an elegant result on exponential spacings

A question on X validated I spotted in the train back from Lyon got me desperately seeking a reference in Devroye’s Generation Bible despite the abyssal wireless and a group of screeching urchins a few seats away from me… The question is about why

$\sum_{i=1}^{n}(Y_i - Y_{(1)}) \sim \text{Gamma}(n-1, 1)$

when the Y’s are standard exponentials. Since this reminded me immediately of exponential spacings, thanks to our Devroye ~~fan-club~~ reading group in Warwick, I tried to download Devroye’s Chapter V and managed after a few aborts (and a significant increase in decibels from the family corner). The result by Sukhatme (1937) is in plain sight as Theorem 2.3 and is quite elegant as it relies on the fact that

$\sum_{i=1}^n y_i=\sum_{j=1}^n (n-j+1)(y_{(j)}-y_{(j-1)})=\sum_{j=2}^n (y_{(j)}-y_{(1)})$

hence sums up as a mere linear change of variables! (Pandurang Vasudeo Sukhatme (1911–1997) was an Indian statistician who worked on human nutrition and got the Guy Medal of the RSS in 1963.)

This entry was posted on April 19, 2017 at 12:17 am and is filed under Statistics with tags Book, change of variables, cross validated, exponential distribution, favourite books, Guy Medal, Jacobian, Luc Devroye, Lyon, P.V. Sukhatme, RSS, simulation, spacings, train. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

4 Responses to “an elegant result on exponential spacings”

Georges Henry Says:
April 19, 2017 at 9:04 pm

And by the way, I do not understand your identity, which does not seem quite correct.

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- xi'an Says:
  April 19, 2017 at 10:32 pm
  
  Sorry, do you mean, the Fubini like identity…? I do not see a difficulty with it mais il n’y a pire sourd que celui…
  
  Reply
- xi'an Says:
  April 20, 2017 at 5:57 am
  
  Got it!!! Merci
  
  Reply
Georges Henry Says:
April 19, 2017 at 3:43 pm

Without calculation, just recall that

$(Y_{(1)},\ldots,Y_{(n)})\sim (\frac{Y_n}{n},\ \frac{Y_n}{n}+ \frac{Y_{n-1}}{n-1},\cdots, \frac{Y_n}{n}+ \frac{Y_{n-1}}{n-1}+\cdots+Y_1)$ which makes trivial the fact that
$\sum_{k=1}^n(Y_k-Y_{(1)})\sim \sum_{k=1}^{n-1}Y_k.$

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