& Aim: There are various data associated with any events in the world which need to be analyzed. In response to this, many researchers attempt to find appropriate methods that could better fit these data using new models. One of these methods is to introduce new distributions which could better describe available data. During last few years, new distributions have been extended based on existing well-known distributions. Usually, new distributions have more parameters than existing ones. This

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... dition of parameter(s) has been proved useful in exploring tail properties and also for improving the goodness-of-fit of the family under study. Methods & Materials: A new family of distributions is introduced by using truncated log-logistic distribution. Some statistical and reliability properties of the new family are derived. Results: Four special lifetime models of the new family are investigated. We estimate the parameters by maximum likelihood method. The obtained results are validated using a real dataset and it is shown that the new distributions provide a better fit than some other known distributions. Conclusion: We have provided four new distributions. The flexibility of the proposed distributions and increased range of skewness was able to fit and capture features in one real dataset much better than some competitor distributions