1 Introduction

Let \(F_{X}(x)\) and \(F_{Y}(y)\) be the cumulative distribution functions of random variables X and Y, respectively, and a parameter \(\alpha \), \(-1<\alpha <1\), then the probability density function (pdf) of Morgenstern family is given by (see [11])

$$\begin{aligned} f_{X,Y}(x,y)=f_{X}(x)f_{Y}(y)\left[ 1+\alpha \left( 2F_X(x)-1\right) (2F_Y(y)-1)\right] . \end{aligned}$$
(1.1)

Here, \(f_{X}(x)\) and \(f_{Y}(y)\) are the pdfs of X and Y, respectively. Also, the parameter \(\alpha \) measures the association between X and Y.

Kamps [10] introduced the concept of generalized order statistics (gos) as a unified models of ordered random variables such as ordinary order statistics, sequential order statistics, progressive type II censoring, record values, and Pfeifers records. Let \(n\in \mathbb {N}\), \(k>0\), \(m_r\in \mathbb {R}\), \(M_r=\sum _{j=r}^{n-1}m_{j}\) be the parameters such that \(\gamma _{r}=k + n - r+M_r>0\) for all \(1\le r\le n\). Also, let \(\tilde{m}=(m_1,\dots ,m_{n-1})\). The random variables \(X(1, n,\tilde{m}, k),X(2, n,\tilde{m}, k), \dots ,X(n, n,\tilde{m}, k)\) are called gos based on the absolutely continuous cumulative distribution function (cdf) F with pdf f, if their joint pdf is given by

$$\begin{aligned}&f_{1,2,\ldots ,n}(x_{1},x_{2},\ldots ,x_{n})\\&\quad =k\left( \prod \limits _{j=1}^{n-1}\gamma _{j}\right) \left( \prod \limits _{i=1}^{n-1}\big (1-F(x_{i})\big )^{m_i}f(x_{i})\right) \big (1-F(x_{n})\big )^{k-1}f(x_{n}), \end{aligned}$$

on the cone \(F^{-1}(0)<x_{1}\le \cdots \le x_{n}<F^{-1}(1)\) of \(\mathrm {R}^{n}\).

An important special case in the concept of gos is choosing \(m_i\) according to \(m_i=m\). In this paper, we consider this special case and denote the random variables \(X(r, n,\tilde{m}, k)\) by X(r, n, m, k), \(r=1,\dots ,n\).

The marginal pdf of rth generalized order statistic, X(r, n, m, k) is given by

$$\begin{aligned} f_{X(r, n, m, k)}(x)=\frac{C_{r}}{(r-1)!} [1-F(x)]^{\gamma _{r}-1}f(x)g^{r-1}_{m}\big (F(x)\big ), \end{aligned}$$
(1.2)

where \(C_{r}=\prod \limits _{j=1}^{r}\gamma _{j}\). Also, the joint pdf of rth and sth gos, \(\varvec{X}(r,s)=(X(r, n, m, k),X(s, n, \) m, k)), \(1\le r<s\le n\) is given by

$$\begin{aligned} f_{\varvec{X}(r,s)}(x,y)= & {} \frac{C_{s}}{(r-1)!(s-r-1)!} [1-F(x)]^{m}f(x)g^{r-1}_{m}\left( F(x)\right) \nonumber \\&\times [h_{m}(F(y))-h_{m}\left( F(x)\right) ]^{s-r-1}[1-F(y)]^{s-r-1}f(y),\; \; \; x<y,\nonumber \\ \end{aligned}$$
(1.3)

where \( g_{m}(t)=h_{m}(t)-h_{1}(t), \;\; t\in (0,1)\) and

$$\begin{aligned}h_{m}(t)=\left\{ \begin{array}{lc} -(m+1)^{-1}(1-t)^{m+1}&{} \ \ \ \ \ \ \ \;\; m\ne -1, \\ -\log (1-t) &{} \ \ \ \ \ \ \ \;\; m=-1. \end{array} \right. \end{aligned}$$

With taking \(m=0\) and \(k=1\), the random variable X(r, n, m, k) reduces to the r-th order statistics, and with taking \(m=-1\) and \(k=1\), the random variable X(r, n, m, k) reduces to r-th upper record value. For more details and some applications of gos, reader can refer to [1–5].

Let \((X_{1},Y_{1}),(X_{2},Y_{2}),\ldots ,(X_{n},Y_{n})\) be n pairs of independent random variables from a bivariate population with joint pdf f(x, y). If the X-variates are arranged in increasing order as \(X(1, n, m, k)\le X(2, n, m, k)\le \dots \le X(n, n, m, k)\), then Y-variates paired (not necessarily in increasing order) with these gos’s are called the concomitants of gos and denoted by \(Y_{[1,n,m,k]},Y_{[2,n,m,k]},\ldots ,Y_{[n,n,m,k]}\). Concomitants of gos are studied by [6–8]. For Morgenstern family with pdf given by (1.1), the pdf of the concomitant of rth- gos \(Y_{[r,n,m,k]}\),\( 1\le r\le n\), is given by [6] as follows:

$$\begin{aligned} g_{[r,n,m,k]}(y)= & {} f_Y(y)\left[ 1+\alpha C^{*}(r,n,m,k)(2F_Y(y)-1)\right] \nonumber \\= & {} f_{1:1}(y)+\alpha C^{*}(r,n,m,k)[f_{2:2}(y)-f_{1:1}(y)], \end{aligned}$$
(1.4)

where

$$\begin{aligned} C^{*}(r,n,m,k)=1-2\frac{C_{r}}{C_{r}(1)},\;\;C_{r}(1)=(\gamma _{1}+1)(\gamma _{2}+1)\ldots (\gamma _{r}+1), \end{aligned}$$

and \(f_{i:n}(y)\) is the pdf of \(Y_{i,n}\), the ith order statistic of a random sample of size n of Y. Note that \(g_{[r,n,m,k]}(y)\) depends only on the marginal distribution of Y and the distribution of \(Y_{2,2}\).

