Generalized exponential function and initial value problem for conformable dynamic equations

1.   Introduction

In 1995, Kamps presented the notion of generalized order statistics (GOS), which is the unification of different models of ascendingly ordered random variables (RVs). The GOS incorporates significant and well-known concepts that have been discussed individually in the statistical literature. Many models of ascendingly ordered RVs, such as sequential order statistics, progressive Type-Ⅱ (PT-Ⅱ) censored order statistics, ordinary order statistics (OOS), record values, and Pfeifer's record model, are theoretically contained in the GOS model.

Assume $F(.)$ to be an arbitrary continuous cumulative distribution function (CDF) with probability density function (PDF) $f(.)$. Assume also $k > 0,$ $n\in N$, and $\tilde{m} = (m_{1}, m_{2}, \cdots, m_{n-1})\in \Re^{n-1}$ to be the parameters such that $\gamma_{n} = k$ and $\gamma_{i} = k+n-i+M_{i}$, for $i = 1, \cdots, n-1$, where $M_{i} = \sum_{\iota = i}^{n-1}m_{\iota}$. Then, the RVs $X_{i:n, \tilde{m}, k}, i = 1, \cdots, n$, are said to be GOS, if their joint PDF is given by

where $\bar {F}(.) = 1-F(.)$ and $F^{-1}(0) < x_{1}\leq\cdots\leq x_{n} < F^{-1}(1).$

Several models of arranged RVs can be considered special instances of GOS. $m_{1} = m_{2} = \cdots = m_{n-1} = m$; $\gamma_{i} = k+(n-i)(m+1)$, $i = 1, \cdots, n$ corresponds to m-generalized order statistics (m-GOS), $\gamma_{i} = n-i+1$ $(m_{i} = 0, k = 1)$ corresponds to OOS, and $m_{i} = -1; \gamma_{i} = k$, $i = 1, \cdots, n,$ $k\in N$ corresponds to k-recored values. Also, for $m_{i} = R_{i},$ $n = m_{0}+\sum_{\nu = 1}^{m_{0}}R_{\nu}$, $R_{\nu}\in N,$ and $\gamma_{i} = n-\sum_{\nu = 1}^{i-1}R_{\nu}-i+1,$ $1\leq i\leq m_{0},$ where $m_{0}$ denotes the fixed number of failure of units to be observed, the model reduces to PT-Ⅱ censored order statistics.

Under the condition $\gamma_{i}\neq\gamma_{j},$ $i, j = 1, \cdots, n-1, \, i\neq j$, Kamps and Cramer ^[23] derived the PDF of $X_{r:n, \tilde{m}, k}$, $1\leq r \leq n$ as

and the joint PDF of $X_{r:n, \tilde{m}, k}$ and $X_{s:n, \tilde{m}, k}$, $r, s = 1, \cdots, n, \, r < s$ as

where $C_{r-1} = \prod_{i = 1}^{r}\gamma_{i},$ $a_{i, r} = \prod_{\stackrel{\imath = 1}{\imath\neq i}}^{r} \frac{1}{\gamma_{\imath}-\gamma_{i}}, \, 1\leq i\leq r\leq n$, and $a_{j, s}^{(r)} = \prod_{\stackrel{\jmath = r+1}{\jmath\neq j}}^{s} \frac{1}{\gamma_{\jmath}-\gamma_{j}},$ $r+1\leq j\leq s\leq n.$

It can be shown that for $m_{1} = m_{2} = \cdots = m_{n-1} = m\neq-1$ (Khan and Khan ^[24]),

and

Therefore, the PDF of $X_{r:n, \tilde{m}, k}$ given in (1.2) reduces to

and the joint PDF of $X_{r:n, \tilde{m}, k}$ and $X_{s:n, \tilde{m}, k}$ given in (1.3) reduces to

where $C_{r-1} = \prod_{i = 1}^{r}\gamma_{i},$ $\gamma_{i} = k+(n-i)(m+1),$ $h_{m}(x) = \begin{cases}-\frac{1}{m+1}(1-x)^{m+1}, &m\neq-1, \\ -ln(1-x), &m = -1, \end{cases}$ and $g_{m}(x) = h_{m}(x)-h_{m}(0),$ $x\in[0, 1)$ (see Kamps ^[22]).

