Edgeworth Series

Approximate a distribution in terms of a Normal Distribution. Let

$$\phi(t)\equiv {1\over \sqrt{2\pi}} e^{-t^2/2},$$

then

$$f(t)=\phi(t)+{\textstyle{1\over 3!}}\gamma_1 \phi^{(3)}(t)+\... ...(4)}(t)+{10{\gamma_1}^2\over 6!} \phi^{(6)}(t)}\right]+\ldots.$$ (1)

See also Cornish-Fisher Asymptotic Expansion, Gram-Charlier Series

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 935, 1972.

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, p. 108, 1951.