On estimating population characteristics from record-breaking observations II. Nonparametric results

TLDR

Record-breaking observations, represented as successive minima and the number of trials needed to achieve them, arise in life testing, stress testing, and quality control, and efficient estimation of the underlying distribution is only possible in a parametric framework when only a single sequence of records is available. The study aims to estimate population quantiles nonparametrically from record-breaking data. The authors derive a nonparametric maximum‑likelihood estimator of the underlying distribution by replicating the record‑observation process and compare its performance to two competing estimators. The estimator is shown to be strongly uniformly consistent and its asymptotic distribution is characterized.

Abstract

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 X1,K1,X2,K2, …, where X1,X2, … are successive minima and K1,K2, … 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 Samaniego and Whitaker [9]). 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 F̂ 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 F̂ is compared to that of two possible competing estimators.