In this paper, we study and derive the properties of concomitants of gos in MTBRD. This paper is organized as follows: In Sect. 2, we study the properties of the marginal distributions of concomitants for gos from MTBRD. In Sect. 3, we consider two concomitants of gos in MTBRD and obtain the properties of their joint distributions. In Sect. 4, we derive the best linear unbiased estimator (BLUE) for a parameter of MTBRD based on concomitants of first n record values.

2 Concomitants of gos in MTBRD

The Rayleigh distribution applies as an important model in noise theory, height of the sea waves, wave length, wave induce pitch, weapon testing, and flight testing. It is reasonable to construct bivariate Rayleigh distribution when the marginal distributions are Rayleigh. By considering Rayleigh distribution [denoted by \(R(\sigma )\)] with the cdf,

$$\begin{aligned} F(x)=1-\exp \left( -\frac{x^{2}}{2\sigma ^{2}}\right) , ~~ x>0, ~~ \sigma >0. \end{aligned}$$

A new member of bivariate distribution is MTBRD with pdf given by

$$\begin{aligned}&f_{X,Y}(x,y)=\frac{xy}{\sigma _{1}^{2}\sigma _{2}^{2}}\exp \left( -\frac{x^{2}}{2\sigma _{1}^{2}}-\frac{y^{2}}{2\sigma _{2}^{2}}\right) \nonumber \\&\quad \,\times \,\left[ 1+\alpha \left( 2\exp \left( -\frac{x^{2}}{2\sigma _{1}^{2}}\right) -1\right) \left( 2\exp \left( -\frac{y^{2}}{2\sigma _{2}^{2}}\right) -1\right) \right] , x,y>0.\quad \end{aligned}$$
(2.1)

In this section, we consider the concomitants of gos in MTBRD and obtain the properties of their marginal distributions. Also, some recurrence relations between moments of concomitants are presented.

2.1 Marginal Distribution of Concomitants

Using (1.4), the pdf and cdf of \(Y_{[r,n,m,k]}\) for MTBRD is obtained as

$$\begin{aligned} g_{[r,n,m,k]}(y)=\frac{y}{\sigma _{2}^{2}}\exp \left( -\frac{y^{2}}{2\sigma _{2}^{2}}\right) \left[ \delta _{1,r}+\delta _{2,r}\exp \left( -\frac{y^{2}}{2\sigma _{2}^{2}}\right) \right] , \end{aligned}$$
(2.2)

and

$$\begin{aligned} G_{[r,n,m,k]}(y)=\delta _{1,r}\left[ 1-\exp \left( -\frac{y^{2}}{2\sigma _{2}^{2}}\right) \right] +\frac{\delta _{2,r}}{2}\left[ 1-\exp \left( -\frac{y^{2}}{\sigma _{2}^{2}}\right) \right] , \end{aligned}$$

where

$$\begin{aligned} \delta _{1,r}=1+\alpha C^{*}(r,n,m,k),\;\;\; \delta _{2,r}=-2\alpha C^{*}(r,n,m,k) \end{aligned}$$

Obviously, we can find that

$$\begin{aligned} g_{[r,n,m,k]}(y)= & {} \delta _{1,r}f_Y(y)+\frac{\delta _{2,r}}{2}f_V(y)=\left[ 1-\frac{\delta _{2,r}}{2}\right] f_Y(y)+\frac{\delta _{2,r}}{2}f_V(y)\nonumber \\= & {} f_Y(y)+\frac{\delta _{2,r}}{2}\left[ f_V(y)-f_Y(y)\right] , \end{aligned}$$
(2.3)

where \(f_Y(y)\) and \(f_V(y)\) are pdf’s of random variables Y and V with \(R(\sigma _{2})\) and \(R\left( \frac{\sigma _{2}}{\sqrt{2}}\right) \), respectively.

Remark 2.1

With taking \(m = 0\) and \(k = 1\) in (2.2), the pdf and the hazard rate function of r-th concomitant of order statistic from MTBRD are given by

$$\begin{aligned}&g_{[r:n]}(y)=\frac{y}{\sigma _{2}^{2}}\exp \left( -\frac{y^{2}}{2\sigma _{2}^{2}}\right) \nonumber \\&\quad \times \left[ 1-\alpha \left( \frac{n-2r+1}{n+1}\right) +2\alpha \left( \frac{n-2r+1}{n+1}\right) \exp \left( -\frac{y^{2}}{2\sigma _{2}^{2}}\right) \right] , 1\le r\le n, \nonumber \\ \end{aligned}$$
(2.4)

and

$$\begin{aligned} h_{[r:n]}(y)=\frac{g_{[r:n]}(y)}{1-G_{[r:n]}(y)}=\frac{\frac{y}{\sigma _{2}^{2}}\left[ 1-\alpha \left( \frac{n-2r+1}{n+1}\right) \left( 1-\exp \left( -\frac{y^{2}}{2\sigma _{2}^{2}}\right) \right) \right] }{1-\alpha \left( \frac{n-2r+1}{n+1}\right) \left( 1-2\exp \left( -\frac{y^{2}}{2\sigma _{2}^{2}}\right) \right) }. \end{aligned}$$

Note that \(h_{[r:n]}(y)\) can be applied in reliability theory when the random variables defined above are taken as the lifetimes of system components or human lives at certain ages.

