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A decomposition for the exponential dispersion model generated by the invariant measure on the hyperboloid

Research paper by M. Casalis, G. Letac, H. Massam

Indexed on: 01 Oct '93Published on: 01 Oct '93Published in: Journal of Theoretical Probability



Abstract

Forθ =θ0,θ1,θ2) andx=(x0, x1, x2) in R3, define [θ,x] =θ0x0 −θ1x1 −θ2x2,C = {x∈ℝ3:x0 > 0 and [x, x]>0},R(x)=([x, x])1/2 forx inC andH1={x∈C: x0>0,R(x)=1}. Define the measure σ onH1 such that if ϑ is inC and к=R(ϑ), then ∫ exp (−[θ,x])ω(dx = (κ exp κ)−1. Therefore, σ is invariant under the action ofSO↑(1, 2), the connected component ofO(1, 2) containing the identity. We first prove that there exists a positive measure ωλ in ℝ3 such that its Laplace transform is (κ exp κ)−γ if and only if γ>1. Finally, for γ⩾1 and ϑ inC, denotingP(θ,ωλ)(dx) = (κ exp λ)−γ exp (−[θ,x])ωλ(dx, we show that ifY0,...,Yn aren+1 independent variables with densityP(θ,ωλ),j=0,...,n and ifSk =X0 + ... +Xk andQk =R(Sk) −R(Sk−1) −R(Yk),k=1,...,n, then then+1 statisticsDn = [θ/κ,Sk] −Rk − 1),Q1,...,Qn are independent random variables with the exponential (к) or gamma (1,1/к) distribution.