David ^[8] introduced the concept of concomitants of order statistics (COS), but Yang ^[47] described the general theory of COS. Concomitants are important in selection and prediction issues, ranked set sampling, parameter estimation, and the characterization of parent bivariate distributions. For a brief overview of the uses of the concomitants of ordered RVs, see Veena and Thomas ^[46] and the references therein. For a review of fundamental findings on COS, see Daivd and Nagaraja ^[9]. Furthermore, for some of the recent works on COS, we refer to Philip and Thomas ^[36,37,38], Kumar et al. ^[29], Barakat et al. ^[3], and Koshti and Kamalja ^[28].

Several authors have investigated the concomitants of GOS (CGOS), including Ahsanullah and Nevzorov ^[1], Beg and Ahsanullah ^[4], El-Din et al. ^[13], Domma and Giordano ^[11], Hanif and Shahbaz ^[15], Shahbaz and Shahbaz ^[40], Tahmasebi et al. ^[44], Alawady et al. ^[2], and Kamal et al. ^[20]. Let $(X_{i}, Y_{i})$, $i = 1, \cdots, n$ be a random sample from a bivariate distribution function $F_{X, Y}(x, y)$. When the $X$-variates are ordered in ascending order as $X_{1:n, \tilde{m}, k}\leq X_{2:n, \tilde{m}, k}\leq X_{3:n, \tilde{m}, k}\leq\cdots\leq X_{n:n, \tilde{m}, k}$, then $Y$-variates paired (not necessarily in ascending order) with these GOS are called the CGOS and are indicated by $Y_{[r:n, \tilde{m}, k]},$ $r = 1, \cdots, n$. The PDF of $Y_{[r:n, \tilde{m}, k]}$ is given by (Ahsanullah and Nevzorov, ^[1])

where $f(y|x)$ is the conditional PDF of $Y$ given $X$ and $f_{r:n, \tilde{m}, k}(x)$ is defined in (1.2).

Moreover, the joint PDF of $Y_{[r:n, \tilde{m}, k]}$ and $Y_{[s:n, \tilde{m}, k]},$ $r, s = 1, \cdots, n, \, r < s$ is given by

where $f_{r, s:n, \tilde{m}, k}(x_{1}, x_{2})$ is given in (1.3).

One of the most notable applications of COS is in ranked set sampling (RSS). RSS is considered a beneficial sampling strategy for improving estimation efficiency and precision if the variable under study is expensive to measure or difficult to get, yet inexpensive and simple to rank. RSS was proposed by McIntyre ^[31] and then supported by Takahasi and Wakimoto ^[45] through mathematical theory. The procedure for RSS is described as follows:

$1)$ Randomly choose $n^{2}$ units from the population under study, then divide them into $n$ sets of $n$ units.

$2)$ Order the elements of each set without making actual measurements.

$3)$ Choose and quantify the $i^{th}$ minimum from the $i^{th}$ set, $i = 1, \cdots, n$, to create a new set of size $n$, known as the RSS.

$4)$ If a large sample size is required, repeat the above three steps $d$ times (cycles) until a sample of size $n d$ is obtained.

For a comprehensive review of the theory and applications of RSS, see Chen et al. ^[7]. In some practical applications, the study variable, say, $Y$, is more difficult to measure, whereas an auxiliary variable $X$ associated with $Y$ is easily quantifiable and may be precisely arranged. In this situation, Stokes ^[42] created another RSS technique, which is as follows:

$1)$ At random, choose $n$ independent bivariate sets of size $n$.

$2)$ Take note of the value of the auxiliary variable on each of these units.

