Open AccessRandom Polygons and Optimal Extrapolation Estimates of pi
Author(s) -
Shasha Wang,
WenQing Xu,
Jitao Liu
Publication year - 2021
Publication title -
statistics, optimization and information computing
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.297
H-Index - 12
eISSN - 2311-004X
pISSN - 2310-5070
DOI - 10.19139/soic-2310-5070-1193
Subject(s) - extrapolation , mathematics , convergence (economics) , random variate , convergence of random variables , combinatorics , random variable , statistics , economics , economic growth
We construct optimal extrapolation estimates of π based on random polygons generated by n independent points uniformly distributed on a unit circle in R2. While the semiperimeters and areas of these random n-gons converge to π almost surely and are asymptotically normal as n → ∞, in this paper we develop various extrapolation processes to further accelerate such convergence. By simultaneously considering the random n-gons and suitably constructed random 2n-gons and then optimizing over functionals of the semiperimeters and areas of these random polygons, we derive several new estimates of π with faster convergence rates. These extrapolation improvements are also shown to be asymptotically normal as n → ∞.