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.
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
SN - 0020-3157
VL - 67
SP - 515
EP - 540
JO - Annals of the Institute of Statistical Mathematics
JF - Annals of the Institute of Statistical Mathematics
IS - 3
ER -