$3)$ From the $i^{th}$ set of size $n$, choose the variable $Y$ associated with the $i^{th}$ smallest $X$, $i = 1, \cdots, n.$

The resulting set of $n$ units is known as the RSS. Consider $(X_{(i:n)_{i}}, Y_{[i:n]_{i}})$, $i = 1, \cdots, n$ to be the pair chosen from the $i^{th}$ set, where $X_{(i:n)_{i}}$ is the $i^{th}$ order statistics of the auxiliary variate in the $i^{th}$ set and $Y_{[i:n]_{i}}$ is the measurement made on the $Y$ variate associated with $X_{(i:n)_{i}}$. $Y_{[i:n]_{i}}$ is obviously the concomitant of the $i^{th}$ order statistics resulting from the $i^{th}$ sample. Numerous authors in the literature have considered the estimation of parameters of the various bivariate distributions using RSS and its modifications. Some work in this area is by Chacko and Thomas ^[5], Philip and Thomas ^[36,37], Koshti and Kamalja ^[26], Irshad et al. ^[16,17], and Dong et al. ^[12].

COS and higher moments of multivariate distributions have received a lot of attention in recent years. Most of the literature on concomitants is concentrated on symmetric distributions such as multivariate normal (Sheikhi et al., ^[41]; Chaumette and Vincent, ^[6]) or multivariate elliptical (Jamalizadeh and Balakrishnan, ^[18]). Skewed distributions have gained a lot of interest recently in the literature since many datasets encountered in reality have some degree of skewness. In this regard, the distribution theory of COS from skew distributions has been investigated by several authors, including Hanif and Shahbaz ^[15], Shahbaz and Shahbaz ^[40], Tahmasebi et al. ^[44], Shahbaz et al. ^[39], and Kamal et al. ^[20]. In this article, we consider the bivariate generalized Weibull (BGW) distribution and the CGOS arising from it. There are numerous reasons for considering this particular bivariate distribution. Due to the presence of four parameters, the joint PDF of the BGW distribution is quite flexible and can take on various shapes depending on the shape parameter. The joint PDF, joint CDF, and conditional PDF for the BGW distribution are all in closed forms, making them appropriate for usage in practice. The univariate marginals of this distribution are able to analyze various types of hazard rates. In addition, it can be utilized for modeling bivariate lifetime data in a variety of scenarios. So far, no results on CGOS arising from the BGW distribution have been found in the literature. Thus, the current study aims to develop the distribution theory of CGOS originating from the BGW distribution and apply it to associated inference problems.

The article is structured as follows: In Section 2, we provide a brief overview of the BGW distribution and some of its properties. In Section 3, we present the marginal PDF as well as the explicit expressions for the single moments of CGOS from the BGW distribution. The joint PDF of CGOS from the BGW distribution is also obtained in Section 3. Furthermore, the explicit expressions for the product moments of CGOS are derived. Section 4 presents the best linear unbiased (BLU) estimator of the parameter of the study variable contained in the BGW distribution using Stokes's RSS and some of the other modified RSS schemes. In Section 5, we apply the results to a real dataset. In Section 6, conclusions are provided.

2.   BGW distribution

A bivariate RV $(X, Y)$ is said to follow a BGW distribution if its PDF is given by (Pathak et al. ^[34])

where $x, y\geq 0, \, \alpha, \beta_{1}, \beta_{2} > 0$, $0 < \theta\leq 1$, $\omega(x, y; \phi) = \frac{x^{\alpha}}{\beta_{1}}+\frac{y^{\alpha}}{\beta_{2}},$ and $\phi = (\alpha, \beta_{1}, \beta_{2})$. The BGW distribution includes the bivariate generalized exponential distribution (refer to Mirhosseini et al. ^[32]) and the bivariate generalized Rayleigh distribution (refer to Pathak and Vellaisamy ^[35]) as sub-models. The conditional PDF of $Y$ given $X = x$ is (Pathak et al. ^[34])

The RV $X\sim EW(\alpha, \beta_{1}, \theta)$ is a member of the exponentiated Weibull (EW) distribution with PDF

and CDF

Similarly, $Y\sim EW(\alpha, \beta_{2}, \theta)$. A series expansion of the PDF of the BGW distribution is given by

Pathak et al. ^[34] showed that the product moments of the BGW distribution are

If we make the transformation, $U = \frac{X}{\beta_{1}^{\star}}$ and $V = \frac{Y}{\beta_{2}^{\star}}$, $\beta_{i}^{\star} = \beta_{i}^{1/\alpha}, i = 1, 2$, the standard BGW distribution has the joint PDF as

It is clear that the variables $U$ and $V$ have the standard EW distribution as marginal functions with PDFs are, respectively, given by

3.   Distribution theory of CGOS from BGW distribution

In this part, we obtain the distributions and moments of CGOS arising from the BGW distribution.

