Unbiased Estimation Of Reliability Function Based On A Single Observation From A Mixture Of Two Distributions

In the past three decades,the mixture distribution model has developed rapidly and is widely used.The contaminated data usually obeys the mixture of multiple distribution functions.It is believed that the data from a distribution function is contaminated by some data from other distribution functions.We desire to obtain the relevant properties of the contaminated distribution,such as its reliability function.Because of the limitation of financial resources and manpower,we can only obtain a small amount of data.It is not suitable to to use maximum likelihood estimation and other methods with good large sample properties.Therefore,it is necessary to study the characteristics of fatigue life distributions in small sample.In extreme cases,we can even get only one sample.In this case,it is difficult to do statistical inference.The work in this paper is to obtain unbiased estimation of the reliability function based on a single observation.In this paper,in the case of the three-parameter Weibull distribution and the Gamma distribution,we derive two unbiased estimations of the reliability function of the Weibull distribution and Gamma distribution based on one single observation from corresponding mixed distribution.In the case of the three-parameter Weibull distribution,we further discuss the unbiased estimations of the variances of the unbiased estimations of the reliability function.Next,we obtain the Bayesian estimation of the reliability function of the two-parameter Weibull distribution.Finally,by doing some simulations,we prove the unbiasedness of the given unbiased estimations,and compare the unbiased estimation with the Bayesian estimation in the case of two-parameter Weibull distribution.