Abstract
The k-out-of-n model is commonly used in reliability theory. In this model the failure of any component of the system does not influence the components still at work. Sequential k-out-of-n systems have been introduced as an extension of k-out-of-n systems where the failure of some component of the system may influence the remaining ones. We consider nonparametric estimation of the cumulative hazard function, the reliability function and the quantile function of sequential k-out-of-n systems. Furthermore, nonparametric hypothesis testing for sequential k-out-of-n-systems is examined. We make use of counting processes to show strong consistency and weak convergence of the estimators and to derive the asymptotic distribution of the test statistics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andersen P.K., Borgan O., Gill R.D., Keidling N. (1993). Statistical Models Based on Counting Processes. Springer, New York
Bordes L. (2004). Nonparametric estimation under progressive censoring. Journal of Statistical Planning and Inference 119, 171–189
Cramer E., Kamps U. (1996). Sequential order statistics and k-out-of-n systems with sequentially adjusted failure rates. Annals of the Institute of Statistical Mathematics 48(3): 535–549
Cramer E., Kamps U. (2001a). Estimation with sequential order statistics from exponential distributions. Annals of the Institute of Statistical Mathematics 53(2): 307–324
Cramer E., Kamps U. (2001b). Sequential k-out-of-n systems. In: Balakrishnan N., Rao C.R. (eds), Handbook of Statistics, Vol 20. Amsterdam, Elsevier, pp. 301–372
Cramer E., Kamps U. (2003). Marginal distributions of sequential and generalized order statistics. Metrika 58(3): 293–310
Dorado C., Hollander M., Sethuraman J. (1997). Nonparametric estimation for a general repair model. The Annals of Statistics 25(3): 1140–1160
Doss H., Gill R.D. (1992). An elementary approach to weak convergence for quantile processes, with applications to censored survival data. Journal of the American Statistical Association 100, 869–877
Gill R.D., Johansen S. (1990). A survey of product-integration with a view toward application in survival analysis. The Annals of Statistics 18(4): 1501–1555
Guilbaud O. (2004). Exact non-parametric confidence, prediction and tolerance intervals with progressive type-II censoring. Scandinavian Journal of Statistics 31(2): 265–281
Harrington D.P., Fleming T.R. (1982). A class of rank test procedures for censored survival data. Biometrika 69, 133–143
Hollander M., Peña E.A., (1995). Dynamic reliability models with conditional proportional hazards. Lifetime Data Analysis 1 (4): 377–401
Kamps U. (1995). A Concept of Generalized Order Statistics. Stuttgart: Teubner
Karr A.F. (1991). Point Processes and their Statistical Inference. New York: Marcel Dekker
Kvam P.H., Peña E.A. (2005). Estimating load-sharing properties in a dynamic reliability system. Journal of the American Statistical Association 100, 262–272
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Beutner, E. Nonparametric inference for sequential k-out-of-n systems. Ann Inst Stat Math 60, 605–626 (2008). https://doi.org/10.1007/s10463-007-0115-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-007-0115-7