The following theorem gives the exact formula of the jpdf of non-identical bivariate order statistics.
Theorem 1
The jpdf of non-identical bivariate order statistics is given by
$${{} \begin{aligned} f_{k,k':n}(\underline{w})\,=\,\sum_{\theta,\varphi=0}^{1}\sum_{r=r_{**}}^{r^{**}}\sum_{\rho_{\theta,\varphi,r}}\, \Pi_{j=1}^{\theta}{F}^{.,1}_{i_{j}}(\underline{w})\Pi_{j=\theta+1}^{1}(f_{2,i_{j}}(y)\,-\,{F}^{.,1}_{i_{j}}(\underline{w})) \Pi_{j=2}^{\varphi+1}{F}^{1,.}_{i_{j}}(\underline{w})\\ \times\Pi_{j=\varphi+2}^{2}(f_{1,i_{j}}(x)-{F}^{1,.}_{i_{j}}(\underline{w}))\Pi_{j=3}^{k-\theta-r+1} (F_{1,i_{j}}(x)-{F}_{i_{j}}(\underline{w}))\Pi_{j=k-\theta-r+2}^{k-\theta+1}F_{i_{j}}(\underline{w})\\ \times \Pi_{j=k-\theta+2}^{k+k'-\theta-\varphi-r}(F_{2,i_{j}}(y)-{F}_{i_{j}}(\underline{w})) \Pi_{j=k+k'-\theta-\varphi-r+1}^{n}G_{i_{j}}(\underline{w})+\sum_{r=0\vee(k+k'-n-1)}^{(k-1)\wedge(k'-1)}\sum_{\rho_{r}}f_{j}(\underline{w})\\ \Pi_{j=2}^{k-r}(F_{1,i_{j}}(x)-{F}_{i_{j}}(\underline{w})) \times\Pi_{j=k-r+1}^{k}F_{i_{j}}(\underline{w})\Pi_{j=k+1}^{k+k'-r}(F_{2,i_{j}}(y)-{F}_{i_{j}}(\underline{w}))\Pi_{j=k+k'-r+1}^{n}G_{i_{j}}(\underline{w}), \end{aligned}} $$
where \(r_{**}=0\vee (k+k'-\theta -\varphi -n), r^{**}= (k-\theta -1)\wedge (k'-\varphi -1),\ \sum _{\rho }\) denotes summation subject to the condition ρ, and \(\sum _{\rho _{\theta _{1},\theta _{2},\varphi _{1},\varphi _{2},\omega,r}}\) denotes the set of permutations of i1,...,in such that \(i_{j_{1}}<...< i_{j_{n}}.\)
Proof
A convenient expression of \(f_{k,k':n}(\underline {w})\) may derived by noting that the compound event E={x<Xk:n<x+δx, y<Yk:n<y+δy} may be realized as follows: r;φ1;s1;θ1;ω;θ2;s2;φ2 and t observations must fall respectively in the regions I1=(−∞,x]∩(−∞,y];I2=(x, x+δx]∩(−∞,y];I3=(x+δx,∞]∩(−∞,y];I4=(−∞,x]∩(y, y+δy];I5=(x, x+δx]∩(y, y+δy];I6=(x+δx,∞]∩(y, y+δy];I7=(−∞,x]∩(y+δy,∞);I8=(x, x+δx]∩(x+δx,∞);and I9=(x+δx,∞)∩(y+δy,∞) with the corresponding probability \(P_{ij}=P({\underline {W}}_{j}\in I_{i}), i=1,2,...,9\). Therefore, the joint density function \(f_{k,k':n}(\underline {w})\) of \(\phantom {\dot {i}\!}(X_{k:n},Y_{k^{\prime }:n})\) is the limit of \(\frac {P(E)}{\delta x\delta y}\) as δx,δy→0, where P(E) can be derived by noting that \(\theta _{1}+\theta _{2}+\omega =\varphi _{1}+\varphi _{2}+\omega =1;\ r+\theta _{1}+s_{2}=k-1;\ r+\varphi _{1}+s_{1}=k'-1;\ r,\theta _{1},s_{2},\varphi _{1},\omega,\theta _{2},s_{1},\varphi _{2},t \geq 0;\ P_{1j}=F_{j}(\underline {w}),P_{2j}=F_{j}^{1,.