The Power Rayleigh distribution (PRD) is a new extension of the standard one-parameter Rayleigh distribution. To employ this distribution as a life model in the analysis of reliability and survival data, we focused on the statistical inference for the parameters of the PRD under the adaptive Type-II censored scheme. Point and interval estimates for the model parameters and the corresponding reliability function at a given time are obtained using likelihood, Bootstrap and Bayesian estimation methods. A simulation study is conducted in different settings of the life testing experiment to compare and evaluate the performance of the estimates obtained. In addition, the estimation procedure is also investigated in real lifetimes data. The results indicated that the obtained estimates gave an accurate and efficient estimation of the model parameters. The Bootstrap estimates are better than the estimates obtained by the likelihood estimation approach, and estimates obtained using the Markov Chain Monte Carlo method by the Bayesian approach under both the squared error and the general entropy loss functions have priority over other point and interval estimates. Under the adaptive Type-II censoring scheme, concluding results confirmed that the PRD can be effectively used to model the lifetimes in survival and reliability analysis.

In this paper, a new generalization of the one parameter Rayleigh distribution called the Power Rayleigh (PRD) was employed to model the life of the tested units in the step-stress accelerated life test. Under progressive Type-I interval censored data, the cumulative exposure distribution was considered to formulate the life model, assuming the scale parameter of PRD has the inverse power function at each stress level. Point estimates of the model parameters were obtained via the maximum likelihood estimation method, while interval estimates were obtained using the asymptotic normality of the derived estimators and the bootstrap resampling method. An extensive simulation study of $ k = 4 $ levels of stress in different combinations of the life test under different progressive censoring schemes was conducted to investigate the performance of the obtained point and interval estimates. Simulation results indicated that point estimates of the model parameters are closest to their initial true values and have relatively small mean squared errors. Accordingly, the interval estimates have small lengths and their coverage probabilities are almost convergent to the 95% significance level. Based on the Fisher information matrix, the D-optimality and the A-optimality criteria are implemented to determine the optimal design of the life test by obtaining the optimum inspection times and optimum stress levels that improve the estimation procedures and give more efficient estimates of the model parameters. Finally, the developed inferential procedures were also applied to a real dataset.

In this study, the effects of meteorological factors, i.e., atmospheric temperature and humidity, the pollutant oxides i.e., sulfur dioxide (SO2) and nitrogen dioxide (NO2) are examined in connection to particulate matter 2.5 (PM2.5) concentrations, which is an important air pollutant that is harmful to both human health and the environment. It is essential to comprehend the variables that cause PM2.5 levels to increase in urban areas and that contribute to efficient air quality management plans. The dataset is taken from multiple monitoring sites throughout the city of Islamabad on meteorological factors along with pollutant oxides from year 2019 to 2023 containing particulate matter, nitrogen dioxide, sulfur dioxide, atmospheric temperature, and humidity from Pakistan Environmental Protection Agency, comprising the observations of PM2.5 and associated measures, i.e., NO2, SO2, atmospheric temperature, and humidity, examined using a statistical regression model. The main goal is to assess how well NO2, SO2, atmospheric temperature, and humidity can predict changes in PM2.5 concentrations. The mean value of PM2.5 is 35.26 µg/m³ (SD = 21.16 µg/m³), and the model explains 44% of the variation in particulate matter concentration. The overall model is statistically significant (F Statistic = 274.52, p < 0.01), and the study showed a statistically significant link between the response variable (PM2.5) and the predictors i.e., NO2, SO2, atmospheric temperature, and humidity (p-values < 0.01). These findings enable that air quality management strategies can offer a better comprehension of the dynamics of air pollution by offering insights into the variables impacting PM2.5 concentrations. Policymakers and environmental organizations can build focused measures to reduce and regulate PM2.5 pollution with the help of the implications of this research. It may be possible to lower PM2.5 levels and improve overall air quality by addressing the contributing elements mentioned in this study, such as NO2, SO2, atmospheric temperature, and humidity, thereby protecting public health and the environment.