A generalized Budan-Fourier approach to generalized Gaussian and exponential mixtures

In the literature, finite mixture models were described as linear combinations of probability distribution functions having the form $f(x)=\mathrm{\Lambda }\sum _{i=1}^n{w}_i{f}_i(x)$, $x\in \mathbb{R}$, where $w_i$ were positive weights, $\mathrm{\Lambda }$ was a suitable normalising constant, and $f_i(x)$ were given probability density functions. The fact that $f(x)$ is a probability density function followed naturally in this setting. Our question was: if we removed the sign condition on the coefficients $w_i$, how could we ensure that the resulting function was a probability density function?

The solution that we proposed employed an algorithm which allowed us to determine all zero-crossings of the function $f(x)$. Consequently, we determined, for any specified set of weights, whether the resulting function possesses no such zero-crossings, thus confirming its status as a probability density function.

In this paper, we constructed such an algorithm which was based on the definition of a suitable sequence of functions and that we called a generalized Budan-Fourier sequence; furthermore, we offered theoretical insights into the functioning of the algorithm and illustrated its efficacy through various examples and applications. Special emphasis was placed on generalized Gaussian mixture densities.