  In studies on lifetimes, occasionally, the population contains statistical
units that are born before the data collection has started. Left-truncated are
units that deceased before this start. For all other units, the age at the
study start often is recorded and we aim at testing whether this second
measurement is independent of the genuine measure of interest, the lifetime.
Our basic model of dependence is the one-parameter Gumbel-Barnett copula. For
simplicity, the marginal distribution of the lifetime is assumed to be
Exponential and for the age-at-study-start, namely the distribution of birth
dates, we assume a Uniform. Also for simplicity, and to fit our application, we
assume that units that die later than our study period, are also truncated. As
a result from point process theory, we can approximate the truncated sample by
a Poisson process and thereby derive its likelihood. Identification,
consistency and asymptotic distribution of the maximum-likelihood estimator are
derived. Testing for positive truncation dependence must include the
hypothetical independence which coincides with the boundary of the copula's
parameter space. By non-standard theory, the maximum likelihood estimator of
the exponential and the copula parameter is distributed as a mixture of a two-
and a one-dimensional normal distribution. The application are 55 thousand
double-truncated lifetimes of German businesses that closed down over the
period 2014 to 2016. The likelihood has its maximum for the copula parameter at
the parameter space boundary so that the $p$-value of test is $0.5$. The life
expectancy does not increase relative to the year of foundation. Using a
Farlie-Gumbel-Morgenstern copula, which models positive and negative
dependence, finds that life expectancy of German enterprises even decreases
significantly over time.
