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Kyungpook Mathematical Journal 2022; 62(2): 389-405

Published online June 30, 2022

Fisher Information and the Kullback-Leibler Distance in Concomitants of Generalized Order Statistics Under Iterated FGM family

Haroon Mohammed Barakat and Islam Abdullah Husseiny*

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
e-mail : ishusseiny@gamil.com and hmbarakat2@gmail.com

Received: June 4, 2020; Revised: June 9, 2021; Accepted: July 6, 2021

We study the Fisher Information (FI) of m-generalized order statistics (m-GOSs) and their concomitants about the shape-parameter vector of the Iterated Farlie-Gumbel-Morgenstern (IFGM) bivariate distribution. We carry out a computational study and show how the FI matrix (FIM) helps in finding information contained in singly or multiply censored bivariate samples from the IFGM. We also run numerical computations about the FIM for the sub-models of order statistics (OSs) and sequential order statistics (SOSs). We evaluate FI about the mean and the shape-parameter of exponential and power distributions, respectively. Finally, we investigate the Kullback-Leibler distance in concomitants of m-GOSs.

Keywords: Fisher information, concomitants, generalized order statistics, IFGM family, Kullback-Leibler distance

Suppose that we have a random variable (RV) X, which has an absolutely continuous distribution function (DF) F(x;θ) and a probability density function (PDF) F(x;θ), where θ is an unknown parameter (θ may be a single or vector valued parameter), θΘ and θ is the parameter space. Under certain regularity conditions (cf. [2]), the FI about the real parameter θ contained in X is defined by Iθ(X)=Elogf(X;θ)θ2=E2logf(X;θ)θ2.  Several authors have studied FI contained in OSs and record values about the unknown parameter of the given DF F(x;θ). Among those authors are Tukey [27], Mehrotra et al. [23], Park [25], Zheng and Gastwirth [28], Abo-Eleneen and Nagaraja [2], Ahmadi and Arghami [3], Hofmann and Ngaraja [18] , Hofmann [17], and Barakat et al. [12]. The FI plays a valuable role in statistical inference through the Cramer-Rao inequality. The present paper is devoted to study the FI contained in m-GOSs and their concomitants about the shape-parameter vector of the IFGM type bivariate distribution. The model of GOSs was suggested by Kamps [20] as a unified model for ordered RVs, which includes, among others, the following sub-models: OSs, SOSs, record values, k-record values, Pfeifer's records and progressive type II censored OSs. The subclass m-GOSs of GOSs contains many important models of ordered RVs such as OSs, SOSs, lower record values, k-records, and type II censored OSs. Let n,m>1,k>0 and γi=k+(ni)(m+1), i=1,2,...,,n, be parameters. Then the RVs X1,n,m,kX2,n,m,k...Xn,n,m,k are said to be m-GOSs based on an arbitrary continuous DF F with PDF f and the survival function F¯=1F, if their joint PDF is of the form

f1,2,...,n(x1,x2,...xn)=k j=1 n1γj i=1 n1F¯m(xi)f(xi)F¯k1(xn)f(xn),

F1(1)xn...x1F1(0). The marginal PDF of rth m-GOS, Xr,n,m,k,1rn, is given by (cf. [20, 8])

fr,n,m,k(x)=Cr1(r1)!F¯γr1(x)f(x)gmr1(F(x)),

where Cr1= i=1rγi,r=1,2,...,n,gm(x)=hm(x)hm(0),x[0,1) and hm(x)=(1x)m+1(m+1), if m1, while h1(x)=log(1x). For more details about the subject of the GOSs and its applications, see [21, 10, 11, 5, 6]. The concept of concomitants of OSs, or record values, arises when we have two random samples and we sort the members of one of them (e.g., the first sample) according to corresponding values of the second random sample. Specifically, in any data collection, several characteristics may be recorded, where some of them are often considered as primary and others can be observed from the primary data automatically. The latter ones are called concomitants. Concomitants of OSs and record values can arise in several applications. The most striking application of concomitants of OSs and record values arises in selection procedures, where items or subjects may be chosen on the basis of their X characteristic, and an associated characteristic Y that is hard to measure or can be observed only later may be of interest. For more details, see [16, 4, 26]. The concept of concomitants can also be easily extended to the model of GOSs. Kamps [20] derived and studied the distribution of concomitants of the Pfeifer's record values. Also, Bairamov and Eryilmaz [7] considered the concomitants for the model of progressive type II censoring. Generally speaking, the study of concomitants of any model of ordered RVs based on the random vector (X,Y) is strongly related to the bivariate DF that governs the random vector (X,Y). Several authors have considered the concomitants of m-GOSs for different bivariate models, see, for example [1, 13, 14, 15]. One of the most efficient model of the frequently used bivariate models is the IFGM DF. The initiation of this type dates back to Huang and Kotz [19], when they used successive iterations in the original FGM distribution to increase the correlation between components. They showed that just one single iteration can result in tripling the covariance for certain marginals. In this paper, we consider the bivariate FGM with a single iteration, which is defined by

FX,Y(x,y)=FX(x)FY(y)1+λ F¯X(x) F¯Y(y)+ωFX(x)FY(y) F¯X(x) F¯Y(y),

denoted by IFGM(λ,ω). The corresponding PDF is given by

fX,Y(x,y)=fX(x)fY(y)1+λC1(x,y)+ωC2(x,y),

where C1(x,y)=(12FX(x))(12FY(y)), C2(x,y)=FX(x)FY(y)(23FX(x))(23FY(y)) and fX(x) and fY(y) are the PDFs of the RVs X and Y, respectively. When the two marginals FX(x) and FY(y) are continuous, they showed that the natural parameter space θ (the admissible set of the parameters λ and ω that makes FX,Y(x,y) is a DF) is convex, where Θ={(λ,ω):1λ1;λ+ω1;ω3λ+96λ3λ22}. Barakat et al. [13] revisited the family IFGM(λ,ω) and showed that the maximum correlation is higher than previously known. They studied some distributional properties of concomitants of OSs for the family IFGM(λ,ω). Moreover, in that paper the authors gave several applications of this model in reliability theory and showed that, the utilization of the IFGM distribution instead of FGM distribution for studying these applications gives more accurate results. It is worth mentioning that the IFGM model has the same efficiency as the well-known Huang-Kotz FGM model (see [1]), but it is more tractable and flexible.

