For any natural $n$, let $U_1,\dots,U_n$ be independent identically distributed random variables each uniformly distributed on the interval $[0,1]$. As usual, let $U_{n:1}\le\cdots\le U_{n:n}$ denote the corresponding order statistics. Consider
\begin{equation}\label{eq:G_i} G_i:=U_{n:i}-U_{n:i-1} \quad \text{for}\quad i=1,\dots,n+1, \end{equation} where $U_{n:0}:=0$ and $U_{n+1:n+1}:=1$.

One may refer to the $G_i$'s as the gaps or, as it is usually done in the literature, the spacings between the consecutive order statistics. Let now $$ G_{n+1:1}\le\cdots\le G_{n+1:n+1} $$ denote the ordered gaps $G_1,\dots,G_{n+1}$, so that the random sets $\{G_{n+1:1},\dots, G_{n+1:n+1}\}$ and $\{G_1,\dots,G_{n+1}\}$ are the same.

Fisher obtained the distribution of the largest gap, $G_{n+1:n+1}$. Fisher's result was generalized in this note, where the distribution of the $k$th smallest gap, $G_{n+1:k}$, was obtained, for each $k=1,\dots,n+1$.

Does anyone know other references to these or other related results in the literature?


I have just found out that this result is due to Irwin.

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  • $\begingroup$ jstor.org/stable/pdf/2983737.pdf too? $\endgroup$ – Brendan McKay Nov 26 '19 at 9:09
  • $\begingroup$ @BrendanMcKay : Thank you for your comment. Yes, that paper by Barton and David is where I found the reference to Irwin. $\endgroup$ – Iosif Pinelis Nov 26 '19 at 14:42
  • $\begingroup$ Note also that exercise 667 in Whitworth's DDC Exercises precedes Fisher by roughly 30 years. $\endgroup$ – esg Nov 27 '19 at 18:07

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