2. Order Statistics

Note that the cdf of the binomial distribution \(b(n,p)\) is \(F(x) = \sum \limits _{k=0}^{\lfloor x \rfloor } \, { }_{n}\mbox{C}_{k} p^k(1-p)^{n-k}\) for \(\lfloor x \rfloor \le x < \lfloor x \rfloor +1\).

In describing the pmf and cdf in the discrete space, we often use a simpler notation by just specifying the values at non-zero points of the pmf instead of specifying over all real numbers unless the specification for all real points is required.

Exercise 2.15 discusses this observation more specifically in the gamma distribution, a generalization of the exponential distribution.

Here, \(\sum \limits _{j=r}^n \, { }_n\mbox{C}_j x^j (1-x)^{n-j}= \sum \limits _{j=r}^{n-1} \, { }_n\mbox{C}_j x^j (1-x)^{n-j} + x^n= 1\) when \(x=1\).

The pdf (2.3.14) can be obtained also by differentiating the cdf (2.3.13).

Note that \(f_{X_{[r]},X_{[s]}} (x,x) =0\) for \(s \ge r+2\). For \(s=r+1\), the values of \(f_{X_{[r]},X_{[s]}} (x,y)\) expressed as (2.3.31) along the line \(x=y\) depends on \(u(0)\): nonetheless, the values of a two-dimensional function along a line of discontinuities on the two-dimensional plane are practically negligible, which is similar to the fact that the value of a one-dimensional function at discontinuities is practically negligible.

In this expression, \(\int _{-1}^y \int _w^y dv dw\), \(\int _{-1}^x \int _w^y dv dw\), and \(\int _{-1}^x \int _w^1 dv dw\) can be replaced with \(\int _{-1}^y \int _{-1}^v dw dv\), \(\left ( \int _{-1}^x \int _{-1}^v + \int _x^y \int _{-1}^x \right ) dw dv\), and \(\left ( \int _{-1}^x \int _{-1}^v + \int _x^1 \int _{-1}^x \right ) dw dv\), respectively, when the order of integrations over v and w are interchanged.

Note that the value of \(\sum \limits _{s=2}^{n} \sum \limits _{r=1}^{s-1} f_{r,s} (x,x)\) depends on \(u(0)\): nonetheless, the difference is practically negligible.

Here, recollecting that \(n_0 = 0\) and \(n_{k+1} = n+1\), the numbers of integrations in the j-th group of integrations are \(n_1 -1 = n_1-n_0-1 \) from 1 to \(n_1 -1\) when \(j=0\), \(n_2-n_1-1 \) from \(n_1 +1\) to \(n_2 -1\) when \(j=1\), \(\cdots \), \(n-n_k = n_{k+1}-n_k-1\) from \(n_k +1\) to n when \(j=k\). Thus, the total number of integrations is \(\left (n_1-1 \right ) + \left ( n_2 - n_1 -1 \right ) + \cdots + \left ( n - n_k \right ) = n-k\).

When obtaining (2.3.55) from (2.3.52) and (2.3.54), note that \(\frac {u\left (x_{i+1}-x_i \right )}{u\left (x_{i+1}-x_i \right )} =u\left (x_{i+1}-x_i \right )\). In addition, \(\frac {u\left ( x_{r+1}-x_r \right )}{u\left ( x_s - x_r \right )}\) is defined only when \(x_s > x_r\) and is equal to \(u\left ( x_{r+1}-x_r \right )\) because \(x_s \ge x_{r+1}\).

The conditional joint pdf (2.3.55) is meaningful only when \(x_1 < x_2 < \cdots < x_n\). Meanwhile, the joint pdf of the order statistics of an \((s-r-1)\)-dimensional i.i.d. random vector with the marginal pdf \(\bar {f} (x) = \frac { f(x) u\left ( x - x_r \right )u\left ( x_s - x \right )} { F\left ( x_s \right ) - F\left ( x_r \right ) } \) is meaningful only when \(x_{r} < x_{r+1} < \cdots < x_{s}\).

Refer to (1.​E.​7).

Refer to (1.​E.​6).

Specifically, \(f_{\left . X_{[s]} \right | \boldsymbol {X}_{[r]}} \left (\left . x_s \right | \boldsymbol {x}^r \right )\) has the additional term of \(\prod \limits _{i=1}^{r-1} u\left ( x_{i+1} - x_i \right )\). In other words, \(f_{\left . X_{[s]} \right | \boldsymbol {X}_{[r]}} \left (\left . x_s \right | \boldsymbol {x}^r \right )\) is zero unless \(x_1 < x_2 < \cdots < x_r < x_s\) while \(f_{\left . X_{[s]} \right | X_{[r]}} \left (\left . x_s \right | x_r \right )\) is zero unless \(x_r < x_s\).

Refer to Exercise 2.28 as well.

Note here that the distributions of \(X_k\) and \(1-X_k\) are the same because \(X_k\) has the distribution \(U(0, 1)\). Consequently, the distributions of \(-\ln X_k\) and \(-\ln \left (1-X_k \right )\) are also the same.

Detailed steps in the evaluation of the joint cdf’s are provided in section “Proofs and Calculations” in Appendix 1, Chap. 2.

Here, \(f_{|X|{ }_{[q]}} (0) = 0\) for \(q \in \mathbb {J}_{2,\infty }\) because \(G_F(0)=0\). In addition, letting \(u(0)=0\), we have \(f_{|X|{ }_{[1]}} (0) = ng_F(0)u(0)=0\) also.

If \(x > y\) when \(r < s\) with \(r, s \in \mathbb {J}_{1,n}\), we have \(p_{r, s}(x,y)=0\). In section “Order Statistics from Uniform Distributions” in Appendix 2, Chap. 2, we discuss another method of obtaining (2.4.23): refer to (2.A2.29).

The result obtained by changing all the negative signs into positive signs in the determinant of a square matrix is called the permanent of the matrix. The permanent of a matrix \(\boldsymbol {A}\) is usually denoted by \(\,{ }^{+}\left |\boldsymbol {A}\right |{ }^{+}\) or \(\mbox{per}\left [ \boldsymbol {A}\right ]\).