In this paper, we investigate the properties of the FI about the vector (λ,ω)Θ (defined in the model (1.2)-(1.3)) contained in (Xr,n,m,k,Y[r,n,m,k]), i.e.,

mboxI(λ,ω)(Xr,n,m,k,Y[r,n,m,k])= Iλ (Xr,n,m,k ,Y[r,n,m,k] ) Iλ,ω (Xr,n,m,k ,Y[r,n,m,k] ) Iλ,ω (Xr,n,m,k ,Y[r,n,m,k] ) Iω (Xr,n,m,k ,Y[r,n,m,k] ),(λ,ω)Θ.

Moreover, we evaluate the FI about the mean and the shape-parameter of the exponential and power distributions, respectively.

In information theory, the relative entropy is a measure of the distance between the PDFs fX(x) and gY(y), see Kullback-Leibler [22]. This information measure is also known as the Kullback-Leibler distance (K-L distance). Since, in the context of concomitants theory, we often encounter the situation that the highest X-scores may be chosen and we wish to know something about the concomitant Y-scores. For example, the X's might refer to a characteristic in a parent and the Y's to the same characteristic in the offspring. The K-L distance can tell us how much information is lost when we approximate the DF of Y[r,n,m,k] by the DF of Yr,n,m,k. In Section 4, we investigate the K-L distance from Y[r,n,m,k] to Yr,n,m,k. The K-L distance is defined by

K(fX(x),gY(y))= f X(x)logfX(x)gY(x)dx.

2. FIM for (λ,ω) in (Xr,n,m,k,Y[r,n,m,k])

Since the conditional PDF of Y[r,n,m,k] given Xr,n,m,k=x is fY[r,n,m,k]|Xr,n,m,k(y|x)=fY|X(y|x), then the joint PDF of (Xr,n,m,k,Y[r,n,m,k]) is given by

fXr,n,m,k,Y[r,n,m,k](x,y;λ,ω)=Cr1(r1)!fX,Y(x,y;λ,ω)F¯Xγr1(x)gmr1(FX(x)).

On the other hand, with FX(x)=x, 0x1, and FY(y)=y, 0y1, one obtains the copula form of the IFGM(λ,ω). This copula (dependence function) is IFGM(λ,ω) with uniform marginals (i.e., free of any unknown parameters), cf. [24]. Therefore, in order to determine the FIM, I(λ,ω)(Xr,n,m,k,Y[r,n,m,k]), we deal with the copula of (1.2), i.e., when X and Y U(0,1). The following theorem determines this FIM.

Theorem 2.1. Let m1. Furthermore, let X and Y U(0,1) with the joint PDF (1.3). Then, for any 1rn and (λ,ω)ΘΩ, where Ω={(λ,ω): λC1(x,y)+ωC2(x,y)<1,0x,y1} (see Remark 2.1), the FIM about the parameter-vector (λ,ω) is given by

I(λ,ω)(Xr,n,m,k,Y[r,n,m,k])= Iλ (Xr,n,m,k ,Y[r,n,m,k] ) Iλ,ω (Xr,n,m,k ,Y[r,n,m,k] ) Iλ,ω (Xr,n,m,k ,Y[r,n,m,k] ) Iω (Xr,n,m,k ,Y[r,n,m,k] ) =Cr1(m+1)r(r1)! × i=0 j=0i(1) iλjωijijΦ(j+2,ij) i=0 j=0i(1) iλjωij0.3ijΦ(j+1,ij+1) i=0 j=0i(1) iλjωijijΦ(j+1,ij+1) i=0 j=0i(1) iλjωij0.3ijΦ(j,ij+2),

where

Φ(j+2,ij)= h=0 j+2 l=0 ij t=0 ij+l+h(1)h+l+t(2)h+ijl(3)l j+2 h ij l ij+l+h t×β(r,γr+tm+1) s=0 j+2 p=0 ij(1)s+p(2)s+ijp(3)p j+2 s ij p(p+s+ij+1)1.

Proof. From (1.3) and (2.1) we get

logfXr,n,m,k,Y[r,n,m,k](x,y;λ,ω)=logCr1(r1)!+log(1+λC1(x,y)+ωC2(x,y))+(γr1)log(1x)+(r1)log1(1x)m+1m+1,

where C1(x,y)=(12x)(12y) and C2(x,y)=xy(23x)(23y),0x,y1. Consider the matrix

I11(x,y)I12(x,y)I21(x,y)I22(x,y)=2logfXr,n,m,k,Y[r,n,m,k](x,y;λ,ω)λ22logfXr,n,m,k,Y[r,n,m,k](x,y;λ,ω)λω2logfXr,n,m,k,Y[r,n,m,k](x,y;λ,ω)λω2logfXr,n,m,k,Y[r,n,m,k](x,y;λ,ω)ω2.

