Truncating the Exponential with a Uniform Distribution

TL;DR

This paper develops methods for estimating the exponential distribution parameter from truncated data observed within a uniform interval, deriving estimators and analyzing their statistical properties.

Contribution

It introduces a maximum likelihood estimator accounting for truncation and compares it to standard methods, providing theoretical and empirical insights.

Findings

The MLE is consistent and asymptotically normal.

Truncation increases the standard error of the estimator.

The proposed method performs well in social and economic science applications.

Abstract

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.