)(x+i+1)β(x,i)β(r,n−r+1)−(1+pi)i(2x+i+3)(x+i+1)(x+i+2)β(x+1,i)β(r,n−r+1) =(2x+i+1)(x+i+1)β(x,i+1)β(r,n−r+1)−(1+pi)(2x+i+3)(x+q+2)β(x+1,q+1)β(r,n−r+1) =ξ1+x(x+i+1)β(x,i+1)β(r,n−r+1)−(1+pi)(x+1)(x+i+2)β(x+1,i+1)β(r,n−r+1).

Therefore, we can easily show that

ξ2+ξ1=(2+xx+i+1)β(x,i+1)β(r,n−r+1)−(1+pi)(2+x+1x+i+2)β(x+1,i+1)β(r,n−r+1)=ηl:r,n(p,i).

Thus by combining this equality with (27), (28), (29) and (24), at t=1, the proof of the theorem follows immediately.

Corollary 3.3.

For m=1, we get for the family HK-FGM-GE, (see [8])

(30)1Qr,n(2)(p)μ[r+2p:n+2p](k)+1Qr,n(1)(p)μ[r+p:n+p](k)−2μ[r:n](k)=1Qr,n(2)(p)+1Qr,n(1)(p)−2μk(θ2,α2)−λD(k,p,1)pβ(r,n−r+1)∑l=03cl(p)β(r+lp,n−r+1),

where c0(p)=2p, c1(p)=−p(2p+3), c2(p)=p2, c3(p)=p(1+p) and D(k,p,1)=μk(θ2,α2)−μk(θ2,α2(1+p)). The following theorem, which is relying on the representation (27), at t=2, given some recurrence relations satisfied by the kth moments of concomitants of order statistics for any arbitrary distributions.

Theorem 3.3.

For any k=1,2,…., we have

(31)μ[r+2:n](k)−μ[r:n](k)μ[r+1:n](k)−μ[r:n](k)=∑i=1mλimiD(k,p,i)Ωr,n(1)(p,i)(r+1)∑i=1mλimiD(k,p,i)Ωr,n(2)(p,i)

and

(32)μ[r:n−2](k)−μ[r:n](k)μ[r:n−1](k)−μ[r:n](k)=∑i=1mλimiD(k,p,i)Ωr,n(1)(p,i)(n−1)∑i=1mλimiD(k,p,i)Ωr,n(2)(p,i),

where

Ωr,n(1)(p,i)=∑l=0ℵ(i−1)i−1l(−1)llp(lp+2r+1)β(lp+r,n−r+1) −(1+pi)(1+l)p((l+1)p+2r+1)β((l+1)p+r,n−r+1),

and

Ωr,n(2)(p,i)=∑l=0ℵ(i−1)i−1l(−1)llpβ(lp+r,n−r+1)−(1+pi)(1+l)pβ((l+1)p+r,n−r+1).

Proof.

It is easy to check that

Δl:r+1,n⋆⋆(p,i)=β(lp+r+1,n−r)−(1+pi)β((l+1)p+r+1,n−r)β(r+1,n−r) =lp+rn−rβ(lp+r,n−r+1)−(1+pi)(l+1)p1+rn−rβ((l+1)p+r,n−r+1)rn−rβ(r,n−r+1) =Δl:r,n⋆⋆(p,i)+lpβ(lp+r,n−r+1)−(1+pi)(l+1)pβ((l+1)p+r,n−r+1)rβ(r,n−r+1).

Therefore, we get

Sr+1,n(2)(p,i)−Sr,n(2)(p,i)=1rβ(r,n−r+1)

×∑l=0ℵ(i−1)i−1l(−1)llpβ(lp+r,n−r+1)−(1+pi)(1+l)pβ((l+1)p+r,n−r+1).

Moreover, we have

Δl:r+2,n⋆⋆(p,i)=β(lp+r+2,n−r−1)−(1+pi)β((l+1)p+r+2,n−r−1)β(r+2,n−r−1). =(lp+r)(lp+r+1)(n−r)(n−r−1)β(lp+r,n−r+1)−(1+pi)((l+1)p+r)((l+1)p+r+1)(n−r)(n−r−1)β((l+1)p+r,n−r+1)r(r+1)(n−r)(n−r−1)β(r,n−r+1).

Thus, we get

Sr+2,n(2)(p,i)−Sr,n(2)(p,i)=λr(r+1)β(r,n−r+1)

×∑l=0ℵ(i−1)i−1l(−1)llp(lp+2r+1)β(lp+r,n−r+1)

−(1+pi)(1+l)p((l+1)p+2r+1)β((l+1)p+r,n−r+1)∑i=0n(i−1)(i−1l)].

