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), θ∈Θ 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)=E∂logf(X;θ)∂θ2=−E∂2logf(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∈ℕ,m>−1,k>0 and γi=k+(n−i)(m+1), i=1,2,...,,n, be parameters. Then the RVs X1,n,m,k≤X2,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¯=1−F, if their joint PDF is of the form

F−1(1)≥xn≥...≥x1≥F−1(0). The marginal PDF of rth m-GOS, Xr,n,m,k,1≤r≤n, is given by (cf. [20, 8])

where Cr−1=∏ i=1rγi,r=1,2,...,n,gm(x)=hm(x)−hm(0),x∈[0,1) and hm(x)=−(1−x)m+1(m+1), if m≠−1, while h−1(x)=−log(1−x). 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

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

where C1(x,y)=(1−2FX(x))(1−2FY(y)), C2(x,y)=FX(x)FY(y)(2−3FX(x))(2−3FY(y)) and f_X(x) and f_Y(y) are the PDFs of the RVs X and Y, respectively. When the two marginals F_X(x) and F_Y(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):−1≤λ≤1;λ+ω≥−1;ω≤3−λ+9−6λ−3λ22}. 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.,

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

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

On the other hand, with FX(x)=x, 0≤x≤1, and FY(y)=y, 0≤y≤1, 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 m≠−1. Furthermore, let X and Y U(0,1) with the joint PDF (1.3). Then, for any 1≤r≤n and (λ,ω)∈Θ∩Ω, where Ω={(λ,ω): ∣λC1(x,y)+ωC2(x,y)∣<1,∀0≤x,y≤1} (see Remark 2.1), the FIM about the parameter-vector (λ,ω) is given by

where

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

where C1(x,y)=(1−2x)(1−2y) and C2(x,y)=xy(2−3x)(2−3y),0≤x,y≤1. Consider the matrix

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

and

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

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

where

and

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

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)∈Θ∩Ω, then the set Θ∩Ω≠ϕ, 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,∀0≤x,y≤1}=ϕ. Thus, Ω∪Ω⋆=U (while Ω∩Ω⋆≠ϕ), where U is the universal set and Ω⋆={(λ,ω):∣λC1x,y)+ωC2(x,y) ∣<1,0≤x≤x0,0≤y≤y0,and∣λC1(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)∣,0≤x,y≤1, 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):−1≤x,y,z≤+1}, then (λ0,ω0)∈Ω, otherwise (λ0,ω0)∉Ω. Note that Ω∪Ω⋆=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)∈Θ∩Ω

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,r≤n+12,λ and ω, for n=1,2,...,5,10,λ=−0.99 and different values for ω, for which (λ,ω)∈Θ∩Ω. 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 a∣b∣c, 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.

