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Change point detection is one of the most challenging statistical problems because the number and position of these points are unknown. In this article, we will first introduce the concept of change point and then obtain the parameter estimation of the first-order autoregressive model AR(1); in order to investigate the precision of estimated parameters, we have done a simulation study. The precision and consistency of parameters were evaluated using MSE. The simulation study shows that parameter estimation is consistent. In the sense that as the sample size increases, the MSE of different parameters converges to zero. Next, the AR(1) model with the change point was fitted to Iran's annual inflation rate data (from 1944 to 2022), and the inflation rate in 2023  and 2024 was predicted using it.

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This article introduces a new extension of the log-logistic distribution, and its properties and parameter estimation are studied and analyzed. It is shown that adding a parameter to this distribution makes its shape more symmetric and less skewed as the parameter increases. Unlike the original distribution, the moments of the new distribution and its quantile function always exist. Furthermore, it is demonstrated that the reliability measures, such as the hazard rate function, the mean residual life function, and stochastic orderings, are more flexible in the new distribution. Additionally, the parameters of the distribution are estimated using the LLP and ML methods, and the efficiency and consistency of the estimators are evaluated through simulation studies. Finally, the practical applicability of the model is demonstrated by applying the new model to real-world data from airborne equipment and lung cancer patients.

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In the time series analysis, we may encounter situations where some elements of the model are imprecise quantities. One of the most common situations is the inaccuracy of the underlying observations, usually due to measurement or human errors. In this paper, a new fuzzy autoregressive time series model based on the support vector machine approach is proposed. For this purpose, the kernel function has been used for the stability and flexibility of the model, and the constraints included in the model have been used to control the points. In order to examine the performance and effectiveness of the proposed fuzzy autoregressive time series model, some goodness of fit criteria are used. The results were based on one example of simulated fuzzy time series data and two real examples, which showed that the proposed method performed better than other existing methods.