For a sample of Exponentially distributed durations we aim at point estimation and a confidence interval for its parameter. A duration is only observed if it has ended within a certain time interval, determined by a Uniform distribution. Hence, the data is a truncated empirical process that we can approximate by a Poisson process when only a small portion of the sample is observed, as is the case for our applications. We derive the likelihood from standard arguments for point processes, acknowledging the size of the latent sample as the second parameter, and derive the maximum likelihood estimator for both. Consistency and asymptotic normality of the estimator for the Exponential parameter are derived from standard results on M-estimation. We compare the design with a simple random sample assumption for the observed durations. Theoretically, the derivative of the log-likelihood is less steep in the truncation-design for small parameter values, indicating a larger computational effort for root finding and a larger standard error. In applications from the social and economic sciences and in simulations, we indeed, find a moderately increased standard error when acknowledging truncation.
翻译:对于指向点分布时间的样本,我们的目标是点数估计和其参数的置信度间隔。 只有当它在一个时间间隔内结束, 由统一分布决定, 才会观察到时间间隔。 因此, 数据是一个短暂的经验过程, 在观察样本的一小部分时, 我们就可以用Poisson 过程来估计, 就象我们的应用一样。 我们从点数过程的标准参数中推断出可能性, 承认潜伏样本的大小为第二个参数, 并得出两者的最大可能性估计器。 指数参数估计器的连续性和无光性常态来自M- 估计的标准结果。 我们比较了所观察到的时间长度的设计与简单的随机抽样假设。 从理论上讲, 日志相似值的衍生物在小参数值的计算- 标定中并不那么明显, 这表明在寻找根值方面做了更大的计算努力, 并且有一个更大的标准错误。 在社会和经济科学以及模拟的应用中, 我们确实发现在承认悬浮时, 一种适度的标准错误。