TY - JOUR

T1 - Spacings around an order statistic

AU - Nagaraja, H. N.

AU - Bharath, Karthik

AU - Zhang, Fangyuan

N1 - Publisher Copyright:
© 2014, The Institute of Statistical Mathematics, Tokyo.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015/6/1

Y1 - 2015/6/1

N2 - 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.

AB - 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.

KW - Central order statistics

KW - Extremes

KW - Intermediate order statistics

KW - Poisson process

KW - Spacings

KW - Uniform distribution

UR - http://www.scopus.com/inward/record.url?scp=84928708318&partnerID=8YFLogxK

U2 - 10.1007/s10463-014-0466-9

DO - 10.1007/s10463-014-0466-9

M3 - Article

AN - SCOPUS:84928708318

VL - 67

SP - 515

EP - 540

JO - Annals of the Institute of Statistical Mathematics

JF - Annals of the Institute of Statistical Mathematics

SN - 0020-3157

IS - 3

ER -