Vincent average

In applied statistics, Vincentization{{cite journal |url=https://projecteuclid.org/euclid.aos/1176348676 |title=Vincentization Revisited |last1=Genest |first1=Christian |publisher=The Annals of Statistics|volume=20 |number=2 |pages=1137–1142 |date=1992 |format=PDF|accessdate=5 Sep 2018}} was described by Ratcliff (1979),{{cite journal |url=http://star.psy.ohio-state.edu/wp/pdf/Papers/psychbull79.pdf |title=Group Reaction Time Distributions and an Analysis of Distribution Statistics |last1=Ratcliff |first1=Roger |journal=Psychological Bulletin|volume=86 |number=3 |pages=446–461 |date=1979 |doi=10.1037/0033-2909.86.3.446 |pmid=451109 |accessdate=18 November 2016}} and is named after biologist S. B. Vincent (1912),{{cite journal |title=The function of the viborissae in the behavior of the white rat |last1=Vincent |last2=Burnham |first1=Stella |publisher=Behavior Monographs|volume=1 |date=1912}} who used something very similar to it for constructing learning curves at the beginning of the 1900s. It basically consists of averaging $n\geq 2$ subjects' estimated or elicited quantile functions in order to define group quantiles from which $F$ can be constructed.

To cast it in its greatest generality, let $F_1,\dots, F_n$ represent arbitrary (empirical or theoretical) distribution functions and define their corresponding quantile functions by

: $F_i^{-1}(\alpha) = \inf\{t\in \mathbb{R} : F_i(t)\ge\alpha) \},\quad 0<\alpha\leq 1.$

The Vincent average of the $F_i$ 's is then computed as

: $F^{-1}(\alpha) = \sum w_i F_i^{-1}(\alpha),\quad 0<\alpha\leq 1,\quad i = 1,\ldots,n$

where the non-negative numbers $w_1,\dots,w_n$ have a sum of $1$ .

References

Category:Applied statistics