@article{Bonaccorsi2024, 
author = {Stefano Bonaccorsi and Bernard Hanzon and Giulia Lombardi},
title = {A generalized Budan-Fourier approach to generalized Gaussian and exponential mixtures},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {10},
pages = {26499-26537},
keywords = {finite mixtures, Gaussian mixtures, Exponential-Polynomial-Trigonometric probability density functions},
url = {https://www.sciopen.com/article/10.3934/math.20241290},
doi = {10.3934/math.20241290},
abstract = {In the literature, finite mixture models were described as linear combinations of probability distribution functions having the form  f(x)=Λ∑i=1nwifi(x),  x∈R, where  wi were positive weights,  Λ was a suitable normalising constant, and  fi(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  wi, 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.}
}