Remark 2.2

With taking \(m = -1\) and \(k = 1\) in (2.2), the pdf and the hazard rate function of r-th concomitant of record value \((R_{[r]})\) from MTBRD are given by

$$\begin{aligned} g_{R_{[r]}}(y)= & {} \frac{y}{\sigma _{2}^{2}}\exp \left( -\frac{y^{2}}{2\sigma _{2}^{2}}\right) \nonumber \\&\times \left[ 1+\alpha \left( \frac{2^{r-1}-1}{2^{r-1}}\right) -2\alpha \left( \frac{2^{r-1}-1}{2^{r-1}}\right) \exp \left( -\frac{y^{2}}{2\sigma _{2}^{2}}\right) \right] , \end{aligned}$$
(2.5)

and

$$\begin{aligned} h_{R_{[r]}}(y)=\frac{\frac{y}{\sigma _{2}^{2}}\left[ 1+\alpha \left( \frac{2^{r-1}-1}{2^{r-1}}\right) \left( 1-\exp \left( -\frac{y^{2}}{2\sigma _{2}^{2}}\right) \right) \right] }{1+\alpha \left( \frac{2^{r-1}-1}{2^{r-1}}\right) \left( 1-2\exp \left( -\frac{y^{2}}{2\sigma _{2}^{2}}\right) \right) }. \end{aligned}$$

2.2 Moment Generating Function and Moments of \(Y_{[r,n,m,k]}\)

Using (2.3), the moment generating function (mgf) of \(Y_{[r,n,m,k]}\) is given by

$$\begin{aligned} M_{[r,n,m,k]}(t)= & {} M_Y(t)+\frac{\delta _{2,r}}{2}\left[ M_V(t)-M_Y(t)\right] \nonumber \\= & {} 1+\sigma _{2} t\exp \left( \frac{-\sigma _{2}^{2}t^{2}}{2}\right) \sqrt{\frac{\pi }{2}}\left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2}\right) +1\right] \nonumber \\&+\frac{\delta _{2,r}}{2}\left[ \sigma _{2} t\exp \left( \frac{-\sigma _{2}^{2}t^{2}}{4}\right) \sqrt{\frac{\pi }{2}}\left\{ \frac{1}{\sqrt{2}}\left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2\sqrt{2}}\right) +1\right] \right. \right. \nonumber \\&\left. \left. -\exp \left( \frac{-\sigma _{2}^{2}t^{2}}{4}\right) \left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2}\right) +1\right] \right\} \right] , \end{aligned}$$
(2.6)

where \(M_Y(t)\) and \(M_U(t)\) are mgf of random variables Y and U with \(R(\sigma _{2})\) and \(R\left( \frac{\sigma _{2}}{\sqrt{2}}\right) \), respectively, and \(\mathrm{erf}(x)=\frac{2}{\sqrt{\pi }}\int _{0}^{x}e^{-u^{2}}\mathrm{d}u\). With differentiating (2.6) with respect to t, we get the l-th moment of \(Y_{[r,n,m,k]}\) as

$$\begin{aligned} \mu ^{(l)}_{[r,n,m,k]}= & {} E\left( Y^{l}_{[r,n,m,k]}\right) =E(Y^l)+\frac{\delta _{2,r}}{2}\left[ E(V^l)-E(Y^l)\right] \nonumber \\= & {} \left( \sqrt{2}\sigma _{2}\right) ^{l}\Gamma \left( \frac{l}{2}+1\right) +\frac{\delta _{2,r}}{2}(\sigma _{2})^{l}\Gamma \left( \frac{l}{2}+1\right) \left[ 1-(\sqrt{2})^{l}\right] . \end{aligned}$$
(2.7)

Since (2.7) is a convergent series for any \(l\ge 0\), so all the moments exist for integer values of \(\sigma _{2}\). With putting \(l=1\), we obtain the mean as

$$\begin{aligned} \mu _{[r,n,m,k]}=E\left( Y_{[r,n,m,k]}\right) =\sigma _{2}\sqrt{\frac{\pi }{2}}+\frac{\delta _{2,r}}{4}\sqrt{\pi }\left( 1-\sqrt{2}\right) . \end{aligned}$$
(2.8)

In general, if h(y) is a measurable function of y, then

$$\begin{aligned} E\left( h(Y_{[r,n,m,k]})\right) =E\left( h(Y)\right) +\frac{\delta _{2,r}}{2}\big [E(h(V))-E(h(Y))\big ], \end{aligned}$$

and for \(r=2,\dots ,n\)

$$\begin{aligned}&E\left\{ h(Y_{[r,n,m,k]})\right\} -E\left\{ h(Y_{[r-1,n,m,k]})\right\} \\&\quad =\left[ \frac{\delta _{2,r}-\delta _{2,r-1}}{2}\right] \left[ E\left( h(V)\right) -E\left( h(Y)\right) \right] \\&\quad =4\alpha \left[ \prod \limits _{j=1}^{r}\frac{\gamma _{j}}{\gamma _{j}+1}-\prod \limits _{j=1}^{r-1}\frac{\gamma _{j}}{\gamma _{j}+1}\right] \left[ E(h(V))-E(h(Y))\right] \\&\quad =4\alpha \left[ \frac{\gamma _{1}\gamma _{2}\cdots \gamma _{r-1}}{(\gamma _{1}+1)(\gamma _{2}+1)\cdots (\gamma _{r-1}+1)(\gamma _{r}+1)}\right] \\&\qquad \times \left[ E(h(Y))-E(h(V))\right] . \end{aligned}$$

Remark 2.3

With taking \(m = 0\) and \( k = 1\) in (2.8), the mean of r-th concomitant of order statistic from MTBRD is given as

$$\begin{aligned} \mu _{[r:n]}=\sigma _{2}\sqrt{\frac{\pi }{2}}\left[ 1+\frac{\alpha (n-2r+1)}{n+1} \left( \frac{\sqrt{2}}{2}-1\right) \right] . \end{aligned}$$

Therefore, the following recurrence relations between the means of concomitants of order statistics are obtained as

