Non-negativity and zero isolation for generalized mixtures of densities
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In the literature, finite mixture models are described as linear combinations of probability distribution functions having the form f(x) = Λ∑_i=1^n w_i f_i(x), x ∈ℝ, where w_i are positive weights, Λ is a suitable normalising constant and f_i(x) are given probability density functions. The fact that f(x) is a probability density function follows naturally in this setting. Our question is: what happens when we remove the sign condition on the coefficients w_i? The answer is that it is possible to determine the sign pattern of the function f(x) by an algorithm based on finite sequence that we call a generalized Budan-Fourier sequence. In this paper we provide theoretical motivation for the functioning of the algorithm, and we describe with various examples its strength and possible applications.
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