The four elements of the matrix (2.4) can easily be determined from the relations

2logfXr,n,m,k,Y[r,n,m,k](x,y;λ,ω)λ2=C12(x,y)(1+λC1(x,y)+ωC2(x,y))2, 2logfXr,n,m,k,Y[r,n,m,k](x,y;λ,ω)λω=C1(x,y)C2(x,y)(1+λC1(x,y)+ωC2(x,y))2

and

2logfXr,n,m,k,Y[r,n,m,k](x,y;λ,ω)ω2=C22(x,y)(1+λC1(x,y)+ωC2(x,y))2.

Then, by using (2.1), (2.4), (2.5), (2.6) and (2.7), the FIM about parameter-vector (λ,ω) can be expressed by

I(λ,ω)(Xr,n,m,k,Y[r,n,m,k])= Iλ (Xr,n,m,k ,Y[r,n,m,k] ) Iλ,ω (Xr,n,m,k ,Y[r,n,m,k] ) Iλ,ω (Xr,n,m,k ,Y[r,n,m,k] ) Iω (Xr,n,m,k ,Y[r,n,m,k] )=E I11 (Xr,n,m,k ,Y[r,n,m,k] ) I12 (Xr,n,m,k ,Y[r,n,m,k] ) I21 (Xr,n,m,k ,Y[r,n,m,k] ) I22 (Xr,n,m,k ,Y[r,n,m,k] )=Cr1(r1)!01 0 1 2logf X r,n,m,k ,Y [r,n,m,k] (x,y;λ,ω)λ2 2logf X r,n,m,k ,Y [r,n,m,k] (x,y;λ,ω)λω 2logf X r,n,m,k ,Y [r,n,m,k] (x,y;λ,ω)λω 2logf X r,n,m,k ,Y [r,n,m,k] (x,y;λ,ω)ω2 ×(1+λC1(x,y)+ωC2(x,y))(1x)γr1 1 (1x) m+1 m+1 r1dxdy=Cr1(r1)!01 0 1 C 12(x,y)1+λC 1(x,y)+ωC2(x,y) C 1(x,y)C2(x,y)1+λC 1(x,y)+ωC2(x,y) C 1(x,y)C2(x,y)1+λC 1(x,y)+ωC2(x,y) C22(x,y)1+λC 1(x,y)+ωC2(x,y) (1x)γr1 1 (1x) m+1 m+1 r1dxdy.

The factor (1+λC1(x,y)+ωC2(x,y))1 in the denominator of the integrand in each element of the matrix (2.8) can be expanded as i=0(1)i(λC1(x,y)+ωC2(x,y))i provided (λ,ω)Ω. Moreover, this infinite expansion is uniformly convergent for λC1(x,y)+ωC2(x,y)<1. Thus, we get

Iλ(Xr,n,m,k,Y[r,n,m,k])=Cr1(r1)!i=0(1)i0101C12(x,y)(λC1(x,y)+ωC2(x,y))i×(1x)γr11(1x)m+1m+1r1dxdy=Cr1(r1)!i=0(1)ij=0iijλjωijJ1J2,

where

J1=01 (12x) j+2xij(23x)ij(1x)γr11(1x)m+1 m+1 r1dx =h=0j+2l=0ij(1)l+h(2)h+ijl(3)lj+2hijl×01xh+l+ij(1x)γr11(1x)m+1m+1r1dx =h=0j+2l=0ijt=0ij+l+h(1)l+h+t(2)h+ijl(3)lj+2hijlij+l+ht×01(1x)γr+t11(1x)m+1m+1r1dx

and

J2=01 (12y) j+2yij(23y)ijdy = s=0 j+2p=0ij (1) s+p(2)s+ijp(3)p j+2s ijp01 ys+p+ijdy = s=0 j+2p=0ij (1) s+p(2)s+ijp(3)p j+2s ijp(s+p+ij+1)1.

Now, by making the transformation u=1(1x)m+1m+1 in J1, we get

J1=h=0j+2l=0ijt=0ij+l+h(1)l+h+t(2)h+ijl(3)lj+2hijlij+l+ht×01m+1ur1(1(m+1)u)γr+tm+11du =1(m+1)rh=0j+2l=0ijt=0ij+l+h(1)l+h+t(2)h+ijl(3)lj+2hijlij+l+ht×β(r,γr+tm+1).

Combining (2.9), (2.10), and (2.11), we get Iλ(Xr,n,m,k,Y[r,n,m,k]). The other elements of the matrix (2.2) (i.e., Iλ,ω(Xr,n,m,k,Y[r,n,m,k]) and Iω(Xr,n,m,k,Y[r,n,m,k])) can be obtained by the same procedure. The theorem is proved.

Remark 2.2. Since (λ=0,ω=0)ΘΩ, then the set ΘΩϕ, where ϕ is the empty set. On the other hand, (λ=0,ω=0)Ω, then  {(λ,ω):λC1(x,y)+ωC2(x,y)>1,0x,y1}=ϕ. Thus, ΩΩ=U (while ΩΩϕ), where U is the universal set and Ω={(λ,ω):λC1x,y)+ωC2(x,y)<1,0xx0,0yy0,andλC1(x,y)+ωC2x,y)1,x>x0,y>y0,for some 0<x0,y0<1}. In order to check (λ0,ω0)Ω, for any (λ0,ω0)Θ, draw the function F(x,y;λ0,ω0)=λ0C1(x,y)+ω0C2(x,y),0x,y1, as 3D diagram (x,y,F), by using Mathematica 12. If the curve (surface) of F falls entirely within the cube κ={(x,y,z):1x,y,z+1}, then (λ0,ω0)Ω, otherwise (λ0,ω0)Ω. Note that ΩΩ=U, means that there are only the following possibilities:

• (1) The curve of F falls entirely within the cube κ, represented by the set Ω;

• (2) A portion of that curve falls within κ and the other portion is outside the cube κ, represented by the set Ω.

