Lifetime Estimation of Defective Products from the Imaginal Mixture of Defective and Non-defective Products: The Trunsored Data Model

摘要

In solving the lifetime estimation problem of defective products using samples of size N from a mixture of defective and non-defective products, a new method of estimating the parameters of the underlying distribution function is proposed. We suppose that the ratio of the defective products to the non-defective products is unknown. The proposed method is to use an imaginal mixture model in which the non-defective products will never fail by the prescribed time T.If the non-defective products are dominant in the mixture, we can obtain the maximum likelihood estimates by dealing with the observed samples as truncated data with a conditional likelihood. If the defective products are dominant, we can no longer use the truncated data approach because the estimated sample size could end up being larger than N in which case the non-defective products would be empty. The imaginal mixture model, however, can estimate the parameters in either case.In addition, this model can test whether the non-defective products are empty or not because the likelihood functions for both the cases are of the same kind, whereas we cannot use the likelihood ratio test directly by using the likelihoods from the truncated model and the censored model due to the different kinds of likelihood. Thus, we call this versatile model which can be used for both the truncated and censored data models the trunsored data model. If it is not rejected that nondefective products are empty, we can regard the sampled data as censored, and can obtain smaller confidence intervals of the estimates of the parameters than those obtained by the trunsored model.After the introduction of this new mixture model, we apply it to the actual field data, and show how the proposed method works.