Vincent average

In applied statistics, Vincentization^[1] 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 ${\displaystyle 1}$.

References

1. Genest, Christian (1992). "Vincentization Revisited" (PDF). 20 (2). The Annals of Statistics: 1137–1142. Retrieved 5 Sep 2018. {{cite journal}}: Cite journal requires |journal= (help)
2. Ratcliff, Roger (1979). "Group Reaction Time Distributions and an Analysis of Distribution Statistics" (PDF). Psychological Bulletin. 86 (3): 446–461. doi:10.1037/0033-2909.86.3.446. PMID 451109. Retrieved 18 November 2016.
3. Vincent, Stella; Burnham (1912). "The function of the viborissae in the behavior of the white rat". 1. Behavior Monographs. {{cite journal}}: Cite journal requires |journal= (help)