Figure 1 (Parts a,b,c, and d) shows that how can we apply this check for some values of (λ,ω)Θ. It is worth noting that (λ,ω)Ω, only for boundary values (or close to them) of λ and ω, such as λ=1,+1 and ω=2,3+3.

Figure 1. 3D Diagrams for checking the belonging relationship (λ,ω)ΘΩ

2.1 Discussion

Table 1 displays the FIM I(λ,ω)(Xr,n,m,k,Y[r,n,m,k]) for the models of OSs and SOSs (i.e., I(λ,ω)(Xr,n,0,1,Y[r,n,0,1]) and I(λ,ω)(Xr,n,1,1,Y[r,n,1,1])) as a function of n,rn+12,λ and ω, for n=1,2,...,5,10,λ=0.99 and different values for ω, for which (λ,ω)ΘΩ. The entries were computed by using the FIM (2.2), the relation (2.3) and MATHEMATICA Ver. 12. The infinite series was cut off after 11 terms and this gives a satisfactory accuracy. Table 1 is constructed as a matrix. Namely, every entry in the two parts (i.e. (m,k)=(0,1) and (m,k)=(1,1)) of Table 1 has the form abc, where a=Iλ(Xr,n,m,k,Y[r,n,m,k]), b=Iλ,ω(Xr,n,m,k,Y[r,n,m,k]) and c=Iω(Xr,n,m,k,Y[r,n,m,k]), respectively. The first raw in each of the two parts of Table 1 represents the FIM I(λ,ω)(X,Y) (with elements Iλ(X,Y),Iλ,ω(X,Y) and Iω(X,Y)) in a single pair. Since the FIM I(λ,ω)(X,Y) in a random sample of size n is nI(λ,ω)(X,Y), Table 1 enables us to compute the proportion of the sample FIM I(λ,ω)(Xr,n,m,k,Y[r,n,m,k]) contained in a single pair. For example, in I(λ,ω)(Xr,n,0,1,Y[r,n,0,1]) (i.e., Part 1 of Table 1), when n=10, the FI about λ (i.e., the element Iλ(Xr,n,0,1,Y[r,n,0,1])) in the extreme pair ranges from 23.6%(24%) to 32% of the total FI as λ=0.99. In contrast, the FI in the central pair no more than 2% of what is available in the complete sample in all cases λ=0.99. Moreover, the FI about the vector (λ,ω) (i.e., the element Iλ,ω(Xr,n,0,1,Y[r,n,0,1])) in the extreme pair (actually the second lower extreme, i.e., r=2) ranges from 3.5%(4%) to 11% of the total FI as λ=0.99. In contrast, the FI in the extreme pair is no more than 2% of what is available in the complete sample in all cases λ=0.99. Finally, the FI about ω (i.e., the element Iω(Xr,n,0,1,Y[r,n,0,1])) in the extreme pair is no more than 3% of what is available in the complete sample in all cases λ=0.99. On the other hand, the FI in the central pair (r=4) ranges from 2% to 7.5%(8%) of the total FI as λ=0.99.

FIM for (Xr,n,0,1,Y[r,n,0,1]) and (Xr,n,1,1,Y[r,n,1,1]) about the parameter-vector (λ=0.99,ω).

I(λ,ω)(Xr,n,0,1,Y[r,n,0,1]),λ=0.99

nrω=0ω=0.4ω=0.8ω=1.2ω=1.4
110.1950|0.0620|0.04500.1650|0.0391|0.02610.1540|0.0321|0.02100.1490|0.0287|0.01880.1470|0.0277|0.0182
210.1950|0.0210|0.01100.1900|0.0182|0.00950.1887|0.0175|0.00910.1860|0.0172|0.00910.1850|0.0172|0.0092
310.2430|0.0170|0.00700.2420|0.0168|0.00720.2400|0.0169|0.00730.2380|0.0170|0.00760.2380|0.0172|0.0078
320.0960|0.0270|0.01800.0870|0.0209|0.01410.0830|0.0187|0.01300.0810|0.0175|0.01230.0800|0.0171|0.0122
410.2930|0.0180|0.00600.2910|0.0174|0.00640.2890|0.0176|0.00650.2870|0.0178|0.00680.2870|0.0180|0.0070
420.0960|0.0160|0.01100.0940|0.0150|0.00960.0920|0.0147|0.00970.0915|0.0147|0.00980.0920|0.0148|0.0100
510.3370|0.0180|0.00500.334|0.0179|0.00570.3320|0.0180|0.00580.3310|0.0182|0.00610.3290|0.0184|0.0061
520.1190|0.0150|0.01000.1170|0.0157|0.00900.1160|0.0158|0.00930.1150|0.0161|0.00960.1160|0.0163|0.0098
530.0620|0.0160|0.01100.0600|0.0139|0.01060.0550|0.0130|0.01030.0540|0.0125|0.01030.0550|0.0124|0.0104
1010.4850|0.0170|0.00300.4830|0.0166|0.00320.4790|0.0167|0.00330.4770|0.0167|0.00330.4760|0.0168|0.0034
1020.2550|0.0220|0.00900.2530|0.0213|0.00700.2510|0.0215|0.00720.2490|0.0218|0.00740.2480|0.0220|0.0075
1030.1310|0.0190|0.00900.1290|0.0191|0.00960.1280|0.0200|0.00980.1270|0.0200|0.01030.1260|0.0202|0.0110
1040.0650|0.0140|0.00900.0650|0.0138|0.01020.0640|0.0141|0.01230.0610|0.0145|0.01330.0600|0.0148|0.0115
1050.0360|0.0090|0.00900.0360|0.0086|0.00910.0358|0.0088|0.00950.0352|0.0090|0.00990.0340|0.0090|0.0095
I(λ,ω)(Xr,n,1,1,Y[r,n,1,1]),λ=0.99

