Regularity property of the functional equation of the Dirichlet distribution
The author deals with the functional equation \[ f_0 ( y_1, \dots, y_{n-1}) \prod^n_{j=1} g_j ( z_{i,j} , \dots, z_{k-1, j }) = g_0 ( x_1, \dots, x_{k-1}) \prod_{i=1}^k f_i ( w_{i,1}, \dots, w_{i, n-1}) . \tag{1} \] After the results obtained by \textit{D. Geiger} and \textit{D. Heckerman} [Ann. Stat. 25, No. 3, 1344-1369 (1997; Zbl 0885.62009)], characterizing the Dirichlet distribution, D. Geiger set the following problem: If the unknown functions in (1) are density functions, does it follow that they are positive everywhere and \( C^{\infty} \)? In this paper an elegant affirmative answer to this question is given.
scientific article; zbMATH DE number 1220995