Table 1 . 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
n	r	ω=0	ω=0.4	ω=0.8	ω=1.2	ω=1.4
1	1	0.1950|0.0620|0.0450	0.1650|0.0391|0.0261	0.1540|0.0321|0.0210	0.1490|0.0287|0.0188	0.1470|0.0277|0.0182
2	1	0.1950|0.0210|0.0110	0.1900|0.0182|0.0095	0.1887|0.0175|0.0091	0.1860|0.0172|0.0091	0.1850|0.0172|0.0092
3	1	0.2430|0.0170|0.0070	0.2420|0.0168|0.0072	0.2400|0.0169|0.0073	0.2380|0.0170|0.0076	0.2380|0.0172|0.0078
3	2	0.0960|0.0270|0.0180	0.0870|0.0209|0.0141	0.0830|0.0187|0.0130	0.0810|0.0175|0.0123	0.0800|0.0171|0.0122
4	1	0.2930|0.0180|0.0060	0.2910|0.0174|0.0064	0.2890|0.0176|0.0065	0.2870|0.0178|0.0068	0.2870|0.0180|0.0070
4	2	0.0960|0.0160|0.0110	0.0940|0.0150|0.0096	0.0920|0.0147|0.0097	0.0915|0.0147|0.0098	0.0920|0.0148|0.0100
5	1	0.3370|0.0180|0.0050	0.334|0.0179|0.0057	0.3320|0.0180|0.0058	0.3310|0.0182|0.0061	0.3290|0.0184|0.0061
5	2	0.1190|0.0150|0.0100	0.1170|0.0157|0.0090	0.1160|0.0158|0.0093	0.1150|0.0161|0.0096	0.1160|0.0163|0.0098
5	3	0.0620|0.0160|0.0110	0.0600|0.0139|0.0106	0.0550|0.0130|0.0103	0.0540|0.0125|0.0103	0.0550|0.0124|0.0104
10	1	0.4850|0.0170|0.0030	0.4830|0.0166|0.0032	0.4790|0.0167|0.0033	0.4770|0.0167|0.0033	0.4760|0.0168|0.0034
10	2	0.2550|0.0220|0.0090	0.2530|0.0213|0.0070	0.2510|0.0215|0.0072	0.2490|0.0218|0.0074	0.2480|0.0220|0.0075
10	3	0.1310|0.0190|0.0090	0.1290|0.0191|0.0096	0.1280|0.0200|0.0098	0.1270|0.0200|0.0103	0.1260|0.0202|0.0110
10	4	0.0650|0.0140|0.0090	0.0650|0.0138|0.0102	0.0640|0.0141|0.0123	0.0610|0.0145|0.0133	0.0600|0.0148|0.0115
10	5	0.0360|0.0090|0.0090	0.0360|0.0086|0.0091	0.0358|0.0088|0.0095	0.0352|0.0090|0.0099	0.0340|0.0090|0.0095
		I(λ,ω)(Xr,n,1,1,Y[r,n,1,1]),		λ=−0.99
n	r	ω=0	ω=0.4	ω=0.8	ω=1.2	ω=1.4
1	1	0.1950|0.0618|0.0442	0.1650|0.0391|0.0261	0.1540|0.0321|0.0210	0.1490|0.0287|0.0188	0.1470|0.0277|0.0182
2	1	0.2440|0.0169|0.0072	0.2420|0.0168|0.0072	0.2390|0.0168|0.0073	0.2380|0.0170|0.0075	0.2380|0.0172|0.0077
3	1	0.3370|0.0178|0.0056	0.3340|0.0179|0.0057	0.3320|0.0180|0.0058	0.3300|0.0182|0.0060	0.3290|0.0184|0.0062
3	2	0.1040|0.0158|0.0096	0.1030|0.0152|0.0094	0.1010|0.0151|0.0095	0.1010|0.0152|0.0097	0.1000|0.0153|0.0099
4	1	0.4080|0.0178|0.0044	0.4050|0.0178|0.0045	0.4020|0.0179|0.0046	0.4000|0.0181|0.0047	0.3990|0.0182|0.0048
4	2	0.1590|0.0179|0.0084	0.1580|0.0181|0.0086	0.1560|0.0183|0.0089	0.1560|0.0187|0.0093	0.1550|0.0189|0.0095
5	1	0.4630|0.0169|0.0035	0.4590|0.0170|0.0036	0.4570|0.0171|0.0037	0.4540|0.0172|0.0037	0.4530|0.0173|0.0038
5	2	0.2160|0.0203|0.0076	0.2140|0.0204|0.0078	0.2120|0.0207|0.0079	0.2110|0.0210|0.0083	0.2090|0.0212|0.0085
5	3	0.0880|0.0149|0.0094	0.0870|0.0151|0.0097	0.0870|0.0154|0.0100	0.0870|0.0158|0.0105	0.0860|0.0161|0.0108
10	1	0.6190|0.0125|0.0014	0.6160|0.0125|0.0014	0.6130|0.0125|0.0014	0.6090|0.0126|0.0014	0.6080|0.0126|0.0014
10	2	0.4120|0.0198|0.0037	0.4080|0.0199|0.0037	0.4050|0.0199|0.0038	0.4020|0.0110|0.0038	0.4030|0.0201|0.0039
10	3	0.2690|0.0229|0.0063	0.2670|0.0230|0.0064	0.2640|0.0232|0.0065	0.2620|0.0234|0.0067	0.2610|0.0236|0.0068
10	4	0.1710|0.0225|0.0087	0.1690|0.0227|0.0089	0.1680|0.0230|0.0091	0.1670|0.0235|0.0095	0.1660|0.0237|0.0097
10	5	0.1020|0.0193|0.0103	0.1010|0.0196|0.0106	0.1000|0.0200|0.0111	0.1002|0.0206|0.0115	0.1000|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 r≤n+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 r≤n+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]).