nrω=0ω=0.4ω=0.8ω=1.2ω=1.4
110.1950|0.0618|0.04420.1650|0.0391|0.02610.1540|0.0321|0.02100.1490|0.0287|0.01880.1470|0.0277|0.0182
210.2440|0.0169|0.00720.2420|0.0168|0.00720.2390|0.0168|0.00730.2380|0.0170|0.00750.2380|0.0172|0.0077
310.3370|0.0178|0.00560.3340|0.0179|0.00570.3320|0.0180|0.00580.3300|0.0182|0.00600.3290|0.0184|0.0062
320.1040|0.0158|0.00960.1030|0.0152|0.00940.1010|0.0151|0.00950.1010|0.0152|0.00970.1000|0.0153|0.0099
410.4080|0.0178|0.00440.4050|0.0178|0.00450.4020|0.0179|0.00460.4000|0.0181|0.00470.3990|0.0182|0.0048
420.1590|0.0179|0.00840.1580|0.0181|0.00860.1560|0.0183|0.00890.1560|0.0187|0.00930.1550|0.0189|0.0095
510.4630|0.0169|0.00350.4590|0.0170|0.00360.4570|0.0171|0.00370.4540|0.0172|0.00370.4530|0.0173|0.0038
520.2160|0.0203|0.00760.2140|0.0204|0.00780.2120|0.0207|0.00790.2110|0.0210|0.00830.2090|0.0212|0.0085
530.0880|0.0149|0.00940.0870|0.0151|0.00970.0870|0.0154|0.01000.0870|0.0158|0.01050.0860|0.0161|0.0108
1010.6190|0.0125|0.00140.6160|0.0125|0.00140.6130|0.0125|0.00140.6090|0.0126|0.00140.6080|0.0126|0.0014
1020.4120|0.0198|0.00370.4080|0.0199|0.00370.4050|0.0199|0.00380.4020|0.0110|0.00380.4030|0.0201|0.0039
1030.2690|0.0229|0.00630.2670|0.0230|0.00640.2640|0.0232|0.00650.2620|0.0234|0.00670.2610|0.0236|0.0068
1040.1710|0.0225|0.00870.1690|0.0227|0.00890.1680|0.0230|0.00910.1670|0.0235|0.00950.1660|0.0237|0.0097
1050.1020|0.0193|0.01030.1010|0.0196|0.01060.1000|0.0200|0.01110.1002|0.0206|0.01150.1000|0.0209|0.0118

Also, as example in I(λ,ω)(Xr,n,1,1,Y[r,n,1,1]) (i.e., Part 2 of Table 1), when n=10, the FI about λ (i.e., the element Iλ(Xr,n,1,1,Y[r,n,1,1])) in the extreme pair ranges from 32% to 41% of the total FI, as λ=0.99 . In contrast, the FI in the central pair ranges from 5% to 7% of what is available in the complete sample in all cases λ=0.99. Moreover, the FI about the vector (λ,ω) (i.e., the element Iλ,ω(Xr,n,1,1,Y[r,n,1,1])) in the extreme pair (actually the third lower extreme, i.e., r=3) ranges from 4% to 9% of the total FI, as λ=0.99. In contrast, the FI in the extreme pair is no more than 4% of what is available in the complete sample in all cases λ=0.99. Finally, the FI about ω (i.e., the element Iω(Xr,n,1,1,Y[r,n,1,1])) in the extreme pair is no more than 1% of what is available in the complete sample in all cases λ=0.99. On the other hand, the FI in the central pair ranges from 2% to 6% of the total FI, as λ=0.99.

Another important usage of Table 1 is that it can readily be used to obtain the FI contained in singly or multiply censored bivariate samples from the IFGM(λ,ω) distribution. One just adds up the FI in individual pairs that constitute the censored sample. For example, in I(λ,ω)(Xr,n,0,1,Y[r,n,0,1]) (i.e., the element Iλ(Xr,n,0,1,Y[r,n,0,1])), when n = 10, the FI about λ in the Type II censored sample consisting of the bottom (or the top) three pairs ranges from 45% to 58% (as ω varies over 0,0.4,0.8,1.2,1.4) when λ=0.99. Another example, in I(λ,ω)(Xr,n,1,1,Y[r,n,1,1]) (i.e., the element Iλ(Xr,n,0,1,Y[r,n,0,1])), when n = 10, the FI about λ in the Type II censored sample consisting of the bottom (or the top) three pairs ranges from 70% to 86% (as ω varies over 0,0.4,0.8,1.2,1.4) when λ=0.99. Note that, for the preceding ratios (when n=10), we have in general FI (λ,ω1)<FI (λ,ω2), if ω1<ω2, where FI(λ,ω) denotes to the FI about λ in the Type II censored sample consisting of the bottom three pairs at ω.

From Table 1, the following properties can be extracted for the models of OSs and SOSs:

• 1. In general, for (m,k)=(0,1), or (m,k)=(1,1), we have

Iλ(Xr,n,m,k,Y[r,n,m,k])>Iλ,ω(Xr,n,m,k,Y[r,n,m,k])>Iω(Xr,n,m,k,Y[r,n,m,k]).

