Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-24T11:27:57.999Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Skewness of Order Statistics in Multiple-Outlier Models

Part of: Survival analysis and censored data Sampling theory, sample surveys Nonparametric inference Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Subhash Kochar  and
Maochao Xu
Subhash Kochar*
Affiliation:
Portland State University
Maochao Xu*
Affiliation:
Illinois State University
*
∗Postal address: Department of Mathematics and Statistics, Portland State University, Portland, Oregon 97201, USA.
∗∗Postal address: Department of Mathematics, Illinois State University, Normal, Illinois 61790-4520, USA. Email address: mxu2@ilstu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Kochar and Xu (2009) proved that a parallel system with heterogeneous exponential component lifetimes is more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. In this paper we extend this study to the general k-out-of-n systems for the case when there are only two types of component in the system. An open problem proposed in Pǎltǎnea (2008) is partially solved.

Keywords

Dispersive orderexponential distributiondecreasing failure ratek-out-of-n systemproportional hazard rateright spread orderskewnessstar order

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings 62N05: Reliability and life testing 62G30: Order statistics; empirical distribution functions 62D05: Sampling theory, sample surveys
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 1 , March 2011 , pp. 271 - 284
Copyright
Copyright © Applied Probability Trust 2011 

References

Ahmed, A. N., Alzaid, A., Bartoszewicz, J. and Kochar, S. C. (1986). Dispersive and superadditive ordering. Adv. Appl. Prob. 18, 1019–1022.CrossRefGoogle Scholar
Arnold, B. C. and Groeneveld, R. A. (1995). Measuring skewness with respect to the mode. Amer. Statistician 49, 34–38.Google Scholar
Bagai, I. and Kochar, S. (1986). On tail-ordering and comparison of failures rates. Commun. Statist. Theory Meth. 15, 1377–1388.CrossRefGoogle Scholar
Balakrishnan, N. (2007). Permanents, order statistics, outliers, and robustness. Revista Matemática Complutense 20, 7–107.CrossRefGoogle Scholar
Balakrishnan, N. and Rao, C. R. (eds) (1998a). Handbook of Statistics 16. Elsevier, New York.Google Scholar
Balakrishnan, N. and Rao, C. R. (eds) (1998b). Handbook of Statistics 17. Elsevier, New York.Google Scholar
Bapat, R. B. and Beg, M. I. (1989). Order statistics for nonidentically distributed variables and permanents. Sankhyā A 51, 79–93.Google Scholar
Bapat, R. B. and Kochar, S. C. (1994). On likelihood-ratio ordering of order statistics. Linear Algebra Appl. 199, 281–291.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Bon, J.-L. and Paltanea, E. (2006). Comparison of order statistics in a random sequence to the same statistics with i.i.d. variables. ESAIM Prob. Statist. 10, 1–10.CrossRefGoogle Scholar
David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd edn. Wiley Interscience, Hoboken, NJ.CrossRefGoogle Scholar
Fernández-Ponce, J. M., Kochar, S. C. and Muñoz-Perez, J. (1998). Partial orderings of distributions based on right-spread functions. J. Appl. Prob. 35, 221–228.CrossRefGoogle Scholar
Hu, T., Lu, Q. and Wen, S. (2008). Some new results on likelihood ratio orderings for spacings of heterogeneous exponential random variables. Commun. Statist. Theory Meth. 37, 2506–2515.CrossRefGoogle Scholar
Hu, T., Wang, F. and Zhu, Z. (2006). Stochastic comparisons and dependence of spacings from two samples of exponential random variables. Commun. Statist. Theory Meth. 35, 979–988.CrossRefGoogle Scholar
Hu, T., Zhu, Z. and Wei, Y. (2001). Likelihood ratio and mean residual life orders for order statistics of heterogeneous random variables. Prob. Eng. Inf. Sci. 15, 259–272.CrossRefGoogle Scholar
Karlin, S. (1968). Total Positivity, Vol. I. Stanford University Press.Google Scholar
Khaledi, B. E. and Kochar, S. C. (2001). Stochastic properties of spacings in a single-outlier exponential model. Prob. Eng. Inf. Sci. 15, 401–408.CrossRefGoogle Scholar
Kochar, S. C. (1989). On extensions of DMRL and related partial orderings of life distributions. Commun. Statist. Stoch. Models 5, 235–245.CrossRefGoogle Scholar
Kochar, S. and Xu, M. (2007). Stochastic comparisons of parallel systems when components have proportional hazard rates. Prob. Eng. Inf. Sci. 21, 597–609.CrossRefGoogle Scholar
Kochar, S. and Xu, M. (2009). Comparisons of parallel systems according to the convex transform order. J. Appl. Prob. 46, 342–352.CrossRefGoogle Scholar
Mao, T. and Hu, T. (2010). Equivalent characterizations on orderings of order statistics and sample ranges. Prob. Eng. Inf. Sci. 24, 245–262.CrossRefGoogle Scholar
Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.Google Scholar
Minc, H. (1978). Permanents (Encyclopaedia Math. Appl. 6). Addison-Wesley, Reading, MA.Google Scholar
Oja, H. (1981). On location, scale, skewness and kurtosis of univariate distributions. Scand. J. Statist. 8, 154–168.Google Scholar
Paltanea, E. (2008). On the comparison in hazard rate ordering of fail-safe systems. J. Statist. Planning Inf. 138, 1993–1993.CrossRefGoogle Scholar
Saunders, I. W. and Moran, P. A. P. (1978). On the quantiles of the gamma and F distributions. J. Appl. Prob. 15, 426–432.CrossRefGoogle Scholar
Shaked, M. and Shantikumar, J. G. (1998). Two variability orders. Prob. Eng. Inf. Sci. 12, 1–23.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders and Their Applications. Springer, New York.Google Scholar
Van Zwet, W. R. (1970). Convex Transformations of Random Variables (Math. Centre Tracts 7), 2nd edn. Mathematical Centre, Amsterdam.Google Scholar
Wen, S., Lu, Q. and Hu, T. (2007). Likelihood ratio orderings of spacings of heterogeneous exponential random variables. J. Multivariate Anal. 98, 743–756.CrossRefGoogle Scholar
Xu, M., Li, X., Zhao, P. and Li, Z. (2007). Likelihood ratio order of m-spacings in multiple-outlier models. Commun. Statist. Theory Meth. 36, 1507–1525.CrossRefGoogle Scholar