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) = \Lambda \sum\limits_{i = 1}^n w_i f_i(x) $, $ x \in \mathbb{R} $, where $ w_i $ were positive weights, $ \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.