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Is there a specific formula one can use to compute the differences in order statistics, say $x_i - x_{i-1}$ when the underlying distribution of $x$ is standard normal?

Also what is the asymptotic value of this difference?

kjetil b halvorsen
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Palaash Bhargava
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    You can calculate the difference by subtracting one from the other, as in your formula, without regard for the underlying distribution. Order statistics are a feature of the sample, not the distribution. I am not sure what you are really asking... – jbowman Apr 06 '20 at 18:57
  • Are you perhaps inquiring about finding the distributions of successive differences of order statistics from a Normal sample? – whuber Apr 07 '20 at 21:55

1 Answers1

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Let $W_{i,j:n} = X_{j:n}-X_{i:n},\; 1\leq i<j\leq n$ be the difference between the $i$th and $j$th order statistics (aka the spacings). The pdf of $W_{i,j:n}$ is then given by:

$$ f_{W_{i,j:n}}(w) = \frac{n!}{(i-1)!(j-i-1)!(n-j)!}\times \int_{-\infty}^{\infty}\left\{F(x_{i})\right\}^{i-1}\left\{F(x_{i} + w) - F(x_{i})\right\}^{j-i-1}\times \left\{1-F(x_{i}+w)\right\}^{n-j}f(x_{i})f(x_{i} + w)\;\mathrm{d}x_{i}, \quad 0<w<\infty $$

This formula is given in $[1]$. As far as I know, there is no simple formula for the standard normal.

References

$[1]$ Arnold BC, Balakrishnan N, Nagaraja HN (2008): A First Course in Order Statistics. Siam, Philadelphia.

COOLSerdash
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  • Could you clarify whether $F$ and $f$ are the CDF and PDF of the underlying distribution of $x$ ? Also, is there a specific reason you labeled the integration variable $x_i$ with $i$ ? – Toool May 17 '20 at 11:31
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    @Tool $F$ denots the cdf and $f$ the pdf, yes. Regarding the notation: I've copied the expression without change from the cited reference. I don't know why the authors chose this kind of notation, sorry. – COOLSerdash May 17 '20 at 14:21