• 2. For (m,k)=(0,1), or (m,k)=(1,1), Iλ(Xr,n,m,k,Y[r,n,m,k]) increases with increasing the difference between r and n, for rn+12. For any n, the greatest value of Iλ(Xr,n,m,k,Y[r,n,m,k]) is always obtained at the lower extreme.

• 3. Iλ,ω(Xr,n,0,1,Y[r,n,0,1]) increases with increasing the difference between r>1 and n, for rn+12. For any n, the greatest value of Iλ,ω(Xr,n,0,1,Y[r,n,0,1]) is obtained at the second lower extreme. On the other hand, the greatest value of Iλ,ω(Xr,n,1,1,Y[r,n,1,1]) is attained frequently at r=3, or r=4.

• 4. For (m,k)=(0,1), or (m,k)=(1,1), Iω(Xr,n,m,k,Y[r,n,m,k]) decreases with increasing the difference between r and n, for r<n+12.

• 5. In general, we have Iλ(Xr,n,1,1,Y[r,n,1,1])>Iλ(Xr,n,0,1,Y[r,n,0,1]).

We begin this section by representing a result of Barakat and Husseiny [9] that gives an explicit form of the marginal PDF of the concomitant Y[r,n,m,k] of m-GOS for IFGM. This will enable us to derive and study the FI about mean and the shape parameter of the exponential and power distributions, respectively.

Lemma 3.1. ([9]) Let m1,X~FX and Y~FY. Furthermore, let V1~FX2 (with PDF fV1) and V2~FX3 (with PDF fV2). Then,

f[r,n,m,k](y)=fY(y)1+λD1(r,n,m,k)(12FY(y))+ωD2(r,n,m,k)(2FY(y)3FY2(y))=fY(y)+λD1(r,n,m,k)(fY(y)fV1(y))+ωD2(r,n,m,k)(fV1(y)fV2(y)),

where D1(r,n,m,k)=2 i=1rγiγi+11 and D2(r,n,m,k)=4 i=1rγiγi+13 i=1rγiγi+21.

Theorem 3.2. (FI in Y[r,n,m,k] about E(Y)) Suppose that m ≠-1, and Y has exponential DF with mean 𝜃, then the FI about 𝜃, contained in Y[r,n,m,k], is given by

Iθ(Y[r,n,m,k])=1θ212δ12δ3 27+0 δ 1 2 e w+ δ 3 2 e 5w2 δ 1 δ 3 e 3w δ 1+ δ 2 e w+ δ 3 e 2w w2dw,

where w=yθ,δ1=1(λD1(r,n,m,k)+ωD2(r,n,m,k)),δ2=2(λD1(r,n,m,k)+2ωD2(r,n,m,k)) and δ3=3ωD2(r,n,m,k).

Proof. Let fY(y)=1θexp(yθ),θ>0,y0, by using (3.1) we get the PDF of the concomitant Y[r,n,m,k] based on the exponential distribution as

f[r,n,m,k](y;θ)=1θexp(yθ)1+λD1(r,n,m,k)121exp(yθ )+ωD2(r,n,m,k)21exp(yθ )31exp(yθ )2.

This expression, after some algebra, can be written as

fY[r,n,m,k](y;θ)=1θexp(yθ)δ1+δ2exp(yθ)+δ3exp(2yθ)

where δ1=1(λD1(r,n,m,k)+ωD2(r,n,m,k)), δ2=2(λD1(r,n,m,k)+2ωD2(r,n,m,k)) and δ3=3ωD2(r,n,m,k). Therefore,

log(fY[r,n,m,k](y;θ))θ=1θyθ1+yθδ2exp(yθ)+2δ3exp(2yθ)δ1+δ2exp(yθ)+δ3exp(2yθ)=1θ2yθ1δ1yθδ1+δ2exp(yθ)+δ3exp(2yθ)+δ3yθexp(2yθ)δ1+δ2exp(yθ)+δ3exp(2yθ).

Thus,

logfY[r,n,m,k](y;θ)θ2=1θ24w2+1+ δ12w2 (δ1 +δ2 e w+δ3 e 2w)2 +δ32w2e 4w δ1 +δ2 e w+δ3 e 2w 24w4δ1w2δ1+δ2e w+δ3e 2w+4δ3w2e 2wδ1+δ2e w+δ3e 2w+2δ1wδ1+δ2e w+δ3e 2w 2δ3we 2w δ1+δ2e w+δ3e 2w 2δ1δ3w2e 2w δ1 +δ2 e w +δ3 e 2w 2 ,

where w=yθ. On the other hand, the PDF of the RV W=Y[r,n,m,k]θ is fW(w)=ew(δ1+δ2ew+δ3e2w). Therefore (3.3) yields

Iθ(Y[r,n,m,k])=0 logfW (w) θ2fW(w)dw=1θ2 i=1 10Ji ,

where

J1=40 w2ewδ1+δ2ew+δ3e2w dw=42δ1+δ 2 4+2δ3 27 ,J2=0 e wδ1+δ2ew+δ3e2w dw=1,J3=δ120 w2 e w δ 1 + δ2 e w + δ 3 e 2w dw, J4=δ320 w2 e 5w δ 1 + δ2 e w + δ 3 e 2w dw,J5=40wewδ1+δ2ew+δ3e2w dw=4δ1+δ 2 4+δ3 9 ,J6=4δ10 w2ewdw=8δ1,J7=4δ30 w2e3wdw=8δ3 27,J8=2δ10wewdw=2δ1,J9=2δ30we3wdw=2δ3 9,

and

J10=2δ1δ30 w2e 3w δ 1+ δ2e w+ δ 3e 2wdw.

