Lorentzian vs Gaussian Fitting Functions

In the crystallographic field, usually the X-ray diffraction peaks are fitted using Gaussian or Lorentz functions. Even more, sometimes it is used the Voigt function that is a mixture of both functions. Each parameter of the function has a different meaning related to the internal strains (Strains) or to the cristallite size (Lorentz)

The Lorentzian function has more pronounced tails than a corresponding Gaussian function, and since this is the natural form of the solution to the differential equation describing a damped harmonic oscillator, I think it should be used in all physics concerned with such oscillations, i.e. natural line widths, plasmon oscillations etc.

One side of the question is whether to use a mixture of Cauchy distributions or a mixture of Gaussians to fit some given multimodal function. Sometimes the data is better suited to one or the other.

Another aspect is how fitting a mixture of Cauchy distributions to data which is well approximated by a mixture of Cauchy distributions differs from fitting a mixture of Gaussians to a mixture of Gaussians. Because Cauchy distributions are less localized and have some nice algebraic properties, it can be much easier to fit a mixture of Cauchy distributions.

Suppose you can estimate the probability density function of a distribution. This is common in practice, and you expect that your measurements are not perfect. There are algebraic methods for identifying a mixture of exponential distributions by using the Laplace transform (numerically) to convert a sum of exponentials to a rational function you can identify with Padé approximation. Similar techniques work on mixtures of Cauchy distributions, as I found when I helped a some material scientists with a spectroscopy problem. The Hilbert transform of a Cauchy distribution is another rational function: $H(frac{1}{1+t^2})(s) = frac{s}{1+s^2}$. Just as with the well-used Laplace-Padé method, you can identify the number of Cauchy components. Of course, you could try to use Padé approximation in the first place to identify the Cauchy mixture, but that wouldn't use all of your data, and it would be very vulnerable to noise. Using the Hilbert transform first lets you cancel out the noise and lets you extract useful information from data points far from the peaks of the components.

By contrast, densities collected several FWHMs or standard deviations from the peak of a Gaussian component will have little information about that component. You need exponentially large amounts of data or else the signal will be hidden by the noise. It is difficult to identify the correct number of components in a Gaussian mixture model. The best numerical fits may ignore components of significant amplitudes which are far from the data in favor of splitting components or adding very localized peaks to fit outliers. These may be ameliorated by using regularization, but the lack of signal far from the peaks of the components can't be fixed by massaging the data.

A Short answer: Robustness. The gaussian distribution effectively assumes there are no outliers. If that assumption is wrong, it can give misleading results! But, on the other hand, using the cauchy distribution might be too extreme. A startpoint for you could be: http://en.wikipedia.org/wiki/Robust_statistics

And if you have more questions, a more appropriate place could be: Cross Validated: https://stats.stackexchange.com/?as=1

This depends on what you are trying to model. In my work (vibrational spectroscopy) the Lorentz lineshape is used to model 'pure' vibrational modes, which only undergo homogeneous line-broadening. The Gaussian lineshape is used to model those curves which have additional broadening terms from instrumental effects.

Here is a citation to a paper on this: Robert Meier, Vibrational Spectroscopy 39 (2005) 266–269

You'll have to do some research in your specific field to know for sure which you should be using. It is a very subtle science...