1.3 Distributions of a Single Order Statistic

Let \(X_1, X_2, \cdots X_n\) be a random sample of size \(n\) from a distribution with Cdf \(P(x)\) and \(F_r(x)\) denote the Cdf of the \(r^{\text {th}}\) order statistic \((r = 1, 2, \cdots , n)\). We assume that the distribution from where the sample was drawn is continuous. \begin {align*} F_r(x) & = P_{pro}\left (X_{(r)} \leq x\right )\\ & = P_{pro}\left (\text {at least}\, r\, \text {of the}\, X_i's\, \text {are less tha or equal to }\, x\right )\\ & = \sum ^n_{i = r} \binom {n}{i}\, \left [P(x)\right ]^i\,\left [1 - P(x)\right ]^{n - i} \hspace {0.5cm}\cdots \hspace {0.3cm} 1.3.1 \end {align*}

Special case \[\text {if}\hspace {0.3cm} r = 1, \hspace {0.5cm} F_1(x) = \sum ^n_{i = 1} \binom {n}{i}\, \left [P(x)\right ]^i\, \left [1 - P(x)\right ]^{n - i} = 1 - \left [1 - P(x)\right ]^n\hspace {0.5cm} \cdots \hspace {0.3cm} 1.3.2\] \(\text {if}\hspace {0.5cm} r = n, \hspace {0.5cm} F_n(x) = \left [P(x)\right ]^n\hspace {0.5cm}\cdots \hspace {0.3cm} 1.3.3\)

From the well-known relation between Binomial sums and the incomplete beta function \[F_r(x) = I_{P(x)}(r\,, \, n - r + 1) = \frac {1}{B(r\, , \, n - r + 1)}\, \int _0^{P(x)}t^{r - 1}\, (1 - t)^{n - r}\, dt\hspace {0.5cm}\cdots \hspace {0.3cm}1.3.4\] Thus the probability density function for the \(r^{\text {th}}\) order-statistic \begin {align*} f_r(x) & = \frac {\partial }{\partial x}\,F_r(x)\\ & = \frac {1}{B(r\, , \, n - r + 1)}\, \frac {\partial }{\partial x}\int ^{P(x)}_0t^{r - 1}\, (1 - t)^{n - r}\, dt\\\\ & = \frac {\left (P(x)\right )^{r - 1}\, \left (1 - P(x)\right )^{n - r}}{B(r\, , \, n - r + 1)}\, \frac {d}{dx}\, P(x)\\\\ f(x) & = \frac {p(x)\, \left [P(x)\right ]^{r -1}\, \left [1 - P(x)\right ]^{n - r}}{B(r\, , \, n - r + 1)}. \end {align*}

\[B(r\, , \, n - r + 1) = \frac {\Gamma (r)\, \Gamma (n - r + 1)}{\Gamma (n + 1)} = \frac {(r - 1)!\, (n - r)!}{n!}.\]

\[\therefore \,\, f_r(x) = \frac {n!}{(r - 1)!\, (n - r)!}\, p(x)\, \left [P(x)\right ]^{r - 1}\, \left [1 - P(x)\right ]^{n - r}\hspace {0.5cm}\cdots \hspace {0.3cm} 1.3.5.\]

\[r = 1, \hspace {0.5cm} f_1(x) = n\, p(x)\, \left [1 - P(x)\right ]^{n - 1}\]

\[r = n, \hspace {0.5cm} f_n(x) = n\, p(x)\, \left [P(x)\right ]^{n - 1}\]

u001XXX.5
(1(3(5))) — Figure 1.2: Densities of the first, third and fifth order statistics of a sample of size \(n=5\) from the uniform distribution on \((0,1)\); that is, the \(\operatorname {Beta}(r,\,n-r+1)\) densities for \(r=1,3,5\). The extreme order statistics are concentrated near the ends of the range and are strongly skewed, while the central one is broad and symmetric. The order statistics are neither independent nor identically distributed, which is the point of the chapter.

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