Therefore, we get Iθ(Y[r,n,m,k])=1θ212δ12δ3 27+J3+J4+J10.

Theorem 3.3. (FI in Y[r,n,m,k] about the shape parameter of power distribution)

Suppose that Y has power DF FY(y)=yα,α>0,0y1, then the FI in Y[r,n,m,k] about α is given by

Iα(Y[r,n,m,k])=1α21δ248δ327+Elogδ1+δ2Y[r,n,m,k]α+δ3Y[r,n,m,k]2α α2,

where δ1=1+λD1(r,n,m,k), δ2=2(ωD2(r,n,m,k)λD1(r,n,m,k)) and δ3=3ωD2(r,n,m,k).

Proof. Let fY(y)=αyα1,α>0,0y1, by using (3.1) we get the PDF of Y[r,n,m,k] based on the power distribution as follow

f[r,n,m,k](y;α)=αyα11+λD1(r,n,m,k)(12yα)+ωD2(r,n,m,k)(2yα3y2α).

This expression, after some algebra, can be written as

f[r,n,m,k](y;α)=αyα1δ1+δ2yα+δ3y2α,

where

δ1=1+λD1(r,n,m,k),δ2=2(ωD2(r,n,m,k)λD1(r,n,m,k)) and δ3=3ωD2(r,n,m,k). Therefore,

logf[r,n,m,k](y;α)α=1α+logy+δ2yαlogy+2δ3y2αlogyδ1+δ2yα+δ3y2α.

Then,

logf[r,n,m,k](y;α)α2=1α2+(logy)2+δ2yαlogy+2δ3y2αlogyδ1+δ2yα+δ3y2α2+2logyα+2αδ2yαlogy+2δ3y2αlogyδ1+δ2yα+δ3y2α+2δ2yαlogy+2δ3y2αlogyδ1+δ2yα+δ3y2αlogy.

Thus, (3.4) yields

Iα(Y[r,n,m,k])=01 logf Y[r,n,m,k] (y;α) α2f[r,n,m,k](y;α)= i=16Ji ,

where

J1=1α201αyα1δ1+δ2yα+δ3y2αdy=1α2,J2=α01yα1δ1+δ2yα+δ3y2α(logy)2dy=27(8δ1+δ2)+8δ3108α2,J3=α01yα1δ2yαlogy+2δ3y2αlogy2δ1+δ2yα+δ3y2αdy,J4=201yα1(δ1+δ2yα+δ3y2α)logydy=36δ1+9δ2+4δ318α2,J5=201yα1δ2yαlogy+2δ3y2αlogydy=9δ2+8δ318α2,

and

J6=2α01 y α1δ2yαlogy+2δ3y2αlogylogydy=27δ2+16δ3 54α2 .

Therefore, we get Iα(Y[r,n,m,k])=1α2114δ2827δ3+J3.

We compute the FI in Y[r,n,0,1] and Y[r,n,1,1] about E(Y)=θ by using formula (3.2). Table 2 provides Iθ(Y[r,n,0,1]) and Iθ(Y[r,n,1,1]) values for n=5,15,λ=0.99,ω=1.8,1.6,1.4,1.2,0,0.4,0.8, for θ=1. Moreover, Table 2 reveals that the greatest values of FI are almost obtained at the maximum OSs.2022-07-18

FI in Y[r,n,0,1] and Y[r,n,1,1] for θ at θ=1. where E(Y)=θ.

Iθ(Y[r,n,0,1])λ=0.99Iθ(Y[r,n,1,1])λ=0.99

nrω=-1.8ω=-1.6ω=-1.4ω =-1.2ω=0nrω=-1.8ω=-1.6ω=-1.4ω=-1.2ω=0
510.76940.77060.77300.77660.827510.75640.76230.76920.77700.8497
520.90140.89280.88530.87910.870520.80020.79660.79430.79360.8275
531.0451.0381.0321.0261.0000530.92050.90960.90000.89190.8756
541.1621.1651.1681.1721.196541.0581.0501.0441.0371.008
551.3121.3101.3141.3221.454551.2271.2311.2381.2451.323
1510.77870.78660.79520.80440.87891510.82930.83690.84500.85340.9149
1520.75290.75660.76140.76760.83761520.76990.77770.78620.79560.8737
1530.78360.78100.78000.78070.82611530.75090.75620.76270.77040.8475
1540.83850.83010.82350.81880.83481540.75670.75840.76150.76610.8323
1550.90190.88940.87870.86970.85871550.77490.77690.77610.77700.8264
1560.96520.95070.93780.92650.89531560.81410.80740.80250.79950.8287
1571.0231.0090.99660.98490.94271570.85690.84640.83770.83100.8391
1581.0741.0631.0521.0421.0001580.90510.89150.87980.87000.8577
1591.1181.1111.1041.0981.0661590.95600.94060.92690.91500.8852
15101.1561.1541.1531.1511.14115101.0070.99190.97780.96500.9229
15111.193 1.1961.1991.2021.22515111.058 1.0441.0311.0190.9729
15121.2331.2391.2461.2541.31715121.105 1.0961.0871.0781.039
15131.2871.2911.2981.3071.41915131.152 1.1491.1461.1431.128
15141.3681.3601.3611.3681.53015141.206 1.2111.2161.2211.259
15151.4971.4611.4421.4391.65415151.340 1.3361.3391.3461.495

Upon relying on the definition in relation (1.4), we get the following theorem:

Theorem 4.1. Let m1 and Y[r,n,m,k] is concomitants of rth m-GOS in IFGM family, then the K-L distance from Y[r,n,m,k] to Yr,n,m,k is given by

