# Optimal Jittered Sampling for two Points in the Unit Square

Florian Pausinger, Manas Rachh, Stefan Steinerberger

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

## Abstract

Jittered Sampling is a refinement of the classical Monte Carlo sampling method. Instead of picking $n$ points randomly from $[0,1]^2$, one partitions the unit square into $n$ regions of equal measure and then chooses a point randomly from each partition. Currently, no good rules for how to partition the space are available. In this paper, we present a solution for the special case of subdividing the unit square by a decreasing function into two regions so as to minimize the expected squared $\mathcal{L}_2-$discrepancy. The optimal partitions are given by a \textit{highly} nonlinear integral equation for which we determine an approximate solution. In particular, there is a break of symmetry and the optimal partition is not into two sets of equal measure. We hope this stimulates further interest in the construction of good partitions.
Original language English 55-61 Statistics & Probability Letters 132 27 Sep 2017 https://doi.org/10.1016/j.spl.2017.09.010 Published - 01 Jan 2018

• math.NA
• math.OC

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