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(This article was first published on <strong><a href="https://xianblog.wordpress.com/2018/06/08/maximal-spacing-around-order-statistics-2/"> R – Xi'an's Og</a></strong>, and kindly contributed to <a rel="nofollow" href="https://www.r-bloggers.com/">R-bloggers)</a> </div>
<p><strong><img data-orig-file="https://xianblog.files.wordpress.com/2018/06/riddler0106.jpeg" data-orig-size="525,327" data-medium-file="https://xianblog.files.wordpress.com/2018/06/riddler0106.jpeg?w=300" data-attachment-id="36291" data-image-meta="{" data-image-description="" src="https://xianblog.files.wordpress.com/2018/06/riddler0106.jpeg?w=450&h=280&fit=450%2C280&resize=450%2C280" data-permalink="https://xianblog.wordpress.com/2018/06/08/maximal-spacing-around-order-statistics-2/riddler0106/" data-large-file="https://xianblog.files.wordpress.com/2018/06/riddler0106.jpeg?w=450&amp;h=280&amp;fit=450%2C280&amp;resize=450%2C280" alt="" data-image-title="riddler0106" sizes="(max-width: 450px) 100vw, 450px" width="450" data-comments-opened="1" class="wp-image-36291" data-recalc-dims="1" height="280">T</strong>he <a rel="nofollow" href="https://fivethirtyeight.com/features/when-will-your-house-collapse-should-you-take-a-construction-contract/" target="_blank">proposed solution</a> of the <a rel="nofollow" href="https://fivethirtyeight.com/features/how-hard-is-it-to-find-a-running-buddy/" target="_blank">riddle from the Riddler</a> discussed <a rel="nofollow" href="https://xianblog.wordpress.com/2018/05/17/maximal-spacing-around-order-statistics/" target="_blank">here a few weeks</a> ago is rather approximative, in that the distribution of</p>
<p><img alt="Delta_n=max_i,min_j,|X_{i}-X_{j}|" title="Delta_n=max_i,min_j,|X_{i}-X_{j}|" src="https://s0.wp.com/latex.php?latex=%5CDelta_n%3D%5Cmax_i%5C%2C%5Cmin_j%5C%2C%7CX_%7Bi%7D-X_%7Bj%7D%7C&bg=000000&%23038;fg=B0B0B0&%23038;s=0" class="latex"></p>
<p>when the n-sample is made of iid Normal variates is (a) replaced with the distribution of one arbitrary minimum and (b) the distribution of the minimum is based on an assumption of independence between the absolute differences. Which does not hold, as shown by the above correlation matrix (plotted via <a rel="nofollow" href="https://cran.r-project.org/web/packages/corrplot/vignettes/corrplot-intro.html" target="_blank">corrplot</a>) for N=11 and 10⁴ simulations. One could think that this correlation decreases with N, but it remains essentially 0.2 for larger values of N. (On the other hand, the minima are essentially independent.)</p>
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