Classroom note: A note on beta function, parseval identity, and a family of integrals in non-parametric regression

Abstract

A family of integrals over infinite intervals appears frequently in the statistical literature on non-parametric regression. This note presents two simple transform methods, based on beta function and Parseval identity, to explicitly evaluate these integrals. These methods will benefit researchers and practitioners working in non-parametric regression, whenever they encounter similar types of problems to solve.

Acknowledgments

We thank an anonymous referee for helpful comments and suggestions.