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Estimating component characteristics from system failure-time data. (English) Zbl 1186.62122
Summary: Suppose that failure times are available from a random sample of \(N\) systems of a given, fixed design with components which have i.i.d. life times distributed according to a common distribution \(F\). The inverse problem of estimating \(F\) from the data on observed system life times is considered. Using the known relationship between the system and component life time distributions via signature and domination theory, the nonparametric maximum likelihood estimator \(\widehat{\overline F}_N(t)\) of the component survival function \({\overline F}(t)\) is identified and shown to be numerically accessible in any application of interest. The asymptotic distribution of \(\widehat{\overline F}_N(t)\) is also identified, facilitating the construction of approximate confidence intervals for \({\overline F}(t)\) for \(N\) sufficiently large. Simulation results for samples of size \(N=50\) and \(N=100\) for a collection of five parametric life time models demonstrate the utility of the recommended estimator. Possible extensions beyond the i.i.d. framework are discussed in the concluding section.

62N05 Reliability and life testing
62G05 Nonparametric estimation
62N02 Estimation in survival analysis and censored data
62E20 Asymptotic distribution theory in statistics
62G15 Nonparametric tolerance and confidence regions
65C60 Computational problems in statistics (MSC2010)
Full Text: DOI
[1] Barlow, Statistical Theory of Reliability (1981)
[2] Boland, Mathematical and Statistical Methods in Reliability pp 89– (2003) · doi:10.1142/9789812795250_0007
[3] Boyles, Estimating a survival curve based on nomination sampling, J Am Stat Assoc 81 pp 1034– (1986) · Zbl 0657.62039
[4] Boyles, On estimating component reliability for systems with random redundancy levels, IEEE Trans Reliability R-36 pp 403– (1987) · Zbl 0624.62091
[5] Cohen, Statistics, testing and defense acquisition (1998)
[6] Costa Bueno, A note on the component lifetime estimation of a multistate monotone system through the System lifetime, Adv Appl Probability 20 pp 686– (1988) · Zbl 0652.60093
[7] Guess, Estimating system and component reliabilities under partial information on cause of failure, J Stat Plann Inference 29 pp 75– (1991) · Zbl 0850.62722
[8] Kvam, On the inadmissibility of empirical averages as estimators in ranked set sampling, J Stat Plann Inference 36 pp 39– (1993) · Zbl 0772.62005
[9] Kvam pp 215– (1993)
[10] Kvam, Nonparametric maximum likelihood estimation based on ranked set samples, J Am Stat Assoc 89 pp 526– (1994) · Zbl 0803.62030
[11] Meilijson, Estimation of the lifetime distribution of the parts from the autopsy statistics of the machine, J Appl Prob 18 pp 829– (1981) · Zbl 0471.62100
[12] Miyakawa, Analysis of incomplete data in competing risks model, IEEE Trans Reliability TR-33 pp 293– (1984)
[13] Moeschberger, Life tests under competing cause of failure and the theory of competing risks, Biometrics 27 pp 909– (1971)
[14] Rao, Linear statistical inference and its applications (1973) · Zbl 0256.62002
[15] Samaniego, On the closure of the IFR class under the formation of coherent systems, IEEE Trans Reliability TR-34 pp 69– (1985) · Zbl 0585.62169
[16] Samaniego, System signatures and their applications in engineering reliability (2007) · Zbl 1154.62075
[17] Satyanarayana, A new topological formula and rapid algorithm for reliability analysis of complex networks, IEEE Trans Reliability TR-30 pp 82– (1978) · Zbl 0409.90039
[18] Stokes, Characterization of a ranked set sample with applications to estimating distribution functions, J Am Stat Assoc 83 pp 374– (1988) · Zbl 0644.62050
[19] Usher, Maximum likelihood analysis of component reliability using masked system life-test data, IEEE Trans Reliability TR-37 pp 550– (1988) · Zbl 0658.62119
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