3.1.   Marginal PDF and single moments of CGOS

Suppose $(X_{i}, Y_{i})$ and $(U_{i}, V_{i})$ are random samples of size $n$ each originating from the BGW distribution and the standard BGW distribution, with PDFs provided by (2.1) and (2.7), respectively. Let $V_{[r:n, \tilde{m}, k]}$ be the concomitant of the $r^{th}$ GOS $U_{r:n, \tilde{m}, k}$. Then, the PDF and the $p^{th}$ moments of $V_{[r:n, \tilde{m}, k]},$ $r = 1, \cdots, n$ are given by the following two theorems:

Theorem 1 If $V_{[r:n, \tilde{m}, k]}$ is the concomitant of the $r^{th}$ GOS from the standard BGW distribution, then the PDF of $V_{[r:n, \tilde{m}, k]}$, for $r = 1, \cdots, n,$ is given by

where $\delta_{\theta}(j, \gamma_{i}) = \binom{\theta}{j}\sum_{\tau = 0}^{\gamma_{i}-1}(-1)^{j+\tau +1}\binom{\gamma_{i}-1}{\tau} B(j, \theta \tau +1)$ and $B(., .)$ is the complete beta function.

Proof. Using the PDF of $U_{r:n, \tilde{m}, k}$ (1.2) in (1.6), the PDF of the $r^{th}$ CGOS $V_{[r:n, \tilde{m}, k]}$ is given as

In view of (2.7) and (2.8), we get

where $z = u^{\alpha}$. Now, by using Eq (3.3121) in Gradshteyn and Ryzhik ^[14] to compute the integral in (3.3), we obtain the result given in (3.1). □

Corollary 1. Taking $m_{1} = m_{2} = \cdots = m_{n-1} = m\neq-1$ in (3.1), the PDF of the $r^{th}$ concomitant of m-GOS from the standard BGW distribution is given by

where $\gamma_{i} = k+(n-i)(m+1).$

Remark 1. When $m = 0$ and $k = 1$ in (3.4), we get the PDF of the $r^{th}$ COS from the standard BGW distribution as

where $C_{r:n} = \frac{n!}{(r-1)!(n-r)!}$.

Theorem 2. Under the conditions of Theorem 1, the $p^{th}$ moment of $V_{[r:n, \tilde{m}, k]}$ is

Proof. Using (3.1), the $p^{th}$ moment of $V_{[r:n, \tilde{m}, k]}$ is given as

where $z = v^{\alpha}$. Then, after integration, we get (3.6). □

Corollary 2. Taking $m_{1} = m_{2} = \cdots = m_{n-1} = m\neq-1$ in (3.6), the $p^{th}$ moment of the concomitant of m-GOS is given by

Remark 2. Let $m = 0$ and $k = 1$ in (3.8), then the $p^{th}$ moment of COS is

3.2.   Joint PDF and product moments of CGOS

Let $V_{[r:n, \tilde{m}, k]}$ and $V_{[s:n, \tilde{m}, k]}$, $r, s = 1, \cdots, n, r < s$ be the concomitants of the $r^{th}$ and $s^{th}$ GOS from the standard BGW distribution. Then, the joint PDF and the product moments of $V_{[r:n, \tilde{m}, k]}$ and $V_{[s:n, \tilde{m}, k]}$ are given by the following two theorems:

Theorem 3. The joint PDF of concomitants $V_{[r:n, \tilde{m}, k]}$ and $V_{[s:n, \tilde{m}, k]}$, $r, s = 1, \cdots, n, r < s$ is given by

where

where ${}_3 F_{2}(a_{1}, \, a_{2}, \, a_{3};\, b_{1}, \, b_{2};\, x)$ denotes the hypergeometric function defined by

and $(c)_{\ell} = c (c+1)\cdots (c+\ell-1)$ is the ascending factorial.