Therefore,

μ[r+2:n](k)−μ[r:n](k)μ[r+1:n](k)−μ[r:n](k)=∑i=1mλimiD(k,p,i)Ωr,n(1)(p,i)(r+1)∑i=1mλimiD(k,p,i)Ωr,n(2)(p,i).

Similarly, we have

Δl:r,n−1⋆⋆(p,i)=β(lp+r,n−r)−(1+pi)β((l+1)p+r,n−r)β(r,n−r). =lp+nn−rβ(lp+r,n−r+1)−(1+pi)(l+1)p+nn−rβ((l+1)p+r,n−r+1)nn−rβ(r,n−r+1) =Δl:r,n⋆⋆(p,i)+lpβ(lp+r,n−r+1)−(1+pi)(l+1)pβ((l+1)p+r,n−r+1)nβ(r,n−r+1).

Consequently, we get

Sr,n−1(2)(p,i)−Sr,n(2)(p,i)=λnβ(r,n−r+1)

×∑l=0ℵ(i−1)i−1j(−1)llpβ(lp+r,n−r+1)−(1+pi)(1+l)pβ((l+1)p+r,n−r+1).

Moreover,

Δl:r,n−2⋆⋆(p,i)=β(lp+r,n−r−1)−(1+pi)β((l+1)p+r,n−r−1)β(r,n−r−1) =(lp+n)(lp+n−1)(n−r)(n−r−1)β(lp+r,n−r+1)−(1+pi)((l+1)p+n)((l+1)p+n−1)(n−r)(n−r−1)β((l+1)p+r,n−r+1)n(n−1)(n−r)(n−r−1)β(r,n−r+1) =Δl:r,n⋆⋆(p,i) +lp(lp+2n−1)β(lp+r,n−r+1)−(1+pi)(l+1)p((l+1)p+2n−1)β((l+1)p+r,n−r+1)n(n−1)β(r,n−r+1).

Thus,

Sr+2,n(2)(p,i)−Sr,n(2)(p,i)=λn(n−1)β(r,n−r+1)

×∑l=0ℵ(i−1)i−1j(−1)llp(lp+2n−1)β(lp+r,n−r+1)

−(1+pi)(1+l)p((l+1)p+2n−1)β((l+1)p+r,n−r+1)∑i=0n(i−1)(i−1j)].

Therefore,

μ[r:n−2](k)−μ[r:n](k)μ[r:n−1](k)−μ[r:n](k)=∑i=1mλimiD(k,p,i)Ωr,n(1)(p,i)(n−1)∑i=1mλimiD(k,p,i)Ωr,n(2)(p,i).

This completes the proof of the theorem.

Corollary 3.4.

At m=1, we get for the HK-FGM-GE family (see [8])

(r+1)μ[r+2:n](k)=(2r+p+1)μ[r+1:n](k)−(r+p)μ[r:n](k)

and

(n+p)μ[r:n](k)=(2n+p−1)μ[r:n−1](k)−(n−1)μ[r:n−2](k).

Theorem 3.4.

For any k=1,2,…, we have

μ[r+2:n](k)+μ[r+1:n](k)−2μ[r:n](k)=∑i=1mλimiD(k,p,i)Φr,n(1)(p,i),

where

Φr,n(1)(p,i)=pr(r+1)β(r,n−r+1)∑ℓ=0ℵ(i−1)i−1ℓ(−1)ℓℓ(ℓp+3r+2)β(ℓp+r,n−r+1)

−(1+pi)(1+ℓ)((ℓ+1)p+3r+2)β((ℓ+1)p+r,n−r+1).

Moreover,

μ[r:n−2](k)+μ[r:n−1](k)−2μ[r:n](k)=∑i=1mλimiD(k,p,i)Φr,n(2)(p,i),

where

Φr,n(2)(p,i)=pn(n−1)β(r,n−r+1)∑ℓ=0ℵ(i−1)i−1ℓ(−1)ℓℓ(ℓp+3n−2)β(ℓp+r,n−r+1)

−(1+pi)(1+ℓ)((ℓ+1)p+3n−2)β((ℓ+1)p+r,n−r+1).

Proof.

The proof of the theorem is similar to the proof of Theorem 3.3, with the exception that the addition operation supersedes the subtraction operation.

Corollary 3.5.

