Statistical estimation of $\pi$: varying choices over dimensions

TL;DR

This paper explores statistical methods for estimating π by analyzing volume ratios of hyperspheres and hypercubes across different dimensions, using Monte Carlo simulations and providing R code for numerical experiments.

Contribution

It introduces a dimension-dependent approach to estimating π and investigates the effects of varying dimensions, including infinite-dimensional considerations, with practical R implementations.

Findings

Estimation accuracy varies with dimension d

Monte Carlo methods effectively estimate π in high dimensions

Infinite-dimensional observations offer new estimation insights

Abstract

This article studies statistical estimation of $\pi$ based on the fact that the ratio of the volumes of a $d$-dimensional hypersphere and a $d$-dimensional hypercube is a certain function of $\pi$, and the function depends on the dimension $d$. The estimation of $\pi$ is carried out for various choices of $d$ (strictly speaking, $d\in\{1, 2, \ldots, 20\}$) using the idea of Monte Carlo simulations. Various intriguing facts are observed, and the estimation of $\pi$ using infinite dimensional observations is outlined. Moreover, the R codes associated with relevant numerical studies are provided.