Statistical estimation of $\ddot{I}€$: varying choices over dimensions

Abstract

This article studies statistical estimation of $\ddot{I}€$ 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 $\ddot{I}€$, and the function depends on the dimension $d$. The estimation of $\ddot{I}€$ is carried out for various choices of $d$ (strictly speaking, $d\in \{1,2,\dots ,20\}$) using the idea of Monte Carlo simulations. Various intriguing facts are observed, and the estimation of $\ddot{I}€$ using infinite dimensional observations is outlined. Moreover, the R codes associated with relevant numerical studies are provided.

8 pages, 4 figures. This is a preliminary draft. The manuscript will be updated further before formal communication