On Integral Theorems and their Statistical Properties
This work addresses density estimation in statistics, offering incremental improvements by optimizing cyclic functions for better estimators.
The authors tackled the problem of density estimation by introducing a class of integral theorems based on cyclic functions and Riemann sums, which provide natural estimators via Monte Carlo methods, resulting in optimal cyclic functions that minimize square integrals as alternatives to the sin function.
We introduce a class of integral theorems based on cyclic functions and Riemann sums approximating integrals. 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 methods. Assessments of the quality of the density estimators can be used to obtain optimal cyclic functions, alternatives to the sin function, which minimize square integrals. Our proof techniques rely on a variational approach in ordinary differential equations and the Cauchy residue theorem in complex analysis.