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The Lindsay transform of natural exponential families
Author(s) -
Kokonendji C. C.,
Seshadri V.
Publication year - 1994
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315588
Subject(s) - mathematics , measure (data warehouse) , convolution (computer science) , exponential function , affine transformation , transformation (genetics) , pure mathematics , gaussian , exponential family , natural exponential family , function (biology) , combinatorics , mathematical analysis , physics , biochemistry , chemistry , quantum mechanics , database , machine learning , evolutionary biology , biology , computer science , artificial neural network , gene
Let μ be an infinitely divisible positive measure on R. If the measure ρ μ is such that x ‐2 [ρ μ (dx)—ρ μ ({0})δ 0 (dx)] is the Lévy measure associated with μ and is infinitely divisible, we consider for all positive reals α and β the measure T α,β (μ) which is the convolution of μ* α and ρ μ *β. For example, if μ is the inverse Gaussian law, then ρ μ is a gamma law with paramter 3/2. Then T α,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.