Proof. Using (1.3) in (1.7), the joint PDF of the $r^{th}$ and $s^{th}$ CGOS $V_{[r:n, \tilde{m}, k]}$ and $V_{[s:n, \tilde{m}, k]}$ is given as

In view of (2.7) and (2.8), we get

where

where $w = e^{-u_{1}^{\alpha}}$ and $B_{w}(., .)$ denotes the incomplete beta function defined by $B_{w}(a_{1}, a_{2}) = \int_{0}^{w} x^{a_{1}-1} (1-x)^{a_{2}-1}dx.$

Now, putting the value of $I(u_{1})$ in (3.12), we get

where $z = u_{1}^{\alpha}$. We know that $B_{w}(a_{1}, a_{2}) = \frac{w^{a_{1}}}{a_{1}}{}_2 F_{1}(a_{1}, \, 1-a_{2};\, a_{1}+1;\, w)$ (see Mathai and Saxena, ^[30]), and

Therefore,

where $t = e^{-z}$. Now, using (3.15), we get the result of (3.10). □

Corollary 3. At $m_{1} = m_{2} = \cdots = m_{n-1} = m\neq-1$ in (3.10), the joint PDF of concomitants $V_{[r:n, m, k]}$ and $V_{[s:n, m, k]}$ of the $r^{th}$ and $s^{th}$ m-GOS for the standard BGW distribution is given by

where $\gamma_{i} = k+(n-i)(m+1).$

Remark 3. For $m = 0$ and $k = 1$ in (3.16), we obtain the joint PDF of the $r^{th}$ and $s^{th}$ COS from the standard BGW distribution as

where $C_{r, s:n} = \frac{n!}{(r-1)!(s-r-1)!(n-s)!}.$

Theorem 4. The product moments of two concomitants $V_{[r:n, \tilde{m}, k]}$ and $V_{[s:n, \tilde{m}, k]}$ are given by

Proof. Using (3.10), the $p^{th}$ and $q^{th}$ moments of $V_{[r:n, \tilde{m}, k]}$ and $V_{[s:n, \tilde{m}, k]}$ are given as

where $z_{i} = v_{i}^{\alpha}, i = 1, 2$. Then, after integration, we get (3.18). □

Corollary 4. Setting $m_{1} = m_{2} = \cdots = m_{n-1} = m\neq-1$ in (3.18), we can get the product moments of two concomitants of m-GOS of the standard BGW distribution.

Remark 4. When $m = 0$ and $k = 1$ in (3.18), we get the product moments of COS as

Remark 5. At $m_{i} = R_{i}, \, n = m_{0}+\sum_{i = 1}^{m_{0}}R_{i},$ and $\gamma_{i} = n-\sum_{\nu = 1}^{i-1}R_{\nu}-i+1,$ $1\leq i\leq m_{0}$ in Theorems 1–4, the results for PT-Ⅱ censored order statistics can be obtained.

Remark 6. From (3.9), the expressions of means and variances of the COS $Y_{[i:n]}, \, i = 1, \cdots, n,$ arising from the BGW distribution, are obtained as follows:

where $Var[V_{[i:n]}] = \mu_{[i:n]}^{(2)}-(\mu_{[i:n]})^{2}$. The expression of the covariances between $Y_{[i:n]}$ and $Y_{[j:n]}$ is given, using (3.9) and (3.20), by

where $Cov(V_{[i:n]}, V_{[j:n]}) = \mu_{[i, j:n]}- \mu_{[i:n]} \mu_{[j:n]}, 1\leq i < j\leq n.$

The means and variances of the COS of the standard BGW distribution for $n = 1, \cdots, 5$ and different values of the parameters $\alpha$ and $\theta$ are calculated in Tables 1 and 2. It can be noted that the condition $\sum_{r = 1}^{n}\mu_{[r:n]}^{j} = n \, \mu_{1:1}^{j}, j = 1, 2$ is satisfied (see David and Nagaraja, ^[10]). In Tables 3–6, we have computed the means and variances of the concomitants of PT-Ⅱ censored order statistics. From Tables 1–6, one can observe that the variances are decreasing with respect to $\alpha$.

