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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.
Statistics & Probability Letters
|June 10, 2026
Summary
Researchers derived a closed-form expression for the full width at half maximum (FWHM) of gamma-shaped functions. This provides a new analytic tool for characterizing bandwidth in scientific applications.
Area of Science:
- Mathematics
- Statistics
- Signal Processing
Background:
- The full width at half maximum (FWHM) is crucial for characterizing unimodal function bandwidth.
- A closed-form expression for the FWHM of gamma-shaped functions is not readily available.
Purpose of the Study:
- Derive and present a closed-form expression for the FWHM of gamma-shaped functions.
- Provide an analytic tool for bandwidth characterization.
Main Methods:
- Utilized the Lambert W function to compute the inverse of the gamma probability density function (PDF).
- Derived an exact analytic expression for the full-width at an arbitrary y-proportion of the maximum (FWyM) of a gamma distribution.
- Obtained the FWHM trivially from the FWyM expression.
Main Results:
- Presented a novel, exact analytic expression for the FWHM of gamma-shaped functions.
- Derived an expression for the octave bandwidth of gamma-shaped functions.
- Compared the FWHM of gamma-shaped functions to their Gaussian approximation.
Conclusions:
- The derived expression offers a valuable method for analyzing the bandwidth of gamma-shaped functions.
- The study provides a practical MATLAB function for computing key quantities.
- This work facilitates a deeper understanding and application of gamma-shaped functions in various scientific fields.
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