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

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.