Testing Truncation Dependence: The Gumbel Copula

In the analysis of left- and double-truncated durations, it is often assumed that the age at truncation is independent of the duration. When truncation is a result of data collection in a restricted time period, the truncation age is equivalent to the date of birth. The independence assumption is then at odds with any demographic progress when life expectancy increases with time, with evidence e.g. on human demography in western civilisations. We model dependence with a Gumbel copula. Marginally, it is assumed that the duration of interest is exponentially distributed, and that births stem from a homogeneous Poisson process. The log-likelihood of the data, considered as truncated sample, is derived from standard results for point processes. Testing for positive dependence must include that the hypothetical independence is associated with the boundary of the parameter space. By non-standard theory, the maximum likelihood estimator of the exponential and the Gumbel parameter is distributed as a mixture of a two- and a one-dimensional normal distribution. For the proof, the third parameter, the unobserved sample size, is profiled out. Furthermore, verifying identification is simplified by noting that the score of the profile model for the truncated sample is equal to the score for a simple sample from the truncated population. In an application to 55 thousand double-truncated lifetimes of German businesses that closed down over the period 2014 to 2016, the test does not find an increase in business life expectancy for later years of the foundation. The $p$-value is $0.5$ because the likelihood has its maximum for the Gumbel parameter at the parameter space boundary. A simulation under the condition of the application suggests that the test retains the nominal level and has good power.