Les familles exponentielles statistiques invariantes par les groupes du cône et du paraboloide de révolution

Abstract

Statistical exponential families invariant with respect to the groups of the cone or the paraboloid of revolution are discussed. B(x, y) denotes the symmetric bilinear form on x0y0 – x1y1 – ·· ·– xdyd on ℝd+1, C denotes the cone of revolution in ℝd+1 {x; B(x,x) > 0 and x0 > 0}, and, for p > ½(d−1), μ p is the positive measure on ℝd+1 defined by its Laplace transform ⨍ exp (B(x,y))μp(dy) = (B(y,y))−p for y on C. More precisely, if p > ½ (d−1) one has and μ½(d−1) concentrated on the boundary ∂C. This paper studies the natural exponential families