Analytical Bounds for Mixture Models in Cauchy–Stieltjes Kernel Families

Fahad Alsharari, Raouf Fakhfakh () and Fatimah Alshahrani
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Fahad Alsharari: Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72311, Saudi Arabia
Raouf Fakhfakh: Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72311, Saudi Arabia
Fatimah Alshahrani: Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Mathematics, 2025, vol. 13, issue 3, 1-9

Abstract: 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 L 2 -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 L 1 -norm. The results are illustrated by some examples from quadratic CSK families.

Keywords: variance function; orthogonal polynomials; mixture models (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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