}(\underline {w})\delta x, P_{3j}=F_{2,j}(y)-F_{j}(x+\delta x,y), P_{4j}= F_{j}^{.,1}(\underline {w})\delta y, P_{5j\cong }F_{j}^{1,1}(\underline {w})\delta x\delta y=f_{j}(\underline {w})\delta x\delta y, P_{6j}\cong (f_{2,j}(y)-F_{j}^{.,1}(\underline {w}+\delta \underline {w}))\delta y,\) where \(f_{2,j}(y)=\frac {\partial F_{2,j}(y)}{\partial y},j=1, 2,...,n,~ \partial \underline {w}=(\delta x,\delta y), \underline {w}+\delta \underline {w}=(x+\delta x,y+\delta y),P_{7j}=F_{1,j}(x)-F_{j}(x,y+\delta y), P_{8j}=(f_{1,j}(x)-F_{j}^{1,.}(\underline {w}+\delta \underline {w}))\delta {x}, P_{9j}=1-F_{1,j}(x+\delta x)-F_{2,j}(y+\delta y)+F_{j}(\underline {w}).\) Thus, we get
$$ {{} \begin{aligned} f_{k,k':n}(\underline{w})\!&=\!\sum_{\theta_{1},\varphi_{1},\theta_{2},\varphi_{2}=0}^{1}\sum_{r=r_{*}}^{r^{*}} \sum_{\rho_{\theta_{1},\theta_{2},\varphi_{1},\varphi_{2},\omega,r}}\,\Pi_{j=1}^{\theta_{1}}P_{4i_{j}}\Pi_{\theta_{1}+1}^{\theta_{1}+\varphi_{1}}P_{2i_{j}} \Pi_{j=\theta_{1}+\varphi_{1}+1}^{\theta_{1}+\varphi_{1}+\theta_{2}}P_{6i_{j}} \Pi_{j=\theta_{1}+\varphi_{1}+\theta_{2}+1}^{\theta_{1}+\varphi_{1}+\theta_{2}+\varphi_{2}}P_{8i_{j}}\\ &\Pi_{j=\theta_{1}+\varphi_{1}+\theta_{2}+\varphi_{2}\,+\,1}^{\theta_{1}+\varphi_{1}+\theta_{2}+\varphi_{2}+\omega}P_{5i_{j}} \Pi_{j=\theta_{1}+\varphi_{1}+\theta_{2}+\varphi_{2}+\omega+1}^{\theta_{2}+\varphi_{1}+\theta_{2}+\omega+k-r\!-1}P_{7i_{j}} \Pi_{j=\theta_{2}+\varphi_{1}+\varphi_{2}+\omega+k\!-r}^{\varphi_{1}\,+\,\theta_{2}+\varphi_{2}+\omega+k-1}P_{1i_{j}} \Pi_{j=\varphi_{1}\,+\,\theta_{2}+\varphi_{2}+\omega\!+k}^{\theta_{2}\!+\varphi_{2}+\omega+k+k'\!-r\,-\,2}P_{3i_{j}}\\ &\Pi_{j=\theta_{2}+\varphi_{2}+\omega+k+k'-r-1}^{n}P_{9i_{j}}, \end{aligned}} $$
(1)
where \(r_{*}=0\vee (k+k'+\theta _{2}+\varphi _{2}+\omega -r-1-n), r^{*}= (k-\theta _{1}-1)\wedge (k'-\varphi _{1}-1),\sum _{\rho }\) denotes summation subject to the condition ρ, and \(\sum _{\rho _{\theta _{1},\theta _{2},\varphi _{1},\varphi _{2},\omega,r}}\) denotes the set of permutations of i1,...,in such that \(i_{j_{1}}<...< i_{j_{n}}\) for each product of the type \(\Pi _{j=j_{1}}^{j_{2}}\). Moreover, if j1>j2, then \(\Pi _{j=j_{1}}^{j_{2}}=1\). But (1) can be written in the following simpler form
