  In clinical and epidemiological research doubly truncated data often appear.
This is the case, for instance, when the data registry is formed by interval
sampling. Double truncation generally induces a sampling bias on the target
variable, so proper corrections of ordinary estimation and inference procedures
must be used. Unfortunately, the nonparametric maximum likelihood estimator of
a doubly truncated distribution has several drawbacks, like potential
non-existence and non-uniqueness issues, or large estimation variance.
Interestingly, no correction for double truncation is needed when the sampling
bias is ignorable, which may occur with interval sampling and other sampling
designs. In such a case the ordinary empirical distribution function is a
consistent and fully efficient estimator that generally brings remarkable
variance improvements compared to the nonparametric maximum likelihood
estimator. Thus, identification of such situations is critical for the simