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. 2020 Oct 15;10:17372. doi: 10.1038/s41598-020-74057-1

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© The Author(s) 2020

Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Analysis of asymptotic properties of pre and post approaches with the k-nn estimator. Left: violin plots⁴² display the distribution of MIF estimates as a function of sample size in relation to the true MIF value indicated by the dashed line. 1e3 estimates comprise each distribution. The trendline intersects the mean of each distribution in order to indicate decreasing bias with increasing number of samples, while the narrowing of the distributions indicate decreased variance with increased sample size. Right: same analysis performed for the post estimator, which exhibits poor bias but considerably smaller variance than the pre estimator. The decrease of this variance with sample size is highlighted by the two trendlines following the 5th and 95th percentiles of each distribution.