1.4 Joint Distribution of Two or More Order Statistics
The joint density of \(X_{(r)}\) and \(X_{(s)}\,\, \left (1\leq r < s < \leq n\right )\) denoted by \(f_{r,s}(x,y)\)
\[P^n_{r-1, s-r-1, n-s} = \frac {n!}{(r - 1)!\, (s - r - 1)!\, (n - s)!}\]
\[f_{r, s}(x,y) = \frac {n!}{(r - 1)!\, (s - r - 1)!\, (n - s)!}\, \left [P(x)\right ]^{r - 1}\, p(x)\, \left [P(y) - P(x)\right ]^{s - r - 1}\, \left [1 - P(y)\right ]^{n - s}\, p(y).\]
The joint probability function for the \(r^{\text {th}}\) and \(s^{\text {th}}\) order statistics with \(r < s\)
\[f_{r,s}(x,y) = \frac {n!\,p(x)\, p(y)}{(r - 1)!\, (s - r - 1)!\, (n - s)!}\, \left [P(x)\right ]^{r - 1}\, \left [P(y) - P(x)\right ]^{s - r - 1}\, \left [1 - P(y)\right ]^{n - s}\hspace {0.3cm} \cdots \hspace {0.3cm}1.4.1(a)\]
If \(\, r = 1, \, \, n = s\hspace {0.5cm} f_{1, n}(x, y) = n\, (n - 1)\, p(x)\, p(y)\, \left [P(y) - P(x)\right ]^{n - 2}\hspace {0.5cm}\cdots \hspace {0.3cm} 1.4.1(b)\)
we can also write the joint pdf of \(X_{n_1}, \, X_{n_2}, \, \cdots \, , \, X_{n_k}\) where \(1\leq n_1 < n_2 <\, \cdots \, < n_k\leq n\)
\begin {align*} f_{X_{(n_1)}\, \cdots \, X_{(n_5)}}(x_1, \, \cdots \, , x_5) = & \frac {n!}{(n_1 - 1)!\, (n_2- n_1-1)!\, (n_3-n_2-1)!\,\cdots \, (n-n_5)!}\\ = & \times \, p(x_1)\, p(x_2)\, \cdots \, p(x_5)\times \left [P(x_1)\right ]^{n_1 - 1}\, \left [P(x_2) - P(x_1)\right ]^{n_2-n_1-1}\\ & \times \left [P(x_3) - P(x_2)\right ]^{n_3 - n_2 - 1}\, \cdots \, \left [1 - P(x_5)\right ]^{n - n_5}. \end {align*}
In general \begin {align*} f_{n_1, n_2, \, \cdots \, , \, n_k}(x_{n_1}, \, x_{n_2}, \, \cdots \, , \, x_{n_k}) = & \frac {n!\,p(x_{n_1})\, p(x_{n_2})\, \cdots \, p(x_{n_k })\, \times \, \left [P(x_{n_1})\right ]^{n_1 - 1}\, }{(n_1 - 1)!\, (n_2-n_1-1)!\, \cdots \, (n_k-n_{k -1}-1)!\, (n-n_k)!}\\ & \times \,\left [P(x_{n_2}) - P(x_{n_1})\right ]^{n_2 - n_1 - 1}\, \left [P(x_{n_3}) - P(x_{n_2})\right ]^{n_3-n_2-1}\, \cdots \cdots \\ & \times \, \left [P(x_{n_k}) - P(x_{n_k - n_1})\right ]^{n_k - n_k-1}\, \left [1 - P(x_{n_k})\right ]^{n -k}\hspace {1.5cm} 1.4.2 \end {align*}
Note 1.4.1. From 1.4.1 we can get the joint cdf \(F_{r, s}(x,y)\) for \(X_{(r)}\) and \(X_{(s)}\) by integration
