On the joint distribution of order statistics from independent non-identical bivariate distributions

Abstract

In this note, the exact joint probability density function (jpdf) of bivariate order statistics from independent non-identical bivariate distributions is obtained. Furthermore, this result is applied to derive the joint distribution of a new sample rank obtained from the rth order statistics of the first component and the sth order statistics of the second component.

Introduction

Multivariate order statistics especially Bivariate order statistics have attracted the interest of several researchers, for example, see [1]. The distribution of bivariate order statistics can be easily obtained from the bivariate binomial distribution, which was first introduced by [2]. Considering a bivariate sample, David et al. [3] studied the distribution of the sample rank for a concomitant of an order statistic. Bairamove and Kemalbay [4] introduced new modifications of bivariate binomial distribution, which can be applied to derive the distribution of bivariate order statistics if a certain number of observations are within the given threshold set. Barakat [5] derived the exact explicit expression for the product moments (of any order) of bivariate order statistics from any arbitrary continuous bivariate distribution function (df). Bairamove and Kemalbay [6] used the derived jpdf by [5] to derive the joint distribution on new sample rank of bivariate order statistics. Moreover, Barakat [7] studied the limit behavior of the extreme order statistics arising from n two-dimensional independent and non-identically distributed random vectors. The class of limit dfs of multivariate order statistics from independent and identical random vectors with random sample size was fully characterized by [8].

Consider n two-dimensional independent random vectors ${{\underline W}_{j}}=(X_{j},Y_{j})$, j=1,2,...,n, with the respective distribution function (df) $F_{j}(\underline w)=F_{j}(x,y)= P(X_{j}\leq x, Y_{j}\leq y), j=1,2,...,n$. Let X_1:n≤X_2:n≤...≤X_n:n and Y_1:n≤Y_2:n≤...≤Y_n:n be the order statistics of the X and Y samples, respectively. The main object of this work is to derive the jpdf of the random vector $Z_{k,k^{\prime }:n}= (X_{n-k+1:n},Y_{n-k^{\prime }+1:n})\phantom {\dot {i}\!},$ where 1≤k, k^′≤n. Let $G_{j}(\underline {w})=P({\underline {W}}_{j}>\underline {w})$ be the survival function of $F_{j}(\underline {w}), j=1,2,...,n$ and let F_1,j(.), F_2,j(.), G_1,j(.)=1−F_1,j(.) and G_2,j(.)=1−F_2,j(.) the marginal dfs and the marginal survival functions of $\Phi _{k,k':n}= P(Z_{k,k':n}\leq \underline {w}),\ F_{j}(\underline {w})$ and $~G_{j}(\underline {w}), j=1,2,...,n,$ respectively. Furthermore, let ${F_{j}}^{1,.}=\frac {\partial F_{j}(\underline {w})}{\partial {x}}$ and ${F_{j}}^{.,1}=\frac {\partial F_{j}(\underline {w})}{\partial {y}}.$ Also, the jpdf of $(X_{n-k+1:n},Y_{n-k^{\prime }+1:n})\phantom {\dot {i}\!}$ is conveniently denoted by $\phantom {\dot {i}\!}f_{k,k^{\prime }:n}(\underline {w}).$ Finally, the abbreviations min(a, b)=a∧b, and max(a, b)=a∨b will be adopted.

The jpdf of non-identical bivariate order statistics

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,...,i_n 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<X_k:n<x+δx, y<Y_k: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,...,i_n 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

$$\begin{array}{*{20}l} {} f_{n,n:n}(\underline{w})=\sum_{\rho_{\theta,\varphi,r}}\, {F}^{.,1}_{i_{1}}(\underline{w}){F}^{1,.}_{i_{2}}(\underline{w})\Pi_{j=3}^{n}F_{i_{j}}(\underline{w})+\sum_{\rho_{r}}f_{i_{3}}(\underline{w}) \Pi_{j=2}^{n}F_{i_{j}}(\underline{w}))(F_{2,i_{n+1}}(y)-{F}_{i_{n+1}}(\underline{w})). \end{array} $$

Joint distribution of the new sample rank of X _r:n and Y _s:n

Consider n two-dimensional independent vectors $\underline {W}_{j}=(X_{j},Y_{j}),j= 1,...,n,$ with the respective df $F_{j}(\underline {W})$ and the jpdf$F_{j}(\underline {W})$. Further assume that (X_n+1,Y_n+1), (X_n+2,Y_n+2),..., (X_n+m,Y_n+m), (m≥1) is another random sample with absolutely continuous df $G^{*}_{j}(x,y), j= 1,...,m$ and jpdf g_j(x, y). We assume that the two samples (X_n+1,Y_n+1),(X_n+2,Y_n+2),...,(X_n+m, Y_n+m), (m≥1) and (X₁,Y₁),(X₂,Y₂),...,(X_n,Y_n) are independent.

For 1≤r, s≤n, m≥1, we define the random variables η₁ and η₂ as follows:

$$\begin{array}{*{20}l} \eta_{1}=\sum_{i=1}^{m}I_{(X_{r:n}-X_{n+i})} \end{array} $$

and

$$\begin{array}{*{20}l} \eta_{2}=\sum_{i=1}^{m}I_{(Y_{s:n}-Y_{n+i})}, \end{array} $$

where I(x)=1 if x>0 and I(x)=0 if x≤0 is an indicator function. The random variables η₁ and η₂ are referred to as exceedance statistics. Clearly η₁ shows the total number of new X observations X_n+1,X_n+2,..., X_n+m which does not exceed a random threshold based on the rth order statistic X_r:n. Similarly, η₂ is the number of new observations Y_n+1,Y_n+2,...,Y_n+m which does not exceed Y_s:n.

