1.6 Conditional Distribution of Order Statistic
\(X_{(r)}\) and \(X_{(s)}\) \[f_r(x) = \frac {n!}{(r -1)!\, (n - r)!}\left (P(x)\right )^{r - 1}\, \left (1 - P(x)\right )^{n - r}\hspace {0.6cm}\cdots \hspace {0.3cm}1.7.1\] \[f_{r,s}(x,y) = \frac {n!\, p(x)\, p(y)}{(r - 1)!\, (s - r - 1)!\, (n - s)!}\left [P(x)\right ]^{r - 1}\, \left [P(y) - P(x)\right ]^{s - r - 1}\, \left [1 - P(y)\right ]^{n - s}\hspace {0.3cm}\cdots \hspace {0.3cm}1.7.2\] we want conditional distribution of \(s\) given \(r\)
\[f_{X_{(s)}/X_{(r)}}(y/x) = \frac {f_{r,s}(x,y)}{f_r(x)} = \frac {(n - r)!\, p(y)\, \left (P(y) - P(x)\right )^{s - r - 1}\, \left (1 - P(y)\right )^{n - s}}{(s - r - 1)!\, (n - s)!\, \, \left (1 - P(x)\right )^{n - r}}.\]
\[f_{s/r}(y/x) = \frac {(n - r)!\, p(y)\, \left (P(y) - P(x)\right )^{s - r - 1}\, \left (1 - P(y)\right )^{n - s}}{(s - r - 1)!\, (n - s)!\, \, \left (1 - P(x)\right )^{n - r}}\hspace {0.6cm}\cdots \cdots \hspace {0.5cm} 1.7.3\]
for \(s = n, \, \, r = 1\) \[f_{n/1}(y/x) = \frac {(n - 1)\, p(y)\ \left (P(y) - P(x)\right )^{n - 2}}{\left (1 - P(x)\right )^{n - 1}}.\]
Exponential \begin {align*} f_{n/1}(y/x) & = \frac {(n - 1)\, \lambda \, e^{-\lambda y}\, \left (e^{-\lambda x} - e^{-\lambda y}\right )^{n - 2}}{\left (e^{-\lambda x}\right )^{n - 1}}\\ & = (n - 1)\, \lambda \, e^{\lambda x (n - 1)}\, e^{-\lambda y}\, \left (e^{-\lambda x} - e^{-\lambda y}\right )^{n - 2}, \hspace {0.3cm} y > x. \end {align*}
* prove that this is a pdf (Bonafide).
The result of 1.7.3 can be stated as follows.
Theorem 1.6.1. For a random sample of size \(n\) from a continuous distribution, the conditional \(X_{(r)} = x \, (r < s)\) is just the distribution of the \((s - r)^{\text {th}}\) order statistic in a random sample of size \(n - r\) drawn from \(\frac {P(y)}{1 - P(x)}, \, y\geq x\). The joint pdf of \(X_{(n_1)}, \, X_{(n_2)}, \, \cdots \, ,\, X_{(n_k)}\) is given by \(n_1< n_2,< \cdots < n_k\). \begin {align*} f_{n_1, n_2, \cdots \ n_k}(x_1, \, x_2, \, \cdots \, , \,x_k) = & \frac {n!\,\, p(x_1)\, p(x_2)\,\, \cdots \,\, p(x_k)}{(n_1 -1)!\, (n_2 - n_1 - 1)!\, (n_3- n_2 - 1)!\cdots (n_k - n_{k - 1} - 1)!\, (n - n_k)!}\\ & \times \left (P(x_1)\right )^{n_1 - 1}\, \left (P(x_2) - P(x_1)\right )^{n_2 - n_1 - 1}\, \cdots \, \left (P(x_k) - P(x_{k - 1}\right )^{n_k - n_{k - 1} - 1}\\ & \times \, \left (1 - P(x_k)\right )^{n - n_k} \end {align*}
It can be verified that
\[f_{X_{(s)}/X_{(1)} = x_{r_1}, \, X_{(r - 1)} = x_{r - 1}\, \cdots \, X_{(1)} = x_1}(y) = f_{X_{(s)}/X_{(r)} = x_r}(y)\]
which establishes that order statistics in a random sample from a continuous distribution for a
Markov chain.
(Use mathematical induction).
Now let \(Z_{(1)} \leq Z_{(2)} \leq \, \cdots \, \leq Z_{(n)}\) denote the order statistics from a random sample of size \(n\) from exponential with mean 1. \[p(z) = e^{-z}\] write the joint pdf of \(Z_{(1)}, \, Z_{(2)}, \, \cdots \, , \, Z_{(n)}\). \[f_{1,2, \cdots \, n}(z_{(1)},\, z_{(2)}, \, \cdots \, , \, z_{(n)}) = n!\, e^{-z_{(1)} - z_{(2)} - --- z_{(n)}}\] which can be written as \[f_{1,2, \cdots \, n}(z_{(1)},\, z_{(2)}, \, \cdots \, , \, z_{(n)}) = n!\, e^{-\sum ^n_{j = 1} \left (n - j + 1\right )\left (z_{(j)} - z_{(j - 1)}\right )}, \hspace {0.3cm}\text {where}\hspace {0.3cm} z_{(0)} = 0.\] making a transformation \[Y_j = \left (n - j + 1\right )\left (Z_{(j)} - Z_{(j - 1)}\right ),\hspace {0.3cm} j = 1, 2,\, \cdots \, , n\hspace {1cm}\cdots \hspace {0.5cm} 1.7.6\] \[P(Y_j) = e^{-y_j}\hspace {1cm} \cdots \cdots \hspace {1cm} 1.7.7\]
We see that the \(Y_j\, '\)s are independent and each \(Y_j\) is exponential with mean 1.
The relation 1.7.6 allows \(Z_{(r)}, \, r = 1, \, 2, \, \cdots \, n\) to be expressed as
\[Z_{(r)} = \sum ^r_{j = 1} \frac {Y_j}{n - j + 1}\hspace {1cm} \cdots \cdots \hspace {1cm} 1.7.8.\]
i.e as a linear function of independent random variables. It follows that \(Z_{(1)}, \, Z_{(2)}, \, \cdots \, , \, Z_{(n)}\) form an additive Markov
chain.
Equations 1.7.6, 1.7. 7 and 1.7.8 have important applications in life testing. Apart from the scale
factor the \(Z_{(j)}\,\)’s can be interpreted as the successive failure lifetimes, of \(n\) items the individual lifetime \(X = \lambda Z\, (\lambda > 0)\)
mean \(\lambda \). The intervals of length \(X_{(j)} - X_{(j - 1)}\) between successive failures are then independently distributed as
\(\frac {\lambda z}{ h - j + 1}\).
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