K(Y[r,n,m,k],Yr,n,m,k)=H(Y[r,n,m,k])+I(Y[r,n,m,k],Yr,n,m,k)=δr,n,m,klog C r1 (r1)!(γr1)(1+λD1(r,n,m,k)2+ωD2(r,n,m,k)3)
(r1)λD1(r,n,m,k)Hn+ωD2(r,n,m,k)HnHn[ 1 m+1]log(1+m),

where

δr,n,m,k=log(1λD1(r,n,m,k)ωD2(r,n,m,k))+2b(r)J0(r,n,m,k)+6c(r)J1(r,n,m,k),Jl(r,n,m,k)=01 zl(a(r)z+b(r)z2+c(r)z3)a(r)+2b(r)z+3c(r)z2dz,l=0,1,Hn=Hn[ 1 m+1]Hn[ 2 m+1],Hn=Hn[ 1 m+1]2Hn[ 2 m+1]+Hn[3 m+1],

Hn[m] denotes the generalized harmonic numbers, which is calculated by Hn[m]= i=1 n1im, and H(Y[r,n,m,k]) is the Shannon entropy that was derived in Theorem 3.2 of [9].

Proof. For simplicity, write D1(r,n,m,k)=Dr(1) and D2(r,n,m,k)=Dr(2), then by using (1.1) and (3.1), the K-L distance from Y[r,n,m,k] to Yr,n,m,k is given by

K(Y[r,n,m,k],Yr,n,m,k)= f [r,n,m,k] (y)log f [r,n,m,k](y) f r,n,m,k(y) dy=H(Y[r,n,m,k])I(Y[r,n,m,k],Yr,n,m,k)=H(Y[r,n,m,k]) f [r,n,m,k] (y)logfr,n,m,k(y)dy
=H(Y[r,n,m,k])log C r1 (r1)!(γr1)A(r1)BE(logfY(Y[r,n,m,k])),

where

A= fY(y)1+λDr(1)(12FY(y))+ωDr(2)(2FY(y)3FY2(y))×logF¯Y(y)dy,B= fY(y)1+λDr(1)(12FY(y))+ωDr(2)(2FY(y)3FY2(y))×log 1 F ¯ Y m+1 (y) m+1 dy.

Upon taking z=FY(y) in A, we get

A=1+λDr(1)2+ωDr(2)3.

On the other hand, by taking the transformation t=1F¯Ym+1(y)m+1 in B, we get the representation

B=B1+λDr(1)(B12B2)+ωDr(2)(2B23B3),

where

B1=01m+1 (1(m+1)t)1m+11logtdt=Hn[ 1 m+1]log(1+m),B2=01m+1 (1(m+1)t)1m+11(1(1(m+1)t)1m+1 )logtdt=122Hn[ 1 m+1 ]+Hn[ 2 m+1 ]log(1+m)

and

B3=01m+1(1(m+1)t)1m+11(1(1(m+1)t)1m+1)2logtdt=Hn[1m+1]+Hn[2m+1]13Hn[3m+1]13log(1+m).

Now, by incorporating the values of B1, B2 and B3 in (4.4), we get

B=Hn[1 m+1]log(1+m)+λDr(1)Hn[1m+1]Hn[2m+1]
+ωDr(2)Hn[1 m+1]2Hn[2 m+1]+Hn[3 m+1].

Finally, upon incorporating the values of (4.3), (4.5) and H(Y[r,n,m,k]) in (4.2), we get (4.1). The theorem is proved.

Table 3 provides the K-L distance from Y[r,n,m,k] to Yr,n,m,k in the models, OSs and SOSs under IFGM. The entries were computed using (4.1). From Table 3, the following properties can be extracted:

K(Y[r,n,0,1],Yr,n,0,1) and K(Y[r,n,1,1],Yr,n,1,1) under IFGM.

K(Y[r,n,0,1],Yr,n,0,1)λ=0.75K(Y[r,n,1,1],Yr,n,1,1)λ=0.75

ωω

nr-1.5-1.0-0.50.51.0nr-1.5-1.0-0.50.51.0
310.73660.67290.61070.49130.434431 1.79391.67221.55191.31661.2019
320.23470.22560.21670.2001 0.1923320.66670.60610.54700.43390.3800
330.68580.63770.59260.51040.4732330.44390.42660.41020.38040.3669
813.46013.28483.11072.76622.5961817.47557.24097.00696.54086.3089
822.61712.39652.17831.74941.5393826.04165.68185.32354.61234.2599
832.03871.86051.68471.34111.1737834.93464.54744.16253.40123.0254
841.60521.51011.41661.23461.1463843.92333.59013.25972.60842.2883
851.38431.36401.34421.30561.2869852.97372.75242.53352.10421.8941
861.52481.52021.51581.50741.5033862.15682.06581.98151.81211.7298
872.25602.15512.05731.87041.7810871.69451.69311.69191.68951.6884
883.99393.62943.27782.60882.2907882.36832.22572.08801.82621.7017

• 1. For fixed n, the maximum value of K(Y[r,n,0,1],Yr,n,0,1) attains at lower and upper extremes. Moreover, its smallest value occurs at the central terms.

• 2. For fixed n, the maximum value of K(Y[r,n,1,1],Yr,n,1,1) occurs at lower extreme terms. In contrast, its smallest value occurs at the central terms.

• 3. For the two models OSs and SOSs the K-L distance decreases with increasing ω.

• 4. Generally, for rn2, we have K(Y[r,n,1,1],Yr,n,1,1)> K(Y[r,n,0,1],Yr,n,0,1).

The authors are grateful to the editors, and the four anonymous reviewers for careful and diligent reading which improved the readability and presentation substantially.

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