  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.
