Limit behaviors for ratios of order statistics from exponential distributions. Issue 24 (11th November 2021)

• Record Type: Journal Article
• Authors: Xu, Shoufang; Mei, Changlin
• Is Part Of: Communications in statistics. Volume 50:Issue 24(2021)
• Page Start: 5918
• Page End: 5928
• Publication Date: 2021-11-11
• Subjects: Exponential distribution -- order statistics -- law of large numbers -- complete convergence -- large deviation principle; 60F15 -- 62G30; Mathematical statistics -- Periodicals; Mathematics; Statistics; 519.2
• DOI: 10.1080/03610926.2020.1737877
• Languages: English
• ISSNs: 0361-0926

Abstract

Abstract: Let { X n i, 1 ≤ i ≤ m n, n ≥ 1 } be an array of independent exponentially distributed random variables with mean λn, and { X n ( 1 ) ≤ X n ( 2 ) ≤ ⋯ ≤ X n ( m n ) } be the order statistics of the random variables { X n i, 1 ≤ i ≤ m n } . Limit properties of the ratios { R nij = X n ( j ) / X n ( i ), 1 ≤ i < j ≤ m n } are investigated for fixed sample size mn = m . Specifically, the weak law of large numbers is derived for R n 1 j under some more general conditions, followed by the establishment of its complete convergence. Furthermore, the large deviation principles for both R n 1 j and R n 2 j are formulated.