3. Rank Statistics

Discussed similarly on the joint pdf (1.​2.​58) in Chap. 1 already, it is usually assumed \(i \ne j\) implicitly in \(p_{R_i, R_j}\) when we say ‘joint’ pmf in the strict sense. In the discussions in this chapter, we will in many cases adopt the concept ‘joint’ in a wider sense as in the joint pmf (3.1.4), taking the case \(i=j\) into account also.

Here, let \(t= F^{-1}\left (\frac {1+v}{2}\right )\). Then, \(v= 2F(t)-1 = G_F(t)\) and, consequently, \(t= G_F^{-1}(v)= F^{-1}\left (\frac {1+v}{2}\right )\) when the pdf \(f(x)\) is an even symmetric function of x.

Refer to Definition 3.​A3.​2 for more detail.

More specifically, the function \(\gamma _L (\alpha ,x)= \int _0^x e^{-t}t^{\alpha -1} dt\) is called the lower incomplete gamma function and the function \(\gamma _U (\alpha ,x)= \int _x^{\infty } e^{-t}t^{\alpha -1} dt\) is called the upper incomplete gamma function.

The rising factorial is also called the ascending factorial, rising sequential product, upper factorial, Pochhammer’s symbol, Pochhammer function, or Pochhammer polynomial, and is the same as Appell’s symbol \((z,n)\).