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arXiv · 1702.05910

Spacings Around An Order Statistic

H. N. Nagaraja·Karthik Bharath·Fangyuan Zhang
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Abstract

We determine the joint limiting distribution of adjacent spacings around a central, intermediate, or an extreme order statistic $X_{k:n}$ of a random sample of size $n$ from a continuous distribution $F$. For central and intermediate cases, normalized spacings in the left and right neighborhoods are asymptotically i.i.d. exponential random variables. The associated independent Poisson arrival processes are independent of $X_{k:n}$. For an extreme $X_{k:n}$, the asymptotic independence property of spacings fails for $F$ in the domain of attraction of Fr\'{e}chet and Weibull ($\alpha \neq 1$) distributions. This work also provides additional insight into the limiting distribution for the number of observations around $X_{k:n}$ for all three cases.

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H. N. Nagaraja, Karthik Bharath, Fangyuan Zhang. 2017-02-20. Spacings Around An Order Statistic. https://arxiv.org/abs/1702.05910

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