Fabius, J. (1973). Two characterizations of the Dirichlet distribution. Ann. of Stat., 1:583--587.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....differential equation the solution of which after appropriate specialization is the general solution of Eq. 4 (Acz el, 66, Section 4. 2, Reduction to differential equations ) Additional characterizations of Dirichlet distributions which are based on other independence assumptions can be found in [Fabius, 1973, James and Mosimann, 1980] 3 The Functional Equation By renaming of variable and function names, Eq. 4 can be written as follows: f 0 (y 1 ; y n Gamma1 ) n Y j=1 g j (z 1;j ; z k Gamma1;j ) g 0 (x 1 ; x k Gamma1 ) k Y i=1 f i ( z i1 y 1 x i ; z ....

Fabius, J. (1973). Two characterizations of the Dirichlet distribution. Ann. of Stat., 1:583--587.

....these representations, as in Equation 4, forms a functional equation. This functional equation is of the type dealt by J arai and consequently, the Theorem by Darroch and Ratcliff holds even without assuming continuous pdfs. Indeed, among other results, this was shown using other techniques, by [Fa73, JM80]. Note that if a pdf is in fact a gpdf, that is, it contains a discrete element, then Lebesgue integrability is not satisfied and this technique is not applicable as is. In this case, one may resort to the functional equations defined by the characteristic functions near the origin. A review of ....

J. Fabius, Two characterizations of the Dirichlet distribution, Annals of Statistics, 1:583-587, 1973.

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