4.   BLU estimator of the parameter $\beta_{2}^{\star}$ of BGW distribution

In this part, we obtain the BLU estimator of $\beta_{2}^{\star}$ involved in the BGW distribution using Stoke's RSS. Assume that $n$ sets of units, each of size $n$, are taken from the BGW distribution with the PDF given in (2.1). Let $X_{(i:n)_{i}}$, $i = 1, \cdots, n$ represent the observation made on the auxiliary variable $X$ in the $i^{th}$ unit of the RSS, and $Y_{[i:n]_{i}}$ represent the measurement performed on the $Y$ variable in the same unit. It is obvious that $Y_{[i:n]_{i}}$ has the same distribution as $Y_{[i:n]}$, the concomitant of the $i^{th}$ order statistics (see David and Nagraja, ^[10], p. 145). From Remark 6, the mean and the variance of $Y_{[i:n]_{i}}$ are given as $E[Y_{[i:n]_{i}}] = \beta_{2}^{\star} \mu_{[i:n]},$ and $Var[Y_{[i:n]_{i}}] = {\beta_{2}^{\star}}^{2} \delta_{i, i:n}, \, 1\leq i \leq n.$ Because the two measurements $Y_{[i:n]_{i}}$ and $Y_{[j:n]_{j}}$ $(i\neq j)$ of $Y$ are based on two independent samples, we have $Cov[Y_{[i:n]_{i}}, Y_{[j:n]_{j}}] = 0.$

Let $\textbf{Y}_{[n]} = (Y_{[1:n]_{1}}, Y_{[2:n]_{2}}, \cdots, Y_{[n:n]_{n}})^{'}$ denote the column vector of COS. Then, the mean vector and the variance-covariance matrix of $\textbf{Y}_{[n]}$ can be written as

and

where $\boldsymbol{\mu} = (\mu_{[1:n]}, \cdots, \mu_{[n:n]})^{'}$ and $\Lambda = diag(\delta_{1, 1:n}, \delta_{2, 2:n}, \cdots, \delta_{n, n:n}).$ If the parameters $\alpha$ and $\theta$ are known, then the combination of (4.1) and (4.2) allow us to apply the generalized Gauss-Markov theorem (see David and Nagraja, ^[10], p. 185). Hence, the BLU estimator $\hat{\beta_{2}^{\star}}$ of $\beta_{2}^{\star}$ is given as

where $a_{i} = \frac{ \mu_{[i:n]}/\delta_{i, i:n}}{\sum_{i = 1}^{n}\mu_{[i:n]}^2/\delta_{i, i:n}}$, and the variance of $\hat{\beta_{2}^{\star}}$ is given by

We have calculated the coefficients $a_{i}$ of $Y_{[i:n]_{i}}, i = 1, \cdots, n$ in $\hat{\beta_{2}^{\star}}$ and $Var[\hat{\beta_{2}^{\star}}]/{\beta_{2}^{\star}}^{2}$ for $n = 1, \cdots, 5$, and different values of the parameters $\alpha$ and $\theta$ are presented in Tables 7 and 8.

A modified RSS approach is presented by Stokes ^[43], wherein only the largest or smallest judgment ranked unit is selected for quantification. Let $n$ random samples each of size $n$ be drawn from the BGW distribution. From each of the $n$ samples, choose the unit for which the measurement on the auxiliary variable $X$ is the smallest (largest) and measure the $Y$ variable associated with it. Then, we call the collection of observations $Y_{[1:n]_{1}}, Y_{[1:n]_{2}}, \cdots, Y_{[1:n]_{n}}$ ($Y_{[n:n]_{1}}, Y_{[n:n]_{2}}, \cdots, Y_{[n:n]_{n}}$) as the lower RSS (LRSS) (upper RSS (URSS)).

