Likelihood Ratio Ordering for Spacings Arising from Multiple-Outlier Models

Document Type : Original Article

Authors
Ebrahim Amini-Seresht 1
Salman Izadkhah 2
1 Department of Statistics‎, ‎Faculty of Science‎, ‎Bu-Ali Sina University‎, ‎Hamedan‎, ‎Iran.
2 Department of Statistics‎, ‎Campus of Bijar‎, ‎University of Kurdistan‎, ‎Bijar‎, ‎Iran‎.
10.22034/jirss.2026.2076023.1150
Abstract
‎‎In this paper, we investigate the stochastic properties of spacings among order statistics derived from a sample of independent, non-negative random variables that are divided into two groups with different distributions. Previous studies have shown that when these distribution functions are exponential distributions with specified hazard rates, the likelihood ratio ordering holds among the spacings under specific conditions. The present work extends these results by considering more general continuous distribution functions. We identify the necessary conditions on the parent distribution functions for preserving the likelihood ratio ordering among spacings in general settings. The comparison results enhance our understanding of stochastic ordering theory and provide valuable insights for applications in reliability‎, ‎survival analysis‎, ‎and related fields‎, ‎aiding in the development of more flexible and accurate statistical models‎.
Keywords
Stochastic order
Likelihood ratio order
Sample spacings
General multiple-outlier models&‌‌‌lrm
Subjects
Balakrishnan N, Rao CR. Handbook of statistics. v. 16: Order statistics: theory and methods. 1998;.
Balakrishnan N, Rao C. Handbook of statistics. v. 17: Order statistics: applications.1998;.
Barmalzan G, Haidari A, Payandeh Najafabadi AT. Stochastic comparisons of sample spacings in single-and multiple-outlier exponential models. Statistics. 2015;49(4):815–830.
Khaledi BE, Kochar S. Some new results on stochastic comparisons of parallel systems. Journal of Applied Probability. 2000;37(4):1123–1128.
Kochar SC, Kirmani S. Some results on normalized spacings from restricted families of distributions. Journal of Statistical Planning and inference. 1995;46(1):47–57.
Kochar SC, Korwar R. Stochastic orders for spacings of heterogeneous exponential random variables. Journal of Multivariate Analysis. 1996;57(1):69–83.
Misra N, van der Meulen EC. On stochastic properties of m-spacings. Journal of Statistical Planning and Inference. 2003;115(2):683–697.
Shaked M, Shanthikumar JG. Stochastic orders. Springer; 2007.
Torrado N, Lillo RE. Likelihood ratio comparisons among spacings related to both one or two samples. Statistics. 2015;49(4):831–841.
Wen S, Lu Q,HuT. Likelihood ratio orderings of spacings of heterogeneous exponential random variables. Journal of Multivariate Analysis. 2007;98(4):743–756.
Xu M, Li X, Zhao P, Li Z. Likelihood ratio order of m-spacings in multiple-outlier models. Communications in Statistics Theory and Methods. 2007;36(8):1507–1525.
Zhao P, Qiao J, Balakrishnan N. Likelihood ratio order of the second spacing in multipleoutlier exponential models. Statistics. 2016;50(1):206–218.
Journal of the Iranian Statistical Society
Volume 24, Issue 2
December 2025
Pages 50-68

  • Receive Date 29 October 2025
  • Revise Date 14 December 2025
  • Accept Date 07 January 2026