Multivariate Digamma Distribution
نویسنده
چکیده
where x = l , 2, . . . ; r > 0 , a > I (ar a + r > 0 , (a)~=a(a+l)...(a-kx--1), and r log F(z)/dz is the digamma or the psi function. The distribution (1.1) will be referred to as DGa (a, r). In this paper we extend DGa (a, r) to a multivariate distribution. The digamma distribution is closely related to the logarithmic series distribution LSr (0) defined by (1.2) P r [ X = x ; 0 ] = 0 ~ / ( l o g ( 1 O ) ) x , x = l , 2 , . . ; 0 < O < l . In fact, if a and r of DGa (a, r) increase indefinitely keeping a/(a+r) =0 constant, then the limit distribution is LSr (O). Conversely, if the parameter 0 of LSr (0) is a random variable with the density (1.3) C(a, ; , ) ( log (1-0))0~-1(1-0) T-1 , 0 < 0 < 1 , which we shall call an end accented beta distribution, then the compounded distribution is DGa (a, r). Digamma distributions can be used
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