At m=1, we have for HK-FGM-GE family (see [8]),

μ[r+2:n](k)+μ[r+1:n](k)−2μ[r:n](k)=−λp(1+p)(p+3r+2)β(r+p,n−r+1)r(r+1)β(r,n−r+1)D(k,p,1).

Moreover, for m=1 we have for the HK-FGM-GE family (see [8]),

μ[r:n−2](k)+μ[r:n−1](k)−2μ[r:n](k)=−λp(1+p)(p+3n−2)β(r+p,n−r+1)n(n−1)β(r,n−r+1)D(k,p,1).

4. CONCOMITANTS OF RECORD VALUES BASED ON GFGM-GE MODEL

A new topic in record values theory is concomitants of record values as analogue to concomitants of order statistics, which was suggested for the first time and studied by [19]. The most important use of concomitants of record values arises in experiments in which a specified characteristic’s measurements of an individual are made sequentially, and only values that exceed or fall below the current extreme value are recorded. So the only observations are bivariate record values, i.e., records and their concomitants. Let {(Xi,Yi)},i=1,2,… be a random sample from the model GFGM-GE (θ1,α1;θ2,α2). When the experimenter interests in studying just the sequence of records of the first component Xi′s the second component associated with the record value of the first one is termed as the concomitant of that record value. The concomitants of record values has many applications, e.g., see [20] and [21]. Some properties from concomitants of record values can be found in [14] and [22]. Let {Rn,n≥1} be the sequence of record values in the sequence of X′s while R[n] be the corresponding concomitant. [19] obtained the pdf of concomitant of nth record value for n≥1, as ℛ[n](y)=∫0∞fY(y|x)hn(x)dx, where hn(x)=1Γ(n)(−log(1−FX(x)))n−1fX(x) is the pdf of Rn. The following theorem gives a useful representation for the pdf ℛ[n](y), as well as the kth moments concomitants of record values based on the GFGM-GE model.

Theorem 4.1.

Let Uj and Vj be defined as in Theorem 3.1. Then,

(33)ℛ[n](y)=fY(y)+∑i=1mλimiS⋆(p,i)−(1+pi)S⋆⋆(p,i) ×∑j=0ℵ(i−1)i−1i(−1)j1jp+1fUj(y)−1+pi(j+1)p+1fVj(y),

where

S⋆(p,i)=∑ℓ=0ℵ(i−1)∑c=0ℵ(ℓp)(−1)ℓ+ci−1jℓpc(c+1)n

and

S⋆⋆(p,i)=∑ℓ=0ℵ(i−1)∑c=0ℵ(p(ℓ+1))(−1)ℓ+ci−1ℓp(ℓ+1)ℓ(ℓ+1)n.

Moreover, if μRn(k)=E(Rnk), k=1,2,…, then,

(34)μRn(k)=μk(θ2,α2)+∑i=1mλimiS⋆(p,i)−(1+pi)S⋆⋆(p,i)D(k;p,i).

Proof.

Clearly, (34) is a simple consequence of (33). Therefore, we have only to prove the relation (33). Now, we have

ℛ[n](y)=fY(y)+∑i=1mλimi(1−FY(y)p(y))i−1[1−(1+pi)F Yp(y)] ×∫0∞(1−F Xp(x))i−1[1−(1+pi)F Xp(x)](−log(1−FX(x)))n−1Γ(n)fX(x)dx =fY(y)+∑i=1mλimi∑j=0ℵ(i−1)i−1i(−1)j1jp+1fUj(y)−1+pi(j+1)p+1fVj(y) ∑ℓ=0ℵ(j−1)i−1ℓ(−1)ℓ1Γ(n)∫0∞F Xℓp(x)(−log(1−FX(x)))n−1fX(x)dx −(1+pi)∑ℓ=0ℵ(i−1)i−1j(−1)ℓ∫0∞F X(ℓ+1)p(x)(−log(1−FX(x)))n−1fX(x)dx.

Upon using the transformation −log(1−FX(x))=t in the above two integrations and applying the binomial theorem on the terms (1−e−t)ℓp and (1−e−t)(ℓ+1)p, in the first and second integrations, respectively, we get the representation (33).

Corollary 4.1.

For m=1, i.e., for HK-FGM-GE model, we get

μRn(k)=μk(θ2,α2)+λS⋆(p,1)−(1+p)S⋆⋆(p,1)D(k;p,1) =μk(θ2,α2)+λ[μk(θ2,α2)−(1+p)μk(θ2,α2p)][1−(1+p)∑ℓ=0ℵ(p)(−1)ℓpℓ(ℓ+1)n].

