Let $r_1 ≤ r_2 ≤ ... ≤ r_N$ denote an ORDERED set of N realizations of real numbers that are uniformly random on the number line from 0 to 1.

Let $R_1 < R_2 < ... < R_N$ denote a set of real numbers that are EVENLY SPACED on the number line from 0 to 1. In other words, $R_i = \frac{i-\frac{1}{2}}{N}$.

Let $e_i = \frac{r_i – R_i}{\frac{1}{N}}$ denote the relative (i.e., normalized by $\frac{1}{N}$ discrepancy between the ith statistical sample and the ith uniformly spaced sample.

QUESTION: what is the PDF for $e_i$?

A numerical experiment shows that the mean of $e_i$ varies in what appears to be an affine manner with respect to i, which shows that the PDF for $e_i$ depends on the value of i. This question pertains to the topic of finite-sampling errors, so it should be well known, but I don’t quite know what search terms would lead to the answer. Citations to publications on the topic would be appreciated.

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    $\begingroup$ The shift by $R_i$ and scaling by $N$ have obvious and easily computed effects on the PDF, so your question is tantamount to asking for the distribution of a uniform order statistic. See stats.stackexchange.com/search?q=uniform+order+statistic. $\endgroup$ – whuber May 29 '18 at 0:41
  • $\begingroup$ This question is also linked with the Kolmogorov Smirnov test in that the distribution of the statistic under the null is the maximum of the $|r_i-i/N|$'s. $\endgroup$ – Xi'an May 29 '18 at 7:39
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    $\begingroup$ As pointed out by W. Huber a few hours ago, the pdf of $e_i$ is the location-scale transform of the Beta$(i,n+1-i)$, i.e.$$e_i=R_i+\epsilon_i/N\qquad \epsilon_i\sim Beta(i,n+1-i)$$ $\endgroup$ – Xi'an May 29 '18 at 8:27

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