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 (Formula Presented), 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.