IEEE TRANSACTIONS ON RELIABILITY, VOL. 63, NO. 2, JUNE 2014 511 Planning Progressive Type-I Interval Censoring Life Tests With Competing Risks Shuo-Jye Wu, Associate Member, IEEE, and Syuan-Rong Huang Abstract—In this article, we investigate some reliability and quality problems when the competing risks data are progressive type-I interval censored with binomial removals. The failure times of the individual causes are assumed to be statistically indepen- dent and exponentially distributed with different parameters. We obtain the estimates of the unknown parameters through a max- imum likelihood method, and also derive the Fisher’s information matrix. The optimal lengths of the inspection intervals are deter- mined under two different criteria. The reliability sampling plans are established under given producer’s and customer’s risks. A Monte Carlo simulation is conducted to evaluate the performance of the estimators, and also some numerical results are presented. Index Terms—D-optimality, exponential distribution, maximum likelihood method, multiple failure modes, reliability sampling plan, variance-optimality. ABBREVIATION MLE maximum likelihood estimate(or) TET total experimental time NOTATION failure time of the -th unit, lifetime of the -th unit under risk , probability density function of joint probability density function of risk and failure time of the -th unit joint cumulative distribution function of risk and failure time of the -th unit cumulative distribution function of parameter of the exponential distribution under risk , total of hazard rates for all risks mean of the exponential distribution number of test units number of risks number of inspections the -th inspection time number of failures at the -th stage due to risk Manuscript received November 20, 2012; revised September 24, 2013; ac- cepted December 04, 2013. Date of publication April 03, 2014; date of current version May 29, 2014. The work was supported in part by the National Science Council of ROC under Grant NSC 100-2118-M-032-002-MY2 and the National Center for Theoretical Sciences (NCTS) of Taiwan. Associate Editor: S. J. Bae. S.-J. Wu is with the Department of Statistics, Tamkang University, Tamsui, New Taipei City 25137, Taiwan (e-mail: shuo@stat.tku.edu.tw). S.-R. Huang is with the Department of Management Sciences, Tamkang Uni- versity, Tamsui, New Taipei City 25137, Taiwan (e-mail: oklaok@gmail.com). Digital Object Identifier 10.1109/TR.2014.2313708 total number of failures at the -th stage total number of failures due to risk total number of failures observed in a life test number of removals at the -th stage number of non-removed surviving units at the beginning of the -th stage probability of a unit to be removed at the -th stage probability of failure in due to risk probability of failure in likelihood function Fisher’s information producer’s risk consumer’s risk lower specification limit, i.e., the critical point for accepting a lot percentile of a standard statistical normal distribution I. INTRODUCTION A product usually consists of many different components with various risk factors so that a product may fail due to one of several causes, called failure modes or competing risks. In certain applications, product lifetime is defined to be the ear- liest occurrence among all these risks. Nelson [25, Chapter 7] enumerated engineering situations when a product fails because of two or more risks. For instance, fatigue specimens of a certain sintered super-alloy can fail from a surface defect or an interior one. In ball bearing assemblies, a ball or the race can fail. A cylindrical fatigue specimen can fail in the cylindrical portion, in the fillet (or radius), or in the grip. A semiconductor device can fail at a junction or at a lead. Some other situations in en- gineering when competing risks occurred can be found in Kim and Bai [18], and Craiu and Lee [11]. In reliability analysis, ignoring the information on causes of failure may result in incorrect inference when improving the reliability of the products. Thus, the data for these competing risks models consist of the failure time, and an indicator vari- able denoting the specific cause of failure of the product. Cox [10] proposed the latent failure model to analyze the data with multiple failure modes. The cause of failure may be assumed to be statistically independent, or statistically dependent. In most situations, it is usually assumed that these competing risks are statistically independent. Although the assumption of statistical dependence may be more realistic, there are some 0018-9529 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.