Spacings around an order statistic

H. N. Nagaraja, Karthik Bharath, Fangyuan Zhang

Research output: Contribution to journalArticle

5 Scopus citations

Abstract

We determine the joint limiting distribution of adjacent spacings around a central, intermediate, or an extreme order statistic $$X_{k:n}$$Xk:n of a random sample of size $$n$$n from a continuous distribution $$F$$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 $$X_{k:n}$$Xk:n. For an extreme $$X_{k:n}$$Xk:n, the asymptotic independence property of spacings fails for $$F$$F in the domain of attraction of Fréchet and Weibull ($$\alpha \ne 1$$α≠1) distributions. This work also provides additional insight into the limiting distribution for the number of observations around $$X_{k:n}$$Xk:n for all three cases.

Original language English 515-540 26 Annals of the Institute of Statistical Mathematics 67 3 https://doi.org/10.1007/s10463-014-0466-9 Published - Jun 1 2015

Keywords

• Central order statistics
• Extremes
• Intermediate order statistics
• Poisson process
• Spacings
• Uniform distribution