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Bandwidth of gamma-distribution-shaped functions via Lambert W function

Author

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  • LoPrete, Anthony
  • Burge, Johannes

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.

Suggested Citation

  • LoPrete, Anthony & Burge, Johannes, 2026. "Bandwidth of gamma-distribution-shaped functions via Lambert W function," Statistics & Probability Letters, Elsevier, vol. 235(C).
    • Handle: RePEc:eee:stapro:v:235:y:2026:i:c:s0167715226000714
    DOI: 10.1016/j.spl.2026.110707
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