\[F_{r,s}(x,y) = \int ^y_{x}\int _{-\infty }^yf_{r,s}(u,v)\, du\, dv\] or we can do it by the following reasoning \begin {align*} F_{r,s}(x,y) & = P(X_{(r)} \leq x\, , \, X_{(s)} \leq y)\\ & = Pro\left (\text {atleast}\, \hspace {0.2cm}r \hspace {0.2cm} \text {of}\, X_i\, 's\leq x\, , \, \text {at least}\, s \, \text {of}\, X_j \,' s\leq y\right )\\ & = \sum _{j = s}^n\sum ^j_{i = r}\frac {n!}{i!\, (j - k)!\, (n - j)!} \times \left [P(x)\right ]^i\, \left [P(y) - P(x)\right ]^{j - i}\, \left [1 - P(y)\right ]^{n - j}\hspace {0.3cm} \cdots \hspace {0.3cm}1.4.3 (a) \end {align*}
If \(\, r = 1, \, s = n\) \begin {align*} F_{1,n}(x,y) & = \sum _{j = n}^n\sum ^j_{i = 1}\frac {n!}{i!\, (j - i)!\, (n-j)!}\, \left [P(x)\right ]^{i}\, \left [P(y) - P(x)\right ]^{j-i}\left [1 - P(y)\right ]^{n - j}\\ & = \sum ^n_{i = 1}\frac {n!}{i!\, (n - i)!}\, \left [P(x)\right ]^i\, \left [P(y) - P(x)\right ]^{n - i}\, \left [1 - P(y)\right ]^{n - n}\\ & = \sum ^n_{i = 1}\binom {n}{i}\, \left [P(x)\right ]^{i}\, \left [P(y) - P(x)\right ]^{n - i}\\ \therefore \hspace {0.3cm} F_{1,n}(x,y) & = \left (P(y)\right )^n - \left (P(y) - P(x)\right )^n \hspace {0.5cm}\cdots \cdots \hspace {0.3cm} 1.4.3 (b)\\ \end {align*}
Example 1.4.2. Let \(X_1, \, X_2, \, \cdots \, \, X_n\) be a random sample from exponential \(p(x) = \lambda \, e^{-\lambda x},\)
\(\, \, P(x) = 1 - e^{-\lambda x}\). Find the following
- (i)
- cdf for \(X_{(r)}\) and also for \(r = 1\), and find \(r = n\).
Solution. From equation 1.3.4 \[F_r(y) = \frac {1}{B(r, n- r + 1)} \int _0^{1 - e^{-\lambda \, y}}t^{r - 1}\, (1 - t)^{n - r}\, dt.\] \begin {align*} r = 1, \, \hspace {0.2cm} F_1(y) & = \frac {1}{B(1, n)} \int _0^{1 - e^{-\lambda \, y}} (1 - t)^{n - r}\, dt\\ & = \frac {\Gamma (n + 1)}{\Gamma (1)\, \Gamma (n)}\cdot \frac {-(1 - t)^{n}}{n}\Bigg |^{1 - e^{-\lambda y}}_0\\ & = 1 - \left (1 - 1 + e^{-\lambda y}\right )^n \end {align*}
\(F_1(y) = 1 - e^{-\lambda n y},\) cdf of the minimum, \(y > 0\).
\begin {align*} F_n(y) = \frac {1}{B(n,1)}\int _0^{1 - e^{-\lambda y}}t^{n - 1}\, dt = \frac {\Gamma (n + 1)}{\Gamma (n)\, \Gamma (1)}\cdot \frac {t^n}{n}\Bigg |^{1 - e^{-\lambda n y}}_0 = \left (1 - e^{-\lambda y}\right )^n \end {align*}
\begin {align*} f_n(y) & = n\, \left (1 - e^{-\lambda y}\right )^{n - 1}\, \left (-e^{-\lambda y}\right )\left (-\lambda \right )\\ & = \lambda n e^{-\lambda y}\, \left (1 - e^{-\lambda y}\right )^{n - 1}\hspace {0.3cm} y > 0. \end {align*} □
- (ii)
- joint pdf for \(X_{(r)}\) and \(X_{(s)}, \, r < s\) also for \(r = 1\) and \(s = n\).