The random variable ζ₁=η₁+1 indicates the rank of X_r:n in the new sample X_n+1, X_n+2,..., X_n+m, and the random variable ζ₂=η₂+1 indicates the rank of Y_s:n in the new sample Y_n+1,Y_n+2,...,Y_n+m. We are interested in the joint probability mass function of random variables ζ₁ and ζ₂. We will need the following representation of the compound event P(ζ₁=p,ζ₂=q)=P(η₁=p−1,η₂=q−1).

Definition 1

Denote A={X_n+i≤X_r:n},A^c={X_n+i>X_r:n}, B={Y_n+i≤Y_s:n} and B^c={Y_n+i>Y_s:n}. Assume that in a fourfold sampling scheme, the outcome of the random experiment is one of the events A or A^c, and simultaneously one of B or B^c, where A^c is the complement of A.

In m independent repetitions of this experiment, if A appears together with B ℓ times, then A and B^c appear together p−ℓ−1 times. Therefore, B appears together with A^c q−ℓ−1 times and B^c m−p−q+ℓ+2 times. This can be described as follows:

Clearly, the random variables η₁ and η₂ are the number of occurrences of the events A and B in m independent trials of the fourfold sampling scheme, respectively. By conditioning on X_r:n=x and Y_s:n=y, the joint distribution of η₁ and η₂ can be obtained from bivariate binomial distribution considering the four sampling scheme with events A={X_n+i≤x}, B={Y_n+i≤y}, and with respective probabilities

$$\begin{array}{*{20}l} P(AB)=P(X_{n+i}\leq x,Y_{n+i}\leq y),\\ P(AB^{c})=P(X_{n+i}\leq x,Y_{n+i}> y),\\ P(A^{c}B)=P(X_{n+i}> x,Y_{n+i}\leq y),\\ P(A^{c}B^{c})=P(X_{n+i}> x,Y_{n+i}> y). \end{array} $$

Now, we can state the following theorem.

Theorem 2

The joint probability mass function of ζ₁ and ζ₂, is given by

$${{} \begin{aligned} &P(\zeta_{1}=p, \zeta_{2}=q) = P(\eta_{1}= p-1,\eta_{2}=q-1)= \sum_{\ell=max (0, p+q-m-2)}^{min (p-1,q-1)}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\\&\Pi_{j=1}^{\ell}G^{*}_{i_{j}}(x,y) \Pi_{j=\ell+1}^{p-1}[G^{*}_{1,i_{j}}(x)-G^{*}_{i_{j}}(x,y)]\Pi_{j=p}^{q-\ell-1+p}\left[G^{*}_{2,i_{j}}(y)-G^{*}_{i_{j}}(x,y)\right] \Pi_{j=q-\ell+p}^{m+2}\overline{G}^{*}_{1,i_{j}}(x) f_{k,k':n(\underline{w})} dxdy, \end{aligned}} $$

where, $p,q= 1,...,m+1,\ f_{k,\acute {k}:n}(\underline {w})$ is defined in (3).

Proof

Consider the fourfold sampling scheme described in Definition (1). By conditioning with respect to X_r:n=x and Y_s:n=y, we obtain

$$ {{} \begin{aligned} P(\zeta_{1}=p, \zeta_{2}=q) \equiv P(\eta_{1}= p-1,\eta_{2}=q-1)= P\left\{\sum_{i=1}^{m}I_{(X_{r:n}-X_{n+i})}=p-1,I_{(Y_{r:n}-Y_{n+i})}=q-1\right\}\\ =\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}P\left\{\sum_{i=1}^{m}I_{(X_{r:n}-X_{n+i})}=p-1,I_{(Y_{s:n}-Y_{n+i})}=q-1| X_{r:n}=x, Y_{s:n}=y\right\}\\ \times P\{X_{r:n} = x, Y_{s:n} = y \} dxdy\\ =\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}P\left\{\sum_{i=1}^{m}I_{(x-X_{n+i})}=p-1,I_{(y-Y_{n+i})}=q-1\right\}{dF}_{r,s:n}(x,y). \end{aligned}} $$

(5)

□

On the other hand,

$$ {{} \begin{aligned} P\left(\sum_{i=1}^{m}I_{(x-X_{n+i})}=p-1,I_{(y-Y_{n+i})}=q-1\right)=\sum_{\ell=max (0,p+q-m-2)}^{min(p-1,q-1)}\Pi_{j=1}^{\ell}P_{i_{j}}(AB)\Pi_{j=\ell+1}^{p-1}P_{i_{j}}(AB^{c})\\ \Pi_{j=p}^{q-\ell-2+p}P_{i_{j}}\Pi_{j=q-\ell-1+p}^{m}P_{i_{j}}. \end{aligned}} $$

(6)

Substituting (6) in (5), we get

$${{} \begin{aligned} P(\zeta_{1}=p, \zeta_{2}=q) = P(\eta_{1}= p-1,\eta_{2}=q-1)= \sum_{\ell=max (0, p+q-m-2)}^{min (p-1,q-1)}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\Pi_{j=1}^{\ell}G^{*}_{i_{j}}(x,y)\\ \Pi_{j=\ell+1}^{p-1}[G^{*}_{1,i_{j}}(x)-G^{*}_{i_{j}}(x,y)]\Pi_{j=p}^{q-\ell-1+p}[G^{*}_{2,i_{j}}(y)-G^{*}_{i_{j}}(x,y)]\Pi_{j =q-\ell+p}^{m}\overline{G}^{*}_{1,i_{j}}(x) f_{k,k':n(\underline{w})} dxdy, \end{aligned}} $$

where p, q=1,...,m+1. This completes the proof.