On maximum likelihood estimation of the concentration parameter of von Mises–Fisher distributions

Kurt Hornik () and Bettina Grün

Computational Statistics, 2014, vol. 29, issue 5, 945-957

Abstract: Maximum likelihood estimation of the concentration parameter of von Mises–Fisher distributions involves inverting the ratio $$R_\nu=I_{\nu +1} / I_\nu $$ R ν = I ν + 1 / I ν of modified Bessel functions and computational methods are required to invert these functions using approximative or iterative algorithms. In this paper we use Amos-type bounds for $$R_\nu $$ R ν to deduce sharper bounds for the inverse function, determine the approximation error of these bounds, and use these to propose a new approximation for which the error tends to zero when the inverse of $$R_\nu $$ R ν is evaluated at values tending to $$1$$ 1 (from the left). We show that previously introduced rational bounds for $$R_\nu $$ R ν which are invertible using quadratic equations cannot be used to improve these bounds. Copyright The Author(s) 2014

Keywords: Maximum likelihood; Modified Bessel function ratio; Numerical approximation; von Mises–Fisher distribution (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:compst:v:29:y:2014:i:5:p:945-957

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DOI: 10.1007/s00180-013-0471-0

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