### 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 |
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Pages (from-to) | 55-61 |

Journal | Statistics & Probability Letters |

Volume | 132 |

Early online date | 27 Sep 2017 |

DOIs | |

Publication status | Published - 01 Jan 2018 |

### Keywords

- math.NA
- math.OC

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## Cite this

Pausinger, F., Rachh, M., & Steinerberger, S. (2018). Optimal Jittered Sampling for two Points in the Unit Square.

*Statistics & Probability Letters*,*132*, 55-61. https://doi.org/10.1016/j.spl.2017.09.010