Chisini mean

In mathematics, a function f of n variables x1, ..., xn leads to a Chisini mean M if, for every vector ⟨x1, ..., xn⟩, there exists a unique M such that[1]

f(M,M, ..., M) = f(x1,x2, ..., xn).

The arithmetic, harmonic, geometric, generalised, Heronian and quadratic means are all Chisini means, as are their weighted variants.

While Oscar Chisini was arguably the first to deal with "substitution means" in some depth in 1929,[1] the idea of defining a mean as above is quite old, appearing (for example) in early works of Augustus De Morgan.[2][original research?]

See also

References

  1. ^ a b Graziani, Rebecca; Veronese, Piero (2009). "How to Compute a Mean? The Chisini Approach and Its Applications". The American Statistician. 63 (1): 33–36. doi:10.1198/tast.2009.0006. JSTOR 27644090. S2CID 119340091.
  2. ^ De Morgan, Augustus. "Mean" in Penny Cyclopaedia (1839).


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