In statistics, the Cunningham function or Pearson–Cunningham function ω_''m'',''n''(''x'') is a generalisation of a special function introduced by and studied in the form here by . It can be defined in terms of the confluent hypergeometric function ''U'', by :

\displaystyle \omega_(x) = \fracU(m/2-n,1+m,x).

The function was studied by Cunningham in the context of a multivariate generalisation of the

Edgeworth expansion The Gram–Charlier A series (named in honor of Jørgen Pedersen Gram and Carl Charlier), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth) are series that approximate a probability distribution in terms of its cumulants. Th ...

for approximating a

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...

based on its (joint) moments. In a more general context, the function is related to the solution of the constant-coefficient diffusion equation, in one or more dimensions. The function ω_''m'',''n''(''x'') is a solution of the differential equation for ''X'': :

xX''+(x+1+m)X'+(n+\tfracm+1)X.

The special function studied by Pearson is given, in his notation by, :

\omega_(x) =\omega_(x).

Notes

References

* * * * See exercise 10, chapter XVI, p. 353 Special hypergeometric functions Statistical approximations {{statistics-stub