The Parameter Estimation Of Two-parameter Exponential Distribution By Grouped Data

With the development of science and technology, people attach more and more importance to the reliability of production. Because the life of the production is a stochastic phenomenon, it comes to a statistical inference problem that we try to determine the reliability indexes of the production. In order to make clear the life distribution of the production measured and solve the indexes of reliability to give some pieces of advice about improving the reliability of production, we often have to make the life experiment. While making the life experiment on N samples of the productionwhose lives follow a certain distribution, at 0 =T₀ 1 <… k-1 k = ∞ we examine the samples, then we can obtain the observation n_j. which is the number of the invalid samples in [T_j-1, T_j) . At last n_j are the grouped data that we want to gain (herej = 1,2,…,k ). Let we have known that the lives of the samples follow a distribution withunknown parameters, next we need to estimate the unknown parameters of the distribution under grouped data.Since it is easy to obtain grouped data, people pay more and more attention to grouped data. The parameter estimate of many common distribution under grouped data have been constantly brought forward, while few scholars discuss the parameter problem in the two-parameter exponential distribution under grouped data. In this paper, we consider the problem mentioned above and obtain a maximum likelihood estimate (MLE) of the two-parameter exponential distribution with grouped data and a invalid sample datum. We compare this MLE with the approximate maximum likelihood estimate (AMLE) which is brought out by GU YI MING (2001). The comparison confirms that the MLE is better than the AMLE in the case of large sample. Furthermore, the existence and uniqueness of the MLE is prove and the reason that the AMLE maybe can not be worked out is presented at the last of the paper.