In observational studies, the probability of selection of sampling units is not always equal. The recorded observations are biased in this scenario. The unweighted distributions in such situations are not useful until the inclusion probability of each item is same. The theory of weighted distributions offers a unifying approach for these types of conditions because it considers the adjustment bias. Failure to comply with such adjustment may lead to inappropriate results. In this article, an efficient mentoring scheme (Weighted-TBE chart) for time between events (TBE) using weighted exponential distribution has been proposed based on weighted variance (WV) method. A comparison has been established between CC based on weighted and unweighted probability distributions. The performance measure ARL has been calculated using Monte Carlo simulations. The Weighted-TBE chart has provided least values of ARL in the presence of unwanted process variations and proved to be more effective than the existing scheme. Further the proposed control chart has been applied to time between failures data to show its practical applicability.

In practice, finite mixture models were often used to fit various type of observed phenomena, specifically those which are random in nature. In this paper, a finite mixture model based on weighted versions of exponential and gamma distribution is considered and studied. Some mathematical properties of the resulting model are discussed including moment generating function, skewness, kurtosis, survival function, hazard rate function, stochastic ordering, order statistics, Bonferroni and Lorenz curves, Renyi entropy measure and estimation of the model parameters. Two real-life data applications from different fields exhibit the fact that in certain situations, the proposed mixture model might be a better alternative than the existing popular models.

In this paper, we modify the Mahmoud and Mandouh (2013) model by adopting double truncation technique. It is referred to as Double Truncated Transmuted Fréchet (DTTF) distribution. Diverse probabilistic and reliability measures are developed and discussed. The MLEs of parameters are derived and a simulation study is also made. The DTTF distribution is modeled by two real-time datasets and supportive rationalized results provide the evidence that DTTF distribution is a reasonably better fit model than its competing models.