On Exact Inferential Results for a Simple Step-Stress Model Under a Time Constraint
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On Exact Inferential Results for a Simple Step-Stress Model Under a Time Constraint Julian G´orny and Erhard Cramer RWTH Aachen University, Aachen, Germany Abstract In simple step-stress models based on exponential distributions, the distributions of the MLEs are commonly obtained using the moment generating function. In this paper, we propose an alternative method, the so-called expected value approach, introduced in G´ orny (2017) to derive the exact distribution of the MLEs. Moreover, we discuss the benefits of this technique. Further, assuming uniformly distributed lifetimes, we show that the MLEs are also explicitly available and that their distributions are discrete for both the cumulative exposure and the tampered failure rate model. Additionally, we illustrate that confidence regions as well as confidence intervals can be established utilizing a connection to the multinomial distribution. The results are illustrated by an illustrative example as well as simulation results. AMS (2000) subject classification. Primary 62E15, 62F10; Secondary 62F25, 62N05. Keywords and phrases. Simple step-stress model, Type-I censoring, Cumulative exposure model, Tampered failure rate model, Exponential distribution, Expected value approach, Uniform distribution, B-spline
1 Introduction As a special form of accelerated life testing, we consider a simple stepstress model with two stress levels s1 and s2 where the stress level is changed at a previously fixed time τ1 . We further assume the experiment to be terminated at the prefixed time τ2 > τ1 (see Fig. 1). This simple step-stress model under a time constraint has been introduced by Balakrishnan et al. (2009). For exponential lifetimes, they determined the MLEs and derived expressions for the respective distributions. For further reading on TypeI censored simple step-stress models under an exponential assumption, we refer to Balakrishnan and Iliopoulos (2010) and Kateri and Kamps (2015). Two-parameter exponential distributions are discussed in Mitra et al. (2013). A multi-sample approach for Type-I censored simple step-stress models has
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J. G´ orny, E. Cramer
Figure 1: Simple step-stress model with stress change point τ1 and time constraint τ2 been discussed in Kateri et al. (2010). An order restricted inference approach can be found in Balakrishnan et al. (2009) & Samanta et al. (2017) and Samanta et al. (2017). For reviews on simple step-stress models, we refer to Balakrishnan (2009), Balakrishnan and Cramer (2014, Chapter 23), Kateri and Kamps (2017), and the recent monograph Kundu and Ganguly (2017). In order to determine the exact distribution of the MLEs for a Type-I censored simple step-stress model, Balakrishnan et al. (2009) employed the approach of the conditional moment generating function. The resulting expressions for the density functions of the MLEs are based on gamma density functions. Recently, G´orny and Cramer (2017b, Remark 3.10 (i)) established alternative representations for the density functions of the MLEs for the Type-I censored simple step-stress mode
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