Solution. \begin {align*} f_{r,s}(x,y) & = \frac {n!\, p(x)\, p(y)}{(r - 1)!\, (s-r-1)!\, (n - s)!}\left [P(x)\right ]^{r - 1}\, \left [P(y) - P(x)\right ]^{s - r - 1}\, \left [1 - P(y)\right ]^{n - s}\\ & = \frac {n!\, \lambda ^2e^{-\lambda (x + y)}}{(r - 1)!\, (s - r - 1)!\, (n - s)!}\left (1 - e^{-\lambda x}\right )^{r - 1}\, \left (e^{-\lambda } - e^{-\lambda y}\right )^{s - r - 1}\, \left (e^{-\lambda y}\right )^{n - s}\\ & = \frac {n!\, \lambda ^2\, e^{-\lambda (x + y + y(n - s))}}{(r - 1)!\, (s - r - 1)!\, (n - s)!}\left (1 - e^{-\lambda x}\right )^{r - 1}\, \left (e^{-\lambda x} - e^{-\lambda y}\right )^{s - r - 1}. \end {align*}
\(r = 1, \, \, s = n\) \[f_{1,n}(x,y) = n(n - 1)\lambda ^2\, e^{-\lambda (x + y)}\left (e^{-\lambda x} - e^{-\lambda y}\right )^{n - 2}\, , \hspace {0.3cm} 0 < x < y < \infty .\] □
- (iii)
- joint cdf for \(X_{(r)}\) and \(X_{(s)}, \, r < s\) also for \(r = 1\) and \(s = n\).
Solution. \begin {align*} F_{r, s}(x,y) & = \sum ^n_{j = s}\sum ^j_{i = r}\frac {n!}{i!\, (j- i)!\, (n - j)!}\left [P(x)\right ]^i\left [P(y) - P(x)\right ]^{j - i}\, \left [1 - P(y)\right ]^{n - j}\\ & = \sum ^n_{j = s}\sum ^j_{i = r}\frac {n!}{i!\, (j - i)!\, (n - j)!}\left [1 - e^{-\lambda x}\right ]^i\, \left [e^{-\lambda x} - e^{-\lambda y}\right ]^{j - i}\left [e^{-\lambda y}\right ]^{n -j}. \end {align*}
\(r = 1, \, \, s = n\) \begin {align*} F_{1,n}(x,y) & = \sum ^n_{i = 1}\frac {n!}{i!\, (n - i)!\, (n - n)!}\left [1 - e^{-\lambda x}\right ]^{i}\, \left [e^{-\lambda x} - e^{-\lambda y}\right ]^{n - i}\\ & = \sum ^n_{i = 1}\binom {n}{i}\left [1 - e^{-\lambda x}\right ]^i\, \left [e^{-\lambda x} - e^{-\lambda y}\right ]^{n - i}\\ & = \left [1 - e^{-\lambda x} + e^{-\lambda x} - e^{-\lambda y}\right ]^n - \left [e^{-\lambda x} - e^{-\lambda y}\right ]^n\\ F_{1,n}(x,y) & = \left [1 - e^{-\lambda y}\right ]^n - \left [e^{-\lambda x} - e^{-\lambda y}\right ]^n. \end {align*}
\begin {align*} \frac {\partial }{\partial x}F_{1, n}(x,y) & = n \lambda e^{-\lambda x}\left (e^{-\lambda x} - e^{-\lambda y}\right )^{n - 1}\\\\ \frac {\partial ^2}{\partial y \, \partial x}F_{1,n}(x,y) & = n(n - 1)\lambda ^2\, e^{-\lambda (x + y)}\left [e^{-\lambda x} - e^{-\lambda y}\right ]^{n - 2}. \end {align*} □
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