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Statistica Sinica 9(1999), 1029-1052


Giovanni Pistone and Henry P. Wynn

Politecnico di Torino and University of Warwick

Abstract: Computations with cumulants are becoming easier through the use of computer algebra but there remains a difficulty with the finiteness of the computations because all distributions except the normal have an infinite number of non-zero cumulants. One is led therefore to replacing finiteness of computations by ``finitely generated'' in the sense of recurrence relationships. In fact it turns out that there is a natural definition in terms of the exponential model which is that the first and second derivative of the cumulant generating function, K, lie on a polynomial variety. This generalises recent polynomial conditions on variance functions. This is satisfied by many examples and has applications to, for example, exact expressions for variance functions and saddle-point approximations.

Key words and phrases: Computer algebra, cumulants, exponential models.

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