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Finite mixture model is a widely acknowledged model-based clustering method for analyzing data. In this paper, a new finite mixture model via an extension of Birnbaum Saunders distribution is introduced. The new mixture model provide a useful generalization of the heavy-tailed lifetime model since the mixing components cover both skewness and kurtosis. Some properties and characteristics of the model are derived and an expectation and maximization (EM)-type algorithm is developed to compute maximum likelihood estimates. The asymptotic standard errors of the parameter estimates are obtained via offering an information-based approach. Finally, the performance of the methodology is illustrated by considering both simulated and real datasets.

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In recent decades, studying order statistics arising from independent and not necessary identically distributed (INID) random variables has been a remarkable concern for researchers. The cumulative distribution function (CDF) of these random variables (F i:n) is a complex manipulating, long time consuming and a softwareintensive tool that takes considerable time. Therefore, obtain approximations and bounds for F i:n and other theoretical properties of these variables, such as moments, quantiles, characteristic functions, and some related probabilities, has always been the main challenge. Recently, Bayramoglu (2018), Bayramoglu (2018), has introduced a set of CDFs (F [i]), whose definitions are based on a point to point ordering of the original CDFs (F i), that can be used to approximate the CDF of i-th order statistics (F i:n). Here, by using just F [1] and F [n] , we provide new upper and lower bounds for the F i:n. Furthermore, new approximations for F 1:n and F n:n , as well as for other cases, are derived. Comparisons with respect to approximations suggested by Bayramoglu Bayramoglu (2018) are also provided.