$${{} \begin{aligned} P(E)=\sum_{\theta,\varphi=0}^{1}\sum_{r=r_{**}}^{r^{**}}\sum_{\rho_{\theta,\varphi,r}}\,\Pi_{j=1}^{\theta}P_{4i_{j}}\Pi_{j= \theta+1}^{1}P_{6i_{j}}\Pi_{j=2}^{\varphi+1}P_{2i_{j}} \Pi_{j=\varphi+2}^{2}P_{8i_{j}}\Pi_{j=3}^{k-\theta-r+1}P_{7i_{j}}\Pi_{j=k-\theta-r+2}^{k-\theta+1}P_{1i_{j}} \\ \Pi_{j=k-\theta+2}^{k+k'-\theta-\varphi-r}P_{3i_{j}} \Pi_{j=k+k'-\theta-\varphi-r+1}^{n}P_{9i_{j}}+\sum_{r=0\vee(k+k'-n-1)}^{(k-1)\wedge(k'-1)}\sum_{\rho_{r}}P_{5i_{3}}\Pi_{j=2}^{k-r}P_{7i_{j}} \Pi_{j=k-r+1}^{k}P_{1i_{j}} \Pi_{j=k+1}^{k+k'-r}P_{3i_{j}}\Pi_{j=k+k'-r}^{n}P_{9i_{j}}, \end{aligned}} $$
where r∗∗=0∨(k+k′−θ−φ−n),r∗∗=(k−θ−1)∧(k′−φ−1). Therefore,
$$ {{} \begin{aligned} f_{k,k':n}(\underline{w})=\sum_{\theta,\varphi=0}^{1}\sum_{r=r_{**}}^{r^{**}}\sum_{\rho_{\theta,\varphi,r}}\,\Pi_{j=1}^{\theta}P_{4i_{j}} \Pi_{j=\theta+1}^{1}P_{6i_{j}}\Pi_{j=2}^{\varphi+1}P_{2i_{j}} \Pi_{j=\varphi+2}^{2}P_{8i_{j}}\Pi_{j=3}^{k-\theta-r+1}P_{7i_{j}}\\ \Pi_{j=k-\theta-r+2}^{k-\theta+1}P_{1i_{j}}\Pi_{j=k-\theta+2}^{k+k'-\theta-\varphi-r}P_{3i_{j}} \Pi_{j=k+k'-\theta-\varphi-r+1}^{n}P_{9i_{j}}+\sum_{r=0\vee(k+k'-n-1)}^{(k-1)\wedge(k'-1)}\sum_{\rho_{r}}P_{5i_{3}}\Pi_{j=2}^{k-r}P_{7i_{j}}\\ \Pi_{j=k-r+1}^{k}P_{1i_{j}}\Pi_{j=k+1}^{k+k'-r}P_{3i_{j}}\Pi_{j=k+k'-r}^{n}P_{9i_{j}}. \end{aligned}} $$
(2)
Thus, we get
$$ {{} \begin{aligned} f_{k,k':n}(\underline{w})=\sum_{\theta,\varphi=0}^{1}\sum_{r=r_{**}}^{r^{**}}\sum_{\rho_{\theta,\varphi,r}}\, \Pi_{j=1}^{\theta}{F}^{.,1}_{i_{j}}(\underline{w})\Pi_{j=\theta+1}^{1}(f_{2,i_{j}}(y)-{F}^{.,1}_{i_{j}}(\underline{w})) \Pi_{j=2}^{\varphi+1}{F}^{1,.}_{i_{j}}(\underline{w})\\ \Pi_{j=\varphi+2}^{2}(f_{1,i_{j}}(x)-{F}^{1,.}_{i_{j}}(\underline{w}))\Pi_{j=3}^{k-\theta-r+1} (F_{2,i_{j}}(x)-{F}_{i_{j}}(\underline{w}))\Pi_{j=k-\theta-r+2}^{k-\theta+1}F_{i_{j}}(\underline{w}) \Pi_{j=k-\theta+2}^{k+k'-\theta-\varphi-r}(F_{2,i_{j}}(y)-{F}_{i_{j}}(\underline{w}))\\ \Pi_{j=k+k'-\theta-\varphi-r+1}^{n}G_{i_{j}}(\underline{w})+\sum_{r=0\vee(k+k'-n-1)}^{(k-1)\wedge(k'-1)}\sum_{\rho_{r}}f_{i_{3}}(\underline{w}) \Pi_{j=2}^{k-r}(F_{1i_{j}}(x)-{F}_{i_{j}}(\underline{w}))\\ \Pi_{j=k-r+1}^{k}F_{i_{j}}(\underline{w})\Pi_{j=k+1}^{k+k'-r}(F_{2,i_{j}}(y)-{F}_{i_{j}}(\underline{w}))\Pi_{j=k+k'-r+1}^{n}G_{i_{j}}(\underline{w}). \end{aligned}} $$
(3)
□
Hence, the proof.
