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Pré-Publication, Document De Travail Année : 2020

New bounds for the solution and derivatives of the Stein equation for the generalized inverse Gaussian and Kummer distributions

Résumé

For Lipschitz test functions we propose a new bound of the solution of the Stein equation related to the generalized inverse Gaussian (resp. the Kummer) distribution. This bound is derived using the general approach established in Döbler (2015) for distributions satisfying a certain differential equation, and thus is optimal for Lipschitz test functions. The main contribution of this paper is to establish an explicit expression of the bound as a function of the parameters of the distribution in terms of the modified Bessel function of the third kind (resp. the confluent hypergeometric function of the second kind). Under a restriction on the parameters we also obtain an optimal bound for the first derivative of the solution. A recurrence formula is established using the iterative technique developed in Döbler (2015) and Döbler et al (2017) in order to bound derivatives of any order, for test functions smooth enough.
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Dates et versions

hal-02562754 , version 1 (04-05-2020)

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  • HAL Id : hal-02562754 , version 1

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Essomanda Konzou, Efoevi Angelo Koudou, Kossi E Gneyou. New bounds for the solution and derivatives of the Stein equation for the generalized inverse Gaussian and Kummer distributions. 2020. ⟨hal-02562754⟩
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