Based on LRSS and URSS, the BLU estimators $\tilde{\beta}_{2, LRSS}^{\star}$ and $\tilde{\beta}_{2, URSS}^{\star}$ of $\beta_{2}^{\star}$ are

and their variances are

The efficiencies $e_{1}$ of $\tilde{\beta}_{2, LRSS}^{\star}$ and $e_{2}$ of $\tilde{\beta}_{2, URSS}^{\star}$ relative to $\hat{\beta_{2}^{\star}}$ are given by

see, for example, Koshti and Kamalja ^[27] and Philip and Thomas ^[37]. We have computed the efficiencies $e_{1}$ and $e_{2}$ for $n = 2, \cdots, 5$, $\alpha = 1, 2$, and $\theta = 0.50, 0.90$, which are presented in Table 9. From Table 9, it can be observed that:

● The efficiency $e_{1}$ is less than one for all selected values of $\alpha, \theta$, and $n$. So, $\hat{\beta_{2}^{\star}}$ is relatively more efficient than $\tilde{\beta}_{2, LRSS}^{\star}$.

● The efficiency $e_{1}$ decreases as $\alpha$ increases, and for a fixed pair $(n, \alpha)$, $e_{1}$ increases as $\theta$ increases.

● The efficiency $e_{2}$ is greater than one for all selected values of $\alpha, \theta$, and $n$. Thus, $\tilde{\beta}_{2, URSS}^{\star}$ is relatively more efficient than $\hat{\beta_{2}^{\star}}$.

● The efficiency $e_{2}$ increases as $\alpha$ increases, and for a fixed pair $(n, \alpha)$, $e_{2}$ decreases as $\theta$ increases.

5.   Real data application

For illustration purposes, we have considered the American Football League dataset given in Jamalizadeh and Kundu ^[19]. The bivariate dataset represents the game time to the first points scored by kicking the ball between goal posts $(X)$ and the 'game time' by moving the ball into the end zone $(Y)$. Pathak et al. ^[34] demonstrated that the BGW distribution fits this data better than other real-life time models. Here, we generate random samples of size five using forty-two pairs of observations. The samples under RSS schemes are displayed in Table 10.

The estimator of $\beta_{2}^{\star}$ under various RSS schemes is a function of $\alpha$ and $\theta$, which are unknown in this case. Thus, the method of moment estimation can be taken (see Kamalja and Koshti ^[21], for example). To obtain the moment estimators of $\alpha$ and $\theta$, we use the moment equations based on the moments of $Y$-observations and the moment equation based on the correlation between $(X, Y)$. These give $\hat{\alpha} = 3.39821$ and $\hat{\theta} = 0.24259$. Table 11 shows the estimates of $\beta_{2}^{\star}$ under the RSS, LRSS, and URSS schemes. The results show that $\tilde{\beta}_{2, URSS}^{\star}$ has the smallest variance. This is consistent with the findings of the efficiency performance study in Section 4.

6.   Conclusions

In this paper, we have considered the CGOS from the BGW distribution. We have derived the PDFs and moments of CGOS from the BGW distribution. Similar results for order statistics and PT-Ⅱ censored order statistics are presented as special instances. Finally, we have obtained the BLU estimator of the parameter associated with the study variable based on Stoke's RSS. Moreover, a real dataset is used for illustration purposes. The results for higher joint moments can be used to create skewness or kurtosis matrices (Kollo, ^[25]), which have important applications in both independent component analysis and invariant coordinate selection. This could be an interesting topic for future research. It will also be interesting to discuss the problem of predicting intervals for future order statistics and record values using concomitants of order statistics and record values arising from BGW distribution; see, for example, Muraleedharan and Chacko ^[33]. In addition, some information measures, such as the Shannon entropy and extropy, for CGOS can also be investigated.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author would like to acknowledge the Deanship of Graduate Studies and Scientific Research, Taif University for funding this work.

Conflict of interest

The author declares that no conflict of interest exist.