Nonparametric Confidence Intervals and Tolerance Limits Based on Minima and Maxima

Abstract

Let X _i,j (1 ≤ i ≤ k, 1 ≤ j ≤ n _i ) be independent random variables and for a fixed i, X _i,j 's, (1 ≤ j ≤ n _i ) be identically distributed random variables with survival function , where α _i is a known positive constant. Also, suppose M _i and M′ _i , respectively, denote the maximum and minimum of the ith sample. This article investigates the nonparametric confidence intervals for an arbitrary quantile of the distribution F and tolerance limits based on these statistics. Various cases have been studied and in each case, the nonparametric confidence intervals are obtained and exact expressions for the confidence coefficients of these confidence intervals are derived. A data set representing the time of successive failures of the air conditioning system on Boeing 720 jet aircraft is used to illustrate the results. Finally, the accuracy of the proposed procedure has been investigated, when α _i 's are unknown via a simulation study.

Keywords:

• Coverage probability
• Multisampling plan
• Order statistics
• Proportional hazard model
• Record values

Mathematics Subject Classification:

• Primary 62G30
• Secondary 62G15, 62E15

Acknowledgment

The authors would like to thank the anonymous referee for his/her useful comments and careful reading the manuscript that definitely improved the article.

Partial support from the Ordered and Spatial Data Center of Excellence of Ferdowsi University of Mashhad is acknowledged.

Notes

^a and ^b stand for the mean and MSE of (i, j; p).

* stands for the confidence intervals with the shortest width and confidence coefficient more than 95%.

Additional information

Notes on contributors

J. Ahmadi

The second author is a member of the Ordered and Spatial Data Center of Excellence of Ferdowsi University of Mashhad.