On maximum likelihood estimation of the concentration parameter of von Mises–Fisher distributions
Maximum likelihood estimation of the concentration parameter of von Mises–Fisher distributions involves inverting the ratio <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$R_\nu=I_{\nu +1} / I_\nu $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>R</mi> <mi mathvariant="italic">ν</mi> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow> <mi mathvariant="italic">ν</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">/</mo> <msub> <mi>I</mi> <mi mathvariant="italic">ν</mi> </msub> </mrow> </math> </EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$R_\nu $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>R</mi> <mi mathvariant="italic">ν</mi> </msub> </math> </EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$R_\nu $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>R</mi> <mi mathvariant="italic">ν</mi> </msub> </math> </EquationSource> </InlineEquation> is evaluated at values tending to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>1</mn> </mrow> </math> </EquationSource> </InlineEquation> (from the left). We show that previously introduced rational bounds for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$R_\nu $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>R</mi> <mi mathvariant="italic">ν</mi> </msub> </math> </EquationSource> </InlineEquation> which are invertible using quadratic equations cannot be used to improve these bounds. Copyright The Author(s) 2014
