Vincent average

In applied statistics, Vincentization¹ was described by Ratcliff (1979),² and is named after biologist S. B. Vincent (1912),³ who used something very similar to it for constructing learning curves at the beginning of the 1900s. It basically consists of averaging ${\displaystyle n\geq 2}$ subjects' estimated or elicited quantile functions in order to define group quantiles from which ${\displaystyle F}$ can be constructed.

To cast it in its greatest generality, let ${\displaystyle F_{1},\dots ,F_{n}}$ represent arbitrary (empirical or theoretical) distribution functions and define their corresponding quantile functions by

${\displaystyle F_{i}^{-1}(\alpha )=\inf\{t\in \mathbb {R} :F_{i}(t)\geq \alpha )\},\quad 0<\alpha \leq 1.}$

The Vincent average of the ${\displaystyle F_{i}}$'s is then computed as

${\displaystyle F^{-1}(\alpha )=\sum w_{i}F_{i}^{-1}(\alpha ),\quad 0<\alpha \leq 1,\quad i=1,\ldots ,n}$

where the non-negative numbers ${\displaystyle w_{1},\dots ,w_{n}}$ have a sum of ${\displaystyle 1}$.