$$\begin{aligned}&\mu _{[r+1:n]}=\frac{\mu _{[r+2:n]}+\mu _{[r:n]}}{2} =\mu _{[r:n]}+\frac{\sigma _{2}\sqrt{\pi }\alpha }{n+1}(\sqrt{2}-1),\\&\mu _{[r:n]}-\mu _{[r:n-1]}=\frac{\sigma _{2}\sqrt{\pi }\alpha r}{n(n+1)}(1-\sqrt{2}),\\&\mu _{[r:n]}-\mu _{[r-i:n]}=\frac{\sigma _{2}\sqrt{\pi }\alpha i}{n+1} (\sqrt{2}-1), \;\;\;1\le i\le r-1,\\&\mu _{[r:n]}-\mu _{[r:n-j]}= \frac{\sigma _{2}\sqrt{\pi }\alpha r j}{(n-j+1)(n+1)}(1-\sqrt{2}),\;\;\; 1\le j\le n-j,\\&\mu =\frac{1}{n}\sum \limits _{r=1}^{n}\mu _{[r:n]}=\sigma _{2} \sqrt{\frac{\pi }{2}},\\&\mu _{[r:n]}=\mu _{[1:n]}+\frac{\sigma _{2}\sqrt{\pi }\alpha (r-1)}{n+1}(\sqrt{2}-1), \;\;\;1\le r\le n,\\&\mu _{[r:n]}=\mu _{[r:r]}+\frac{\sigma _{2}\sqrt{\pi }\alpha r (n-r)}{(r+1)(n+1)}(1-\sqrt{2}),\;\;\; 1\le r\le n,\\&\mu _{[n-r+1:n]}=\mu _{[r:n]}+\sigma _{2}\delta _{r}(\sqrt{2}-1),\\&\mu _{[r:n]}=\mu _{[r\beta :(n+1)\beta -1]},\;\;\;\; \beta \ge 1, \\&\mu _{[r:n]}=\sum \limits _{s=n-r+1}^{n}(-1)^{s-n+r-1}\genfrac(){0.0pt}0{s-1}{n-r}\genfrac(){0.0pt}0{n}{s}\mu _{[1:s]}.\nonumber \end{aligned}$$

Remark 2.4

Set \(m =-1\) and \(k=1\) in (2.8), to get the mean of r-th concomitant of record value from MTBRD as

$$\begin{aligned} \mu _{R_{[r]}}=E(R_{[r]})=\sigma _{2}\sqrt{\frac{\pi }{2}}\left[ 1+\alpha \left( \frac{1}{2^{r-1}}-1\right) \left( \frac{\sqrt{2}}{2}-1\right) \right] , \;\; { r=1,\dots ,n }. \end{aligned}$$

An explicit expression of Shannon entropy for concomitants of gos in Morgenstern family is given as (see [12])

$$\begin{aligned} H(Y_{[r,n,m,k]}){=}E[-\log g_{[r,n,m,k]}(Y)]=W_{r,n,m,k}+H(Y)\left( 1{+}\frac{\delta _{2,r}}{2}\right) +\delta _{2,r}\phi _{f}(u),\nonumber \\ \end{aligned}$$
(2.9)

where

$$\begin{aligned}&W_{r,n,m,k}\nonumber \\&=\frac{1}{2\delta _{2,r}}\left\{ \left( 1-\frac{\delta _{2,r}}{2}\right) ^{2} \log \left( 1-\frac{\delta _{2,r}}{2}\right) -\left( 1+\frac{\delta _{2,r}}{2}\right) ^{2}\log \left( 1+\frac{\delta _{2,r}}{2}\right) \right\} +\frac{1}{2},\nonumber \\ \end{aligned}$$
(2.10)

and \(\phi _{f}(u)=\int _{0}^{1}u\log f_{Y}\left( F^{-1}_{Y}(u)\right) \mathrm{d}u.\) Applying this expression for \(Y_{[r:n]}\) in MTBRD, we have

$$\begin{aligned} H(Y_{[r:n]})=Z_{\alpha ,n}(r)+\frac{\alpha (n-2r+1)}{n+1}\left( \log \sqrt{2} -\frac{1}{2}\right) +1-\frac{1}{2}\psi (1)+\log \left( \frac{\sigma _{2}}{\sqrt{2}}\right) , \end{aligned}$$

where

$$\begin{aligned} Z_{\alpha ,n}(r)= & {} \frac{n+1}{8\alpha (n-2r+1)}\left\{ \left( 1-\frac{\alpha (n-2r+1)}{n+1}\right) ^2\left[ 2\log (1-\frac{\alpha (n-2r+1)}{n+1})-1\right] \right. \nonumber \\&\left. -\left( 1+\frac{\alpha (n-2r+1)}{n+1}\right) ^2\left[ 2\log \left( 1+\frac{\alpha (n-2r+1)}{n+1}\right) -1\right] \right\} . \end{aligned}$$

2.3 Some Recurrence Relations

We will present several recurrence relations between pdf’s, moments, and mgf’s of concomitants. From (2.3), for \(r=2,\dots ,n\) we have

$$\begin{aligned}&g_{[r,n,m,k]}(y)-g_{[r-1,n,m,k]}(y)=2\alpha \left[ \prod \limits _{j=1}^{r} \frac{\gamma _{j}}{\gamma _{j}+1}- \prod \limits _{j=1}^{r-1}\frac{\gamma _{j}}{\gamma _{j}+1}\right] \left[ f_{V}(y)-f_{Y}(y)\right] ,\\&g_{[r,n,m,k]}(y)-g_{[r-1,n-1,m,k]}(y)=2\lambda \left[ \prod \limits _{j=1}^{r} \frac{\gamma _{j}}{\gamma _{j}+1}-\prod \limits _{j=1}^{r-1}\frac{\gamma _{j+1}}{\gamma _{j+1}+1}\right] \left[ f_{V}(y)-f_{Y}(y)\right] ,\\&g_{[r-1,n,m,k]}(y)-g_{[r-1,n-1,m,k]}(y)=2\lambda \left[ \prod \limits _{j=1}^{r-1} \frac{\gamma _{j}}{\gamma _{j}+1}-\prod \limits _{j=1}^{r-1}\frac{\gamma _{j+1}}{\gamma _{j+1}+1}\right] \left[ f_{V}(y)-f_{Y}(y)\right] . \end{aligned}$$

Also, \(1\le i\le n-r\) and \(1\le j\le r-1\), we have

$$\begin{aligned}&g_{[r,n,m,k]}(y)-g_{[r,n-i,m,k]}(y)=\alpha [C^{*}(r,n-i,m,k)\\&-C^{*}(r,n,m,k)][f_{V}(y)-f_{Y}(y)],\\&g_{[r,n,m,k]}(y)-g_{[r-j,n,m,k]}(y)=\lambda [C^{*}(r-j,n,m,k)\\&-C^{*}(r,n,m,k)][f_{V}(y)-f_{Y}(y)],\\&g_{[r,n,m,k]}(y)-g_{[r-j,n-i,m,k]}(y)=\lambda [C^{*}(r-j,n-i,m,k)\\&-C^{*}(r,n,m,k)][f_{V}(y)-f_{Y}(y)]. \end{aligned}$$

