The Lindsay transform of natural exponential families

Abstract

AbstractLet μ be an infinitely divisible positive measure on R. If the measure ρμ is such that x‐2[ρμ(dx)—ρμ({0})δ0(dx)] is the Lévy measure associated with μ and is infinitely divisible, we consider for all positive reals α and β the measure Tα,β(μ) which is the convolution of μ*α and ρμ*β. For example, if μ is the inverse Gaussian law, then ρμ is a gamma law with paramter 3/2. Then Tα,β(μ) is an extension of the Lindsay transform of the first order, restricted to the distributions which are infinitely divisible. The main aim of this paper is to point out that it is possible to apply this transformation to all natural exponential families (NEF) with strictly cubic variance functions P. We then obtain NEF with variance functions of the form □ΔP(□Δ), where A is an affine function of the mean of the NEF. Some of these latter types appear scattered in the literature.