Hostname: page-component-5db58dd55d-8mwbx Total loading time: 0 Render date: 2026-05-31T12:45:06.498Z Has data issue: false hasContentIssue false
  • English
  • Français

STOCHASTIC MONOTONICITY OF CONDITIONAL ORDER STATISTICS IN MULTIPLE-OUTLIER SCALE POPULATION

Published online by Cambridge University Press:  16 November 2016

Ebrahim Amini-Seresht  and
Yiying Zhang
Ebrahim Amini-Seresht
Affiliation:
Department of Statistics, Bu-Ali Sina University, Hamedan, Iran E-mail: e.amini64@yahoo.com
Yiying Zhang
Affiliation:
Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong E-mail: zhyy@hku.hk
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper discusses the stochastic monotonicity property of the conditional order statistics from independent multiple-outlier scale variables in terms of the likelihood ratio order. Let X 1, …, X n be a set of non-negative independent random variables with X i , i=1, …, p, having common distribution function F(λ1x), and X j , j=p+1, …, n, having common distribution function F(λ2x), where F(·) denotes the baseline distribution. Let X i:n (p, q) be the ith smallest order statistics from this sample. Denote by $X_{i,n}^{s}(p,q)\doteq [X_{i:n}(p,q)|X_{i-1:n}(p,q)=s]$ . Under the assumptions that xf′(x)/f(x) is decreasing in x∈ℛ+, λ1≤λ2 and s 1≤s 2, it is shown that $X_{i:n}^{s_{1}}(p+k,q-k)$ is larger than $X_{i:n}^{s_{2}}(p,q)$ according to the likelihood ratio order for any 2≤i≤n and k=1, 2, …, q. Some parametric families of distributions are also provided to illustrate the theoretical results.

Keywords

conditional order statisticslikelihood ratio orderpermanentrr 2 dependent

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 31 , Issue 3 , July 2017 , pp. 366 - 380
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1. Avérous, J., Genest, C., & Kochar, S.C. (2005). On the dependence structure of order statistics. Journal of Multivariate Analysis 94: 159–171.Google Scholar
2. Balakrishnan, N. (2007). Permanents, order statistics, outliers, and robustness. Revista Matematica Complutense 20: 7–107.Google Scholar
3. Balakrishnan, N. & Rao, C.R. (1998a). Handbook of statistics. Vol. 16: order statistics: theory and methods. Amsterdam: Elsevier.Google Scholar
4. Balakrishnan, N. & Rao, C.R. (1998b). Handbook of statistics. Vol. 17: Order statistics: applications. Amsterdam: Elsevier.Google Scholar
5. Balakrishnan, N. & Torrado, N. (2016). Comparisons between largest order statistics from multiple-outlier models. Statistics 50: 176–189.CrossRefGoogle Scholar
6. Barlow, R.E. & Proschan, F. (1981). Statistical theory of reliability and life testing: Probability models. To Begin With, Silver Spring, MD.Google Scholar
7. Bickel, P.J. (1967). Some contributions to the theory of order statistics. In Fifth Berkeley Symposium on Mathematics and Statistics. 1: 575–591.Google Scholar
8. Burr, I.W. (1942). Cumulative frequency functions. Annals of Mathematical Statistics 13: 215–232.Google Scholar
9. Colangelo, A., Scarsini, M., & Shaked, M. (2006). Some positive dependence stochastic orders. Journal of Multivariate Analysis 97: 46–78.CrossRefGoogle Scholar
10. David, H.A. and Nagaraja, H.N. (2003). Order Statistics, 3rd ed. Hoboken, New Jersey: John Wiley & Sons.CrossRefGoogle Scholar
11. Dolati, A., Genest, C., & Kochar, S.C. (2008). On the dependence between the extreme order statistics in the proportional hazards model. Journal of Multivariate Analysis 99: 777–786.CrossRefGoogle Scholar
12. Esary, J.D. & Proschan, F. (1972). Relationships among some notions of bivariate dependence. Annals of Mathematical Statistics 43: 651–655.Google Scholar
13. Karlin, S. (1968). Total positivity. Stanford, CA: Stanford University Press.Google Scholar
14. Khaledi, B.-E., Farsinezhad, S., & Kochar, S.C. (2011). Stochastic comparisons of order statistics in the scale model. Journal of Statistical Planning and Inference 141: 276–286.Google Scholar
15. Kochar, S.C. & Torrado, N. (2015). On stochastic comparisons of largest order statistics in the scale model. Communications in Statistics—Theory and Methods 44: 4132–4143.CrossRefGoogle Scholar
16. Lai, C.D. & Xie, M. (2006). Stochastic aging and dependence for reliability. New York: Springer-Verlag.Google Scholar
17. Lehmann, E.L. (1966). Some concepts of dependence. Annals of Mathematical Statistics 37: 1137–1153.Google Scholar
18. Marshall, A.W., Olkin, I., & Arnold, B.C. (2011). Inequalities: theory of majorization and its applications, 2nd ed. New York: Springer-Verlag.CrossRefGoogle Scholar
19. Müller, A. & Stoyan, D. (2002). Comparison methods for stochastic models and risks. New York: John Wiley & Sons.Google Scholar
20. Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. New York: Springer-Verlag.Google Scholar
21. Tukey, J.W. (1958). A problem of Berkson, and minimum variance orderly estimators. Annals of Mathematical Statistics 29: 588–592.CrossRefGoogle Scholar
22. Van lint, J.H. (1981). Notes on Egoritsjev's proof of the van der Waerden conjecture. Linear Algebra and its Applications 39: 1–8.Google Scholar