Combinatorial results on the fitting problems of the multivariate gamma distribution introduced by Prékopa and Szántai

Abstract

Generalizations of the results of an earlier paper of the second author, related to the problem of fitting a multivarite gamma distribution to empirical data, are discussed in the paper. The multivariate gamma distribution under consideration is the one that was introduced in the paper of Prékopa and Szántai (in Water Resources Research, 14:19–24, 1978), some earlier results on the fitting problem were given in the paper of Szántai (in Alkalmazott Matematikai Lapok 10:35–60, 1984). In the present paper it is proved that the necessary conditions given earlier are not sufficient and some further new, mostly computational results are provided, too. Using the more efficient computation tools we are able now to give the sufficient conditions for dimensions 5 and 6 as well. For higher dimensions we have only necessary conditions and the invention of a suitable necessary and sufficient condition remains an open problem when n is greater than 6. The miscellaneousness of the necessary and sufficient conditions obtained in our new project for n=6 indicates that finding necessary and sufficient conditions in general should be a very hard problem.

Appendix

(Figures for the description of extremal directions of $\mathcal{D}_{n}$ in small dimensions.)

Fig. 1

Extremal directions of $\mathcal{D}_{2}$

Fig. 2

Further extremal directions of $\mathcal{D}_{3}$

Fig. 3

Further extremal directions of $\mathcal{D}_{4}$

Fig. 4

Further extremal directions of $\mathcal{D}_{5}$

Fig. 5

Further regular extremal directions of $\mathcal{D}_{6}$

Fig. 6

Irregular extremal directions of $\mathcal{D}_{6}$ (Part 1 of 4)

Fig. 7

Irregular extremal directions of $\mathcal{D}_{6}$ (Part 2 of 4)

Fig. 8

Irregular extremal directions of $\mathcal{D}_{6}$ (Part 3 of 4)

Fig. 9

Irregular extremal directions of $\mathcal{D}_{6}$ (Part 4 of 4)