Moreover, For m=1 and p=1, i.e., for FGM-GE family, we get

μRn(k)=μk(θ2,α2)+λS⋆(1,1)−(1+p)S⋆⋆(1,1)D(k;1,1) =μk(θ2,α2)1−λ(2−(n−1)−1).

Proof.

The proof is obvious, since it follows after simple algebra.

REFERENCES

1.D. Morgenstern, Mitt. Math. Stat., Vol. 8, 1956, pp. 234-235. https://ci.nii.ac.jp/naid/10011938351

2.E. J. Gumbel, J. Am. Stat. Assoc., Vol. 55, 1974, pp. 698-707.

3.D. J. G. Farlie, Biometrica, Vol. 47, 1960, pp. 307-323.

4.J. S. Huang and S. Kotz, Metrika, Vol. 49, 1999, pp. 135-145.

5.N. L. Johnson and S. Kotz, Commun. Stat., Vol. 4, 1975, pp. 415-427.

6.I. Bairamov and K. Bayramoglu, J. Multivar. Anal., Vol. 113, 2013, pp. 106-115.

7.S. Tahmasebi and A. Jafari, Bull. Malays. Math. Sci. Soc., Vol. 38, 2015, pp. 1411-1423.

8.H. M. Barakat, E. M. Nigm, and A. H. Syam, Malays. Math. Sci. Soc., Vol. 42, 2019, pp. 337-353.

9.H. Bekrizadeh, G. A. Parham, and M. R. Zadkarmi, Appl. Math. Sci., Vol. 71, 2012, pp. 3527-3533. http://m-hikari.com/ams/ams-2012/ams-69-72-2012/bekrizadehAMS69-72-2012.pdf

10.R. D. Gupta and D. Kundu, Aust. NZ. J. Stat., Vol. 41, 1999, pp. 173-188.

11.E. L. Lehmann, Math. Stat., Vol. 37, 1966, pp. 1137-1153.

12.A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series: Elementary Functions, Amsterdam, 1998. https://trove.nla.gov.au/work/16532475

13.I. Bairamov, S. Kotz, and M. Becki, J. Appl. Statist., Vol. 28, 2001, pp. 521-536.

14.M. Ahsanullah, Bull. Malays. Math. Sci. Soc., Vol. 32, 2009, pp. 101-117. https://www.academia.edu/29357995

15.H. A. David, Bull. Inter. Statist. Inst., Vol. 45, 1973, pp. 295-300.

16.P. K. Bhattacharya, Ann. Statist., Vol. 2, 1974, pp. 1034-1039.

17.Z. Chen, Z. Bai, and B. Sinha, Ranked Set Sampling: Theory and Applications, New York, 2004.

18.H. A. David and H. N. Nagaraja, N. Balakrishnan and C. R. Rao (editors), Handbook of Statistics, Elsevier, Amsterdam, 1998, pp. 487-513.

19.R. L. Houchens, Record Value Theory and Inference, Ann Arbor, 1984. https://www.worldcat.org/title/record-value-theory-and-inference/oclc/10829025

20.O. M. Bdair and M. Z. Raqab, Bull. Malays. Math. Sci. Soc., Vol. 2, 2013, pp. 23-38. http://www.kurims.kyoto-u.ac.jp/EMIS/journals/BMMSS/pdf/v37n2/v37n2p14.pdf

21.B. C. Arnold, N. Balakrishnan, and H. N. Nagaraja, Records, Wiley Series in Probability and Statistics, Probability and Statistics, New York, 1998.

22.M. Ahsanullah and M. Shakil, Bull. Malays. Math. Sci. Soc., Vol. 36, 2013, pp. 625-635. https://www.emis.de/journals/BMMSS/pdf/acceptedpapers/2010-10-056_R3.pdf

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Journal: Journal of Statistical Theory and Applications
Volume-Issue: 18 - 3
Pages: 309 - 322
Publication Date: 2019/09/19
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.190822.001 How to use a DOI?
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

ris enw bib

TY  - JOUR
AU  - H. M. Barakat
AU  - E. M. Nigm
AU  - M. A. Alawady
AU  - I. A. Husseiny
PY  - 2019
DA  - 2019/09/19
TI  - Concomitants of Order Statistics and Record Values from Generalization of FGM Bivariate-Generalized Exponential Distribution
JO  - Journal of Statistical Theory and Applications
SP  - 309
EP  - 322
VL  - 18
IS  - 3
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.190822.001
DO  - 10.2991/jsta.d.190822.001
ID  - Barakat2019
ER  -

download .riscopy to clipboard