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Bandwidth of gamma-distribution-shaped functions via Lambert W function
Anthony LoPrete1, Johannes Burge1,2,3
1Bioengineering Graduate Group, University of Pennsylvania, Philadelphia, PA, USA.
Abstract:
The full width at half maximum (FWHM) is a useful quantity for characterizing the bandwidth of unimodal functions. However, a closed-form expression for the FWHM of gamma-shaped functions - i.e. functions that are shaped like the gamma distribution probability density function (PDF) - is not widely available. Here, we derive and present just such an expression. To do so, we use the Lambert W function to compute the inverse of the gamma PDF. We use this inverse to derive an exact analytic expression for the full-width at an arbitrary y-proportion of the maximum (FWyM) of a gamma distribution, from which the FWHM follows trivially. (An expression for the octave bandwidth of gamma-shaped functions is also provided.) The FWHM is then compared to the Gaussian approximation of gamma-shaped functions. A MATLAB function is provided that computes the key quantities. A few other related issues are also discussed.
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