Using (2.7) the following recurrence relations between moments of concomitants are valid:

$$\begin{aligned}&\mu ^{(l)}_{[r,n,m,k]}-\mu ^{(l)}_{[r,n-i,m,k]}=\alpha [C^{*} (r,n-i,m,k)\\&\quad -C^{*}(r,n,m,k)](\sigma _{2})^{l}\Gamma \left( \frac{l}{2}+1\right) [1-(\sqrt{2})^{l}],\\&\mu ^{(l)}_{[r,n,m,k]}-\mu ^{(l)}_{[r-j,n,m,k]}=\alpha [C^{*} (r-j,n,m,k)\\&\quad -C^{*}(r,n,m,k)](\sigma _{2})^{l}\Gamma \left( \frac{l}{2}+1\right) [1-(\sqrt{2})^{l}],\\&\mu ^{(l)}_{[r,n,m,k]}-\mu ^{(l)}_{[r-j,n-i,m,k]}=\alpha [C^{*} (r-j,n-i,m,k)\\&\quad -C^{*}(r,n,m,k)](\sigma _{2})^{l}\Gamma \left( \frac{l}{2}+1\right) [1-(\sqrt{2})^{l}], \end{aligned}$$

Furthermore, for \(i,j=1\), we have

$$\begin{aligned} \mu ^{(l)}_{[r,n,m,k]}-\mu ^{(l)}_{[r,n-1,m,k]}&=\left[ \frac{2r\alpha (m+1) \gamma _{2}\gamma _{3}\ldots \gamma _{r}}{(\gamma _{1}+1) (\gamma _{2}+1)\ldots (\gamma _{r}+1)(\gamma _{r+1}+1)}\right] \\&\quad \times (\sigma _{2})^{l} \Gamma \left( \frac{l}{2}+1\right) [1-(\sqrt{2})^{l}],\\ \mu ^{(l)}_{[r,n,m,k]}-\mu ^{(l)}_{[r-1,n,m,k]}&=\left[ \frac{2\alpha \gamma _{1}\gamma _{2}\ldots \gamma _{r-1}}{(\gamma _{1}+1)(\gamma _{2}+1)\ldots (\gamma _{r-1}+1)(\gamma _{r}+1)}\right] \\&\quad \times (\sigma _{2})^{l}\Gamma \left( \frac{l}{2}+1\right) [1-(\sqrt{2})^{l}]. \end{aligned}$$

For \(r=2,\dots ,n\) , \(1\le i_{1}\le i_{2}\le n-r\) and \(1\le j_{1}\le j_{2}\le r-1\), the relation between mgf’s of concomitants are

$$\begin{aligned}&M_{[r,n,m,k]}(t)- M_{[r,n-i_{1},m,k]}(t)\\&\quad =\alpha \left[ C^{*}(r,n-i_{1},m,k)-C^{*}(r,n,m,k)\right] \\&\qquad \times \left[ \sigma _{2} t\exp \left( \frac{-\sigma _{2}^{2}t^{2}}{4}\right) \sqrt{\frac{\pi }{2}}\left\{ \frac{1}{\sqrt{2}} \left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2\sqrt{2}}\right) +1\right] \right. \right. \\&\qquad \left. \left. -\exp \left( \frac{-\sigma _{2}^{2}t^{2}}{4}\right) \left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2}\right) +1\right] \right\} \right] ,\\ \end{aligned}$$
$$\begin{aligned}&M_{[r,n,m,k]}(t)- M_{[r-j_{1},n,m,k]}(t)\\&\quad =\alpha \left[ C^{*}(r-j_{1},n,m,k)-C^{*}(r,n,m,k)\right] \\&\quad \times \left[ \sigma _{2} t\exp \left( \frac{-\sigma _{2}^{2}t^{2}}{4}\right) \sqrt{\frac{\pi }{2}} \left\{ \frac{1}{\sqrt{2}} \left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2\sqrt{2}}\right) +1\right] \right. \right. \\&\quad \left. \left. -\exp \left( \frac{-\sigma _{2}^{2}t^{2}}{4}\right) \left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2}\right) +1\right] \right\} \right] ,\\ \end{aligned}$$
$$\begin{aligned}&M_{[r,n,m,k]}(t)- M_{[r-j_{1},n-i_{1},m,k]}(t)\\&=\alpha \left[ C^{*}(r-j_{1},n-i_{1},m,k)-C^{*}(r,n,m,k)\right] \\&\quad \times \left[ \sigma _{2} t\exp \left( \frac{-\sigma _{2}^{2}t^{2}}{4}\right) \sqrt{\frac{\pi }{2}} \left\{ \frac{1}{\sqrt{2}}\left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2\sqrt{2}}\right) +1\right] \right. \right. \\&\quad \left. \left. -\exp \left( \frac{-\sigma _{2}^{2}t^{2}}{4}\right) \left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2}\right) +1\right] \right\} \right] ,\\ \end{aligned}$$
$$\begin{aligned}&M_{[r-j_{1},n-i_{1},m,k]}(t)- M_{[r-j_{2},n-i_{2},m,k]}(t)\\&\quad = \alpha \left( C^{*}(r-j_{2},n-i_{2},m,k\right) \\&\qquad -C^{*}(r-j_{1},n-i_{1},m,k)) \left[ \sigma _{2} t\exp \left( \frac{-\sigma _{2}^{2}t^{2}}{4}\right) \right. \\&\quad \sqrt{\frac{\pi }{2}}\left\{ \frac{1}{\sqrt{2}}\left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2\sqrt{2}}\right) +1\right] \right. \\&\left. \left. \quad -\exp \left( \frac{-\sigma _{2}^{2}t^{2}}{4}\right) \left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2}\right) +1\right] \right\} \right] . \end{aligned}$$

