Analytical Bounds for Mixture Models in Cauchy–Stieltjes Kernel Families
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 di...
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2025-01-01
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| author | Fahad Alsharari Raouf Fakhfakh Fatimah Alshahrani |
| author_facet | Fahad Alsharari Raouf Fakhfakh Fatimah Alshahrani |
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| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula>-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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>1</mn></msup></semantics></math></inline-formula>-norm. The results are illustrated by some examples from quadratic CSK families. |
| format | Article |
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| language | English |
| publishDate | 2025-01-01 |
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| spelling | doaj-art-7eb3d3b878ad41b187e893041c169f6c2025-08-20T03:12:35ZengMDPI AGMathematics2227-73902025-01-0113338110.3390/math13030381Analytical Bounds for Mixture Models in Cauchy–Stieltjes Kernel FamiliesFahad Alsharari0Raouf Fakhfakh1Fatimah Alshahrani2Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72311, Saudi ArabiaDepartment of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72311, Saudi ArabiaDepartment of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi ArabiaMixture 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula>-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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>1</mn></msup></semantics></math></inline-formula>-norm. The results are illustrated by some examples from quadratic CSK families.https://www.mdpi.com/2227-7390/13/3/381variance functionorthogonal polynomialsmixture models |
| spellingShingle | Fahad Alsharari Raouf Fakhfakh Fatimah Alshahrani Analytical Bounds for Mixture Models in Cauchy–Stieltjes Kernel Families Mathematics variance function orthogonal polynomials mixture models |
| title | Analytical Bounds for Mixture Models in Cauchy–Stieltjes Kernel Families |
| title_full | Analytical Bounds for Mixture Models in Cauchy–Stieltjes Kernel Families |
| title_fullStr | Analytical Bounds for Mixture Models in Cauchy–Stieltjes Kernel Families |
| title_full_unstemmed | Analytical Bounds for Mixture Models in Cauchy–Stieltjes Kernel Families |
| title_short | Analytical Bounds for Mixture Models in Cauchy–Stieltjes Kernel Families |
| title_sort | analytical bounds for mixture models in cauchy stieltjes kernel families |
| topic | variance function orthogonal polynomials mixture models |
| url | https://www.mdpi.com/2227-7390/13/3/381 |
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