A long testing period is usually required for the life testing of high-reliability products or materials. It is possible to shorten the testing process by using ALTs (accelerated life tests). Due to the fact that ALTs test products in harsher settings than are typical use conditions, the life expectancy of the objects they evaluate is reduced. Censored data in which the specific failure timings of all units assigned to test are not known, or all units assigned to test have not failed, may arise in ALTs for a variety of reasons, including operational failure, device malfunction, expense, and time restrictions. In this paper, we have considered the step stress partially accelerated life test (SSPALT) under two different censoring schemes, namely the type-I progressive hybrid censoring scheme (type-I PHCS) and the type-II progressive censorship scheme (type-II PCS). The failure times of the items are assumed to follow NH distribution, while the tampered random variable (TRV) model is used to explain the effect of stress change. In order to obtain the estimates of the unknown parameters, the maximum likelihood estimation (MLE) approach is adopted. Furthermore, based on the asymptotic theory of MLEs, the approximate confidence intervals (ACIs) are also constructed. The point estimates under two censoring schemes are compared in terms of root mean squared errors (RMSEs) and relative absolute biases (RABs), while ACIs are compared in terms of their lengths and coverage probabilities (CPs). The performance of the estimators has been evaluated and compared under two censoring schemes with various sample sizes through a simulation study. Simulation results show that estimates with type-I PHCS outperform estimates with type-II PCS in terms of RMSEs, RABs, lengths, and CPs. Finally, a real-world numerical example of insulating fluid failure times is presented to show how the approaches will work in reality.

<abstract><p>The initial COVID-19 vaccinations were created and distributed to the general population in 2020 thanks to emergency authorization and conditional approval. Consequently, numerous countries followed the process that is currently a global campaign. Taking into account the fact that people are being vaccinated, there are concerns about the effectiveness of that medical solution. Actually, this study is the first one focusing on how the number of vaccinated people might influence the spread of the pandemic in the world. From the Global Change Data Lab "Our World in Data", we were able to get data sets about the number of new cases and vaccinated people. This study is a longitudinal one from 14/12/2020 to 21/03/2021. In addition, we computed Generalized log-Linear Model on count time series (Negative Binomial distribution due to over dispersion in data) and implemented validation tests to confirm the robustness of our results. The findings revealed that when the number of vaccinated people increases by one new vaccination on a given day, the number of new cases decreases significantly two days after by one. The influence is not notable on the same day of vaccination. Authorities should increase the vaccination campaign to control well the pandemic. That solution has effectively started to reduce the spread of COVID-19 in the world.</p></abstract>

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