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On estimating population characteristics from record-breaking observations. II. Nonparametric results. (English) Zbl 0656.62045
Consider an experiment in which only record-breaking values (e.g., values smaller than all previous ones) are observed. The data available may be represented as \(X_ 1,K_ 1,X_ 2,K_ 2,...\), where \(X_ 1,X_ 2,..\). are successive minima and \(K_ 1,K_ 2,..\). are the numbers of trials needed to obtain new records. Such data arise in life testing and stress testing and in industrial quality-control experiments. When only a single sequence of random records are available, efficient estimation of the underlying distribution F is possible only in a parametric framework [see part I, ibid. 33, 531-543 (1986; Zbl 0605.62027)].
In the present article we study the problem of estimating certain population quantiles nonparametrically from such data. Furthermore, under the assumption that the process of observing random records can be replicated, we derive and study the nonparametric maximum-likelihood estimator \(\hat F\) of F. We establish the strong uniform consistency of this estimator as the number of replications grows large, and identify its asymptotic distribution theory. The performance of \(\hat F\) is compared to that of two possible competing estimators.

MSC:
62G05 Nonparametric estimation
62N05 Reliability and life testing
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