Analytical Bounds for Mixture Models in Cauchy–Stieltjes Kernel Families

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Mixture models are widely used in mathematical statistics and theoretical probability. However, the mixture probability distribution is rarely explicit in its formula. One must then decide whether to keep the parent probability distribution or to obtain an approximation of the mixture probability distribution. In such cases, it is essential to estimate or evaluate the distance between a mixture probability distribution and its parent probability distribution. On the other hand, orthogonal polynomials offer a versatile mathematical tool for approximating, fitting, and analyzing mixture models, facilitating more accurate and efficient modeling in statistics and data science. This article considers mixture models in Cauchy–Stieltjes Kernel (CSK) families. Using a suitable basis of polynomials, we obtain an expression for the distance in the L2-norm between the mixed probability distribution and its parent probability distribution which belongs to a given CSK family. For the distance between the corresponding distribution functions, bounds are derived in L1-norm. The results are illustrated by some examples from quadratic CSK families.