使用Uniform Distribution截断指数
Truncating the Exponential with a Uniform Distribution
摘要：
对于指数分布的持续时间样本，我们的目标是点估计和它的参数的置信区间。持续时间仅在它在由均匀分布确定的特定时间间隔内结束时才被观察到。因此，数据是一个截断的经验过程，当只有一小部分样本被观察到时，我们可以用泊松过程来近似，就像我们的应用程序一样。我们从点过程的标准参数中推导出似然，承认潜在样本的大小作为第二个参数，并推导出两者的极大似然估计。指数参数估计的相合性和渐近正态性来自于M-估计的标准结果。我们比较了设计与一个简单的随机样本假设观察到的持续时间。从理论上讲，对数似然的导数在小参数值的截断设计中不太陡峭，表明寻根的计算工作量较大，标准误较大。在社会和经济科学的应用中，以及在模拟中，我们确实发现，当承认截断时，标准误会适度增加。
英文摘要：
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.
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