In applied statistics, Vincentization was described by Ratcliff (1979), and is named after biologist S. B. Vincent (1912), who used something very similar to it for constructing learning curves at the beginning of the 1900s. It basically consists of averaging n ≥ 2 {\displaystyle n\geq 2} subjects' estimated or elicited quantile functions in order to define group quantiles from which F {\displaystyle F} can be constructed. To cast it in its greatest generality, let F 1 , … , F n {\displaystyle F_{1},\dots ,F_{n}} represent arbitrary (empirical or theoretical) distribution functions and define their corresponding quantile functions by
F i − 1 ( α ) = inf { t ∈ R : F i ( t ) ≥ α ) } , 0 < α ≤ 1. {\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 F i {\displaystyle F_{i}} 's is then computed as
F − 1 ( α ) = ∑ w i F i − 1 ( α ) , 0 < α ≤ 1 , i = 1 , … , n {\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 w 1 , … , w n {\displaystyle w_{1},\dots ,w_{n}} have a sum of 1 {\displaystyle 1} .
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