If we take \(m=0\) and \(k=1\), then the l-th moment and mgf of \(Y_{[r:n]}\) can be deduced from (2.6) and (2.7), respectively as

$$\begin{aligned} E[Y^{l}_{[r:n]}]= & {} \left( \sqrt{2}\sigma _{2}\right) ^{l}\Gamma \left( \frac{l}{2}+1\right) \left[ 1 +\frac{\alpha (n-2r+1)}{n+1}\left( 2^{\frac{-l}{2}}-1\right) \right] ,\\ M_{[r:n]}(t)= & {} \left( 1-\frac{\alpha (n-2r+1)}{n+1}\right) \left[ 1+\sigma _{2} t\exp \left( \frac{-\sigma _{2}^{2}t^{2}}{2}\right) \sqrt{\frac{\pi }{2}}\left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2}\right) +1\right] \right] \\&+\frac{\alpha (n-2r+1)}{n+1}\left[ 1+\frac{\sigma _{2}}{\sqrt{2}} t\exp \left( \frac{-\sigma _{2}^{2}t^{2}}{4}\right) \sqrt{\frac{\pi }{2}}\left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2\sqrt{2}}\right) +1\right] \right] . \end{aligned}$$

If we take \(m=-1\) and \(k=1\), then the l-th moment and mgf of r-th concomitant of record value can be deduced from (2.6) and (2.7), respectively as

$$\begin{aligned} \mu ^{(l)}_{R_{[r]}}= & {} (\sqrt{2}\sigma _{2})^{l}\Gamma \left( \frac{l}{2} +1\right) \left[ 1+\alpha \left( \frac{1}{2^{r-1}}-1\right) \left( 2^{\frac{-l}{2}}-1\right) \right] ,\\ M_{R_{[r]}}(t)= & {} \left( 1-\alpha \left( \frac{1}{2^{r-1}}-1\right) \right) \left[ 1+\sigma _{2} t\exp \left( \frac{-\sigma _{2}^{2}t^{2}}{2}\right) \sqrt{\frac{\pi }{2}}\left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2}\right) +1\right] \right] \nonumber \\&+\alpha \left( \frac{1}{2^{r-1}}-1\right) \left[ 1+\frac{\sigma _{2}}{\sqrt{2}} t\exp \left( \frac{-\sigma _{2}^{2}t^{2}}{4}\right) \sqrt{\frac{\pi }{2}}\left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2\sqrt{2}}\right) +1\right] \right] . \end{aligned}$$

3 Joint Distribution of Two Concomitants

In this Section, we obtain the joint distribution of concomitants of two gos and study their properties.

Let \(Y_{[r,n,m,k]}\) and \(Y_{[s,n,m,k]}\), \(1\le r <s\le n\) be concomitants of the r-th and s-th gos from a Morgenstern family. The joint pdf of \(Y_{[r,n,m,k]}\) and \(Y_{[s,n,m,k]}\) is given by (see [6])

$$\begin{aligned} g_{[r,s,n,m,k]}(y)&=f_{Y}(y_{1})f_{Y}(y_{2})\left[ 1+\alpha \big (1-2F_{Y}(y_{1})\big ) \left[ 2\prod \limits _{i=1}^{r}\frac{\gamma _{i}}{\gamma _{i}+1}-1\right] \right. \nonumber \\&\quad +\alpha \big (1-2F_{Y}(y_{2})\big )\left[ 2\prod \limits _{i=1}^{s}\frac{\gamma _{i}}{\gamma _{i}+1}-1\right] \nonumber \\&\quad +\alpha ^{2}\left( 1-2F_{Y}(y_{1})\right) \left( 1-2F_{Y}(y_{2})\right) \nonumber \\&\quad \times \left[ 4\left\{ \frac{\gamma _{1}\gamma _{2}\cdots \gamma _{r}\gamma _{r+1}\cdots \gamma _{s}}{(\gamma _{1}+2)(\gamma _{2}+2)\cdots (\gamma _{r}+2)(\gamma _{r+1}+1) \cdots (\gamma _{s}+1)}\right\} \right. \nonumber \\&\quad \left. \left. -2\prod \limits _{i=1}^{s}\frac{\gamma _{i}}{\gamma _{i}+1} -2\prod \limits _{i=1}^{r}\frac{\gamma _{i}}{\gamma _{i}+1}+1\right] \right] . \end{aligned}$$
(3.1)

Therefore, the joint pdf of these concomitants for MTBRD is

$$\begin{aligned} f_{[r,s,n,m,k]}(y)= & {} \frac{y_{1}y_{2}}{[\sigma _{1}\sigma _{2}]^{2}} \exp \left( -\frac{y_{1}^{2}}{2\sigma _{1}^{2}}-\frac{y_{2}^{2}}{2\sigma _{2}^{2}}\right) \nonumber \\&\times \left[ 1+\alpha \left( 2\exp \left( -\frac{y_{1}^{2}}{2\sigma _{1}^{2}}\right) -1\right) \left[ 2\prod \limits _{i=1}^{r}\frac{\gamma _{i}}{\gamma _{i}+1}-1\right] \right. \nonumber \\&+\alpha \left( 2\exp \left( -\frac{y_{2}^{2}}{2\sigma _{2}^{2}}\right) -1\right) \left[ 2 \prod \limits _{i=1}^{s}\frac{\gamma _{i}}{\gamma _{i}+1}-1\right] \nonumber \\&+\alpha ^{2}\left( 2\exp \left( -\frac{y_{1}^{2}}{2\sigma _{1}^{2}}\right) -1\right) \left( 2 \exp \left( -\frac{y_{2}^{2}}{2\sigma _{2}^{2}}\right) -1\right) \nonumber \\&\times \left[ 4\left\{ \frac{\gamma _{1}\gamma _{2}\cdots \gamma _{r}\gamma _{r+1} \cdots \gamma _{s}}{(\gamma _{1} +2)(\gamma _{2}+2)\cdots (\gamma _{r}+2)(\gamma _{r+1}+1) \cdots (\gamma _{s}+1)}\right\} \right. \nonumber \\&\left. \left. -2\prod \limits _{i=1}^{s}\frac{\gamma _{i}}{\gamma _{i}+1}-2\prod \limits _{i=1}^{r}\frac{\gamma _{i}}{\gamma _{i}+1}+1\right] \right] . \end{aligned}$$
(3.2)

