Özet:

Abstract To study the heterogeneous nature of lifetimes of certain mechanical or engineering processes, a mixture model of some suitable lifetime distributions may be more appropriate and appealing than simpler models. In this paper, a new mixture family of the lifetime distributions is introduced via harmonic weighted mean of an underlying distribution and the distribution of the proportional hazard model corresponding to the baseline model. The proposed class of distributions includes the general Marshall-Olkin family of distributions as a special case. Some important properties of the proposed model such as survival function, hazard function, order statistics and some results on stochastic ordering are obtained in a general setting. A special case of this new family is considered by employing Weibull distribution as the parent distribution. We derive several properties of the special distribution such as moments,hazard function survival regression and certain characterizations results. Moreover, we estimate the parameters of the model by using frequentist and Bayesian approaches. For Bayesian analysis, five loss functions, namely the squared error loss function (SELF), weighted squared error loss function (WSELF), modified squared error loss function (MSELF), precautionary loss function (PLF), and K-loss function (KLF) are considered. The beta prior as well as the gamma prior are used to obtain the Bayes estimators and posterior risk of the unknown parameters of the model. Furthermore, credible intervals (CIs) and highest posterior density (HPD) intervals are also obtained. A simulation study is presented via Monte Carlo to investigate the bias and mean square error of the maximum likelihood estimators. For illustrative purposes, two real-life applications of the proposed distribution to  Kidney and cancer patients are provided.

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