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Vincent average

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Statistical estimation technique

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} .

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)
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