TY - JOUR
T1 - Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables
AU - Zhao, Peng
AU - Li, Xiaohu
AU - Balakrishnan, N.
PY - 2009/5
Y1 - 2009/5
N2 - Let X1, ..., Xn be independent exponential random variables with respective hazard rates λ1, ..., λn, and let Y1, ..., Yn be independent exponential random variables with common hazard rate λ. This paper proves that X2 : n, the second order statistic of X1, ..., Xn, is larger than Y2 : n, the second order statistic of Y1, ..., Yn, in terms of the likelihood ratio order if and only if λ ≥ frac(1, 2 n - 1) (2 Λ1 + frac(Λ3 - Λ1 Λ2, Λ12 - Λ2)) with Λk = ∑i = 1n λik, k = 1, 2, 3. Also, it is shown that X2 : n is smaller than Y2 : n in terms of the likelihood ratio order if and only if λ ≤ frac(underover(∑, i = 1, n) λi - under(max, 1 ≤ i ≤ n) λi, n - 1) . These results form nice extensions of those on the hazard rate order in Pa ̌lta ̌nea [E. Pa ̌lta ̌nea, On the comparison in hazard rate ordering of fail-safe systems, Journal of Statistical Planning and Inference 138 (2008) 1993-1997].
AB - Let X1, ..., Xn be independent exponential random variables with respective hazard rates λ1, ..., λn, and let Y1, ..., Yn be independent exponential random variables with common hazard rate λ. This paper proves that X2 : n, the second order statistic of X1, ..., Xn, is larger than Y2 : n, the second order statistic of Y1, ..., Yn, in terms of the likelihood ratio order if and only if λ ≥ frac(1, 2 n - 1) (2 Λ1 + frac(Λ3 - Λ1 Λ2, Λ12 - Λ2)) with Λk = ∑i = 1n λik, k = 1, 2, 3. Also, it is shown that X2 : n is smaller than Y2 : n in terms of the likelihood ratio order if and only if λ ≤ frac(underover(∑, i = 1, n) λi - under(max, 1 ≤ i ≤ n) λi, n - 1) . These results form nice extensions of those on the hazard rate order in Pa ̌lta ̌nea [E. Pa ̌lta ̌nea, On the comparison in hazard rate ordering of fail-safe systems, Journal of Statistical Planning and Inference 138 (2008) 1993-1997].
KW - 60E15
KW - 60K10
KW - Hazard rate order
KW - Majorization order
KW - Weakly majorization order
KW - p-larger order
KW - primary
KW - secondary
UR - http://www.scopus.com/inward/record.url?scp=60549111809&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=60549111809&partnerID=8YFLogxK
U2 - 10.1016/j.jmva.2008.09.010
DO - 10.1016/j.jmva.2008.09.010
M3 - Article
AN - SCOPUS:60549111809
SN - 0047-259X
VL - 100
SP - 952
EP - 962
JO - Journal of Multivariate Analysis
JF - Journal of Multivariate Analysis
IS - 5
ER -