Relation (3) may be written in term of permanents (c.f [9]) as follows:
$$ {{} \begin{aligned} f_{k,k':n}(\underline{w})= \sum_{\theta,\varphi=0}^{1}\sum_{r=r_{**}}^{r^{**}}\frac{1}{(k-\theta-r-1)!r!(k'-\varphi-r-1)!(n-k-k'+\varphi+\theta+r-1)!}\\ \begin{array}{ccccccc}\text{Per} [\underline{U}^{.,1}_{1,1}&(\underline{U}^{1}_{.,1}\,-\,\underline{U}^{.,1}_{1,1})&\underline{U}^{1,.}_{1,1}& \left(\underline{U}^{1}_{1,.}\,-\,\underline{U}^{1,.}_{1,1}\right)& \left(\underline{U}_{1,.}\,-\,\underline{U}_{1,1}\right)&\underline{U}_{1,1}&(\underline{U}_{.,1}\,-\,\underline{U}_{1,1})~\\ ~~ {\theta}~ &~ { 1-\theta}~~&~ {\varphi} ~~&~ {1-\varphi}~~&~ {k-\theta-r-1}~&~ {r}&~ {k'-\varphi-r-1}\\ (1-\underline{U}_{1,.}-\underline{U}_{1,.}+\underline{U}_{1,1}){\vphantom{\underline{U}^{.,1}_{1,1}}}]\\~~ {n-k-k'+\theta+\varphi+r-1} \end{array}\\ +\sum_{r=r_{*}}^{r^{*}}\frac{1}{(k-r)!r!(k'-r)!(n-k-k'+r)!} ~{\renewcommand{\arraystretch}{0.6} \begin{array}{cccccccc}\text{Per} [\underline{U}^{1,1}_{1,1}~&~(\underline{U}_{1,.}-\underline{U}_{1,1})~&\underline{U}_{1,1}~&~ (\underline{U}_{.,1}-\underline{U}_{1,1})~&~ (1-\underline{U}_{1,.}-\underline{U}_{1,.}+\underline{U}_{1,1})]\\ ~ {1}~ &~ {k-r}~~&~ {r} ~~&~ {k'-r}~&~ {n-k-k'+r-1}\end{array}}, \end{aligned}} $$
(4)
where \(~\underline U_{1,.}=(F_{11}(x_{1})~~F_{12}(x_{1})~...~ F_{1n}(x_{1}))',\ \underline U_{.,1}=(F_{2,1}(x_{2})~~F_{2,2}(x_{2})~...~ F_{2,n}(x_{2}))',\ \underline U_{1,1}=(F_{1}(\underline x)~~F_{2}(\underline x)~...~ F_{n}(\underline x))'\) and \(\underline 1\) is the n×1 column vector of ones. Moreover, if \({\underline {a}}_{1}, {\underline {a}}_{2},... \) are column vectors, then
$$\begin{array}{cccc} \text{Per}[&{\underline{a}}_1~~&~~{\underline{a}}_2~~&~~...]\\ &~~{i_{1}}~~&~~{i_{2}}~~&~~... \end{array} $$
will denote the matrix obtained by taking i1 copies of \({\underline {a}}_{1},\ i_{2}\) copies of \({\underline {a}}_{2},\) and so on.
Finally, when k=k′=1, in (3), we get
$$\begin{array}{*{20}l} f_{1,1:n}(\underline{w})=\sum_{\rho_{\theta,\varphi,r}}\, (f_{2,i_{1}}(y)-{F}^{.,1}_{i_{1}}(\underline{w})) (f_{1,i_{2}}(x)-{F}^{1,.}_{i_{2}}(\underline{w}))\Pi_{j=3}^{n} G_{i_{j}}(\underline{w})+\sum_{\rho_{r}}f_{i_{3}}(\underline{w})\\ (F_{2,i_{2}}(y)-{F}_{i_{2}}(\underline{w}))\Pi_{j=3}^{n}G_{i_{3}}(\underline{w}). \end{array} $$
Also, for k=k′=n, we get
$$\begin{array}{*{20}l} {} f_{n,n:n}(\underline{w})=\sum_{\rho_{\theta,\varphi,r}}\, {F}^{.,1}_{i_{1}}(\underline{w}){F}^{1,.}_{i_{2}}(\underline{w})\Pi_{j=3}^{n}F_{i_{j}}(\underline{w})+\sum_{\rho_{r}}f_{i_{3}}(\underline{w}) \Pi_{j=2}^{n}F_{i_{j}}(\underline{w}))(F_{2,i_{n+1}}(y)-{F}_{i_{n+1}}(\underline{w})). \end{array} $$