Likelihood-based analysis of doubly-truncated data under the location-scale and AFT model
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Likelihood-based analysis of doubly-truncated data under the location-scale and AFT model Achim Dörre1
· Chung-Yan Huang2 · Yi-Kuan Tseng2 · Takeshi Emura3
Received: 13 July 2018 / Accepted: 16 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Doubly-truncated data arise in many fields, including economics, engineering, medicine, and astronomy. This article develops likelihood-based inference methods for lifetime distributions under the log-location-scale model and the accelerated failure time model based on doubly-truncated data. These parametric models are practically useful, but the methodologies to fit these models to doubly-truncated data are missing. We develop algorithms for obtaining the maximum likelihood estimator under both models, and propose several types of interval estimation methods. Furthermore, we show that the confidence band for the cumulative distribution function has closedform expressions. We conduct simulations to examine the accuracy of the proposed methods. We illustrate our proposed methods by real data from a field reliability study, called the Equipment-S data. Keywords Reliability · Confidence band · Confidence interval · Accelerated life testing · Newton–Raphson algorithm · Weibull distribution
1 Introduction Doubly-truncated data arise in many fields, including economics, engineering, medicine, and astronomy. When studying the lifetime distribution of a technological unit, it may be too time-consuming or operationally not feasible to obtain simple failure data. In this case, it is more convenient to collect field-failure data (Lawless
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00180020-01027-6) contains supplementary material, which is available to authorized users.
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Takeshi Emura [email protected]
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Department of Economics, University of Rostock, Rostock, Germany
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Graduate Institute of Statistics, National Central University, Taoyuan, Taiwan
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Department of Information Management, Chang Gung University, Taoyuan, Taiwan
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2003; Ye and Tang 2016), where any unit is retrospectively collected if it fails within a pre-specified study period. This sampling mechanism yields two types of missing units: left-truncated units that fail before study initiation and right-truncated units that fail after the study end. This sampling design is common in astronomy (Efron and Petrosian 1999), economics (Dörre 2020; Frank and Dörre 2017), medicine (Scheike and Keiding 2006; Moreira and de Uña-Álvarez 2010), and other fields. Analyses of doubly-truncated data require appropriate techniques to account for the sampling design. Figure 1 illustrates the sampling mechanism. Let B be the birth date of a unit and T be its lifetime. Hence, F = B + T is the date of failure. Suppose that data collection begins at the date τs and ends at the date τe . We only collect units whose failures occur in the interval [τs , τe ]. Therefore, the sampling criterion i
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