On Integral Theorems: Monte Carlo Estimators and Optimal Functions

• URL: http://arxiv.org/abs/2107.10947v1
• Date: Thu, 22 Jul 2021 22:25:21 GMT
• Title: On Integral Theorems: Monte Carlo Estimators and Optimal Functions
• Authors: Nhat Ho and Stephen G. Walker
• Abstract summary: We introduce a class of integral theorems based on cyclic functions. The integral theorems provide natural estimators of density functions via Monte Carlo integration. Our proof techniques rely on a variational approach in ordinary differential equations and the Cauchy residue theorem in complex analysis.
• Score: 9.619814126465206
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce a class of integral theorems based on cyclic functions and Riemann sums approximating integrals theorem. The Fourier integral theorem, derived as a combination of a transform and inverse transform, arises as a special case. The integral theorems provide natural estimators of density functions via Monte Carlo integration. Assessments of the quality of the density estimators can be used to obtain optimal cyclic functions which minimize square integrals. Our proof techniques rely on a variational approach in ordinary differential equations and the Cauchy residue theorem in complex analysis.

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