Also, the joint mgf of \(Y_{[r,n,m,k]}\) and \(Y_{[s,n,m,k]}\) is obtained as

$$\begin{aligned} M_{[r,s,n,m,k]}(t_{1},t_{2})= & {} M_{Y}(t_{1})M_{Y}(t_{2}) \nonumber \\&+ \alpha \left[ 2\prod \limits _{i=1}^{r}\frac{\gamma _{i}}{\gamma _{i}+1}-1\right] \left[ M_{Y} (t_{1})M_{Y}(t_{2})-M_{V}(t_{1})M_{Y}(t_{2})\right] \nonumber \\&+\,\alpha [2\prod \limits _{i=1}^{s}\frac{\gamma _{i}}{\gamma _{i}+1}-1] [M_{Y}(t_{1})M_{Y}(t_{2})-M_{Y}(t_{1})M_{V}(t_{2})]\nonumber \\&+\,\alpha ^{2}[M_{Y}(t_{1})-M_{V}(t_{1})][M_{Y}(t_{2})-M_{V}(t_{2})]\nonumber \\&\times \left[ 4\left( \frac{\gamma _{1}\gamma _{2}\cdots \gamma _{r}\gamma _{r+1}\cdots \gamma _{s}}{(\gamma _{1}+2)(\gamma _{2}+2) \cdots (\gamma _{r}+2)(\gamma _{r+1}+1)\cdots (\gamma _{s}+1)}\right) \right. \nonumber \\&\left. -2\prod \limits _{i=1}^{s}\frac{\gamma _{i}}{\gamma _{i}+1}-2\prod \limits _{i=1}^{r}\frac{\gamma _{i}}{\gamma _{i}+1}+1\right] . \end{aligned}$$
(3.3)

Differentiating (3.3) with respect \(t_{1}\) and \(t_{2}\), and putting \(t_{1}=t_{2}=0\), we can obtain the product moments \(E\left\{ Y^{l_{1}}_{[r,n,m,k]}Y^{l_{2}}_{[s,n,m,k]}\right\} =\mu ^{(l_{1},l_{2})}_{[r,s,n,m,k]}\), \(l_{1},l_{2}>0\) as

$$\begin{aligned} \mu ^{(l_{1},l_{2})}_{[r,s,n,m,k]}= & {} \mu ^{l_{1}l_{2}}_{1:1}+ \lambda \left[ 2\prod \limits _{i=1}^{r}\frac{\gamma _{i}}{\gamma _{i}+1}-1\right] \left[ \mu ^{l_{1}}_{1:1}\mu ^{l_{2}}_{1:1}-\mu ^{l_{1}}_{2:2}\mu ^{l_{2}}_{1:1}\right] \nonumber \\&+\lambda \left[ 2\prod \limits _{i=1}^{s}\frac{\gamma _{i}}{\gamma _{i}+1}-1\right] \left[ \mu ^{l_{1}}_{1:1}\mu ^{l_{2}}_{1:1}-\mu ^{l_{1}}_{1:1}\mu ^{l_{2}}_{2:2}\right] \nonumber \\&+\lambda ^{2}\left[ \mu ^{l_{1}}_{1:1}-\mu ^{l_{1}}_{2:2}\right] \left[ \mu ^{l_{2}}_{1:1}-\mu ^{l_{2}}_{2:2}\right] \nonumber \\&\times \left[ 4\left( \frac{\gamma _{1}\gamma _{2}\dots \gamma _{r}\gamma _{r+1}\dots \gamma _{s}}{(\gamma _{1}+2)(\gamma _{2}+2) \dots (\gamma _{r}+2)(\gamma _{r+1}+1)\dots (\gamma _{s}+1)}\right) \right. \nonumber \\&\left. -2\prod \limits _{i=1}^{s}\frac{\gamma _{i}}{\gamma _{i}+1}-2\prod \limits _{i=1}^{r}\frac{\gamma _{i}}{\gamma _{i}+1}+1\right] . \end{aligned}$$
(3.4)

The joint mgf of the concomitants of the r-th and s-th order statistics, \(Y_{[r:n]}\) and \(Y_{[s:n]}\) can be deduced from (3.3) with \(m=0\) and \(k=1\) as

$$\begin{aligned}&M_{Y_{[r:n]},Y_{[s:n]}}(t_{1},t_{2}) \\&\quad = M_{Y}(t_{1})M_{Y}(t_{2}) +\frac{\alpha (n-2r+1)}{n+1}\left[ M_{V}(t_{1})M_{Y}(t_{2})-M_{Y}(t_{1})M_{Y} (t_{2})\right] \nonumber \\&\qquad +\,\frac{\alpha (n-2s+1)}{n+1}\left[ M_{Y}(t_{1})M_{V}(t_{2})-M_{Y}(t_{1}) M_{Y}(t_{2})\right] \nonumber \\&\qquad +\,\frac{\alpha ^{2}}{(n+1)(n+2)}[(n-2s+1)(n+2)-2r(n-2s)][M_{V}(t_{1}) M_{Y}(t_{2})\nonumber \\&\qquad -\,M_{Y}(t_{1})M_{Y}(t_{2})]\nonumber \\&\qquad \times \, [M_{Y}(t_{1})M_{V}(t_{2})-M_{Y}(t_{1})M_{Y}(t_{2})], \end{aligned}$$

where

$$\begin{aligned}&M_{Y}(t)=1+\sigma _{2} t\exp \left( \frac{-\sigma _{2}^{2}t^{2}}{2}\right) \sqrt{\frac{\pi }{2}}\left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2}\right) +1\right] ,\\&M_{V}(t)=1+\frac{\sigma _{2}}{\sqrt{2}} t\exp \left( \frac{- \sigma _{2}^{2}t^{2}}{4}\right) \sqrt{\frac{\pi }{2}}\left[ \mathrm{erf}\left( \frac{\sigma _{2} t}{2\sqrt{2}}\right) +1\right] . \end{aligned}$$

The product moment \(E[Y_{[r:n]}Y_{[s:n]}]=\mu _{[r,s:n]}\) is obtained as

$$\begin{aligned} \mu _{[r,s:n]}=\frac{\pi \sigma _{2}^{2}}{2}\left[ 1+[\tau _{r}+\tau _{s}] \left( \frac{\sqrt{2}}{2}-1\right) +\tau _{r,s}\left( \frac{\pi \sigma _{2}^{2}}{2}\right) \left( \frac{3}{2}-\sqrt{2}\right) \right] , \end{aligned}$$

where

$$\begin{aligned}&\tau _{r}=\frac{\alpha (n-2r+1)}{n+1},\;\;\; \tau _{s}=\frac{\alpha (n-2s+1)}{n+1},\\&\tau _{r,s}=\frac{\alpha ^{2}}{(n+1)(n+2)}[(n-2s+1)(n+2)-2r(n-2s)]. \end{aligned}$$

Therefore, the covariance and correlation between \(Y_{[r:n]}\) and \(Y_{[s:n]}\) are given as

$$\begin{aligned} Cov(Y_{[r:n]},Y_{[s:n]})=\sigma _{2}^{2} \beta _{r,s},\ \ \ \ \rho _{[r,s:n]}=\frac{\pi [\tau _{r,s}-\tau _{r}\tau _{s}] \left( \frac{3}{2}-\sqrt{2}\right) }{(4-\pi )\sqrt{a_ra_s}}, \end{aligned}$$
(3.5)

respectively, where \(\beta _{r,s}=\frac{\pi }{2}\left[ \tau _{r,s}-\tau _{r}\tau _{s}\right] \left( \frac{3}{2}-\sqrt{2}\right) \), \(a_r=1-\frac{\tau _{r}^{2}\pi \left( \frac{3}{2}-\sqrt{2}\right) }{4-\pi }-\frac{\tau _{r}\left[ 2+\pi (\sqrt{2}-2)\right] }{4-\pi }\). Also, the r-th and s-th concomitants are positively correlated and its value decreases as r and s pull apart.

Table 1 The values of coefficients \(c_{}\) and the efficiency \(e_1\)
Full size table

Remark 3.1

Set \(m =-1\) and \(k=1\) in (3.3) and (3.4), we can obtain the joint mgf and product moments of two concomitants of record values of MTBRD.

4 Estimation of \(\sigma _{2}\) Based on Concomitants of Record Values

Suppose \(\mathbf {R}_{[n]}\) denote the vector of concomitants of first n record values as \(\mathbf {R}_{[n]}=(R_{[1]},R_{[1]},\dots ,\) \(R_{[n]})'.\) Then, from (2.7) we can write the mean vector and variance covariance matrix of \(\mathbf {R}_{[n]}\) as

$$\begin{aligned} E(\mathbf {R}_{[n]})=\sigma _{2}\varvec{\xi },\;\;\; D(\mathbf {R}_{[n]})=\sigma _{2}^{2} \mathbf {G}, \end{aligned}$$
(4.1)

respectively, where \(\varvec{\xi }=(\xi _{1},\xi _{2},\ldots ,\xi _{n})'\), \(\mathbf {G}=\left( (a_{i,j})\right) \) and

$$\begin{aligned}&\xi _{i}=\sqrt{\frac{\pi }{2}}\left[ 1+\alpha \left( \frac{1}{2^{i-1}}-1\right) \left( \frac{\sqrt{2}}{2}-1\right) \right] , \;\;1\le i\le n,\\&a_{i,j}=\frac{\pi }{2}\left[ \alpha ^{2}\left( \frac{1}{3^{i}2^{j-i-2}}-\frac{1}{2^{j-1}}-\frac{1}{2^{i-1}}+1\right) -\xi _{i}\xi _{j}\right] \left( \frac{3}{2}-\sqrt{2}\right) . \end{aligned}$$

Now, if \(\alpha \) involved in \(\mathbf {\xi }\) and \(\mathbf {G}\) are known, then by using the method given in ([9], p. 185) the BLUE of \(\sigma _{2}\) is derived as

$$\begin{aligned} \hat{\sigma }_{2}=(\varvec{\xi }'\varvec{G}^{-1}\varvec{\xi })^{-1}\varvec{\xi }'\varvec{G}^{-1}\varvec{R}_{[n]}=\sum _{k=1}^{n}c_{k}R_{[k]}, \end{aligned}$$
(4.2)

where \(c_{k}\), \(k=1,2,\ldots ,n\) are constants. It is clear that \(\hat{\sigma }_{2}\) is a linear function of the concomitants \(R_{[k]}\), \(k=1,2,\ldots ,n\). Also the variance of \(\hat{\sigma }_{2}\) is given by

$$\begin{aligned} \mathrm{Var}(\hat{\sigma }_{2})=(\varvec{\xi }'\varvec{G}^{-1}\varvec{\xi })^{-1}\sigma _{2}^{2}. \end{aligned}$$
(4.3)

We have evaluated the coefficients \(c_{k}\) of \(R_{[k]}\), \(k=1,2,\ldots ,n\) in \(\hat{\sigma }_{2}\) and \(Var(\hat{\sigma }_{2})\) for \(n=2(1)10(5)25\) and \(\alpha =0.25, 0.5, 0.75\) and are presented in Tables 1 and 2. Also, we can make an efficiency comparison of our estimator \(\hat{\sigma }_{2}\), with the well-known unbiased estimator of \(\sigma _{2}\) given by \(\tilde{\sigma }_{2}=\frac{\overline{Y}}{\sqrt{\frac{\pi }{2}}}\). We have obtained the ratio \(e_{1}=\frac{\mathrm{Var}(\tilde{\sigma }_{2})}{\mathrm{Var}(\hat{\sigma }_{2})}\) to measure the efficiency of our estimate \(\hat{\sigma }_{2}\) relative to \(\tilde{\sigma }_{2}\) for \(n=2(1)10(5)25\) and \(\alpha =0.25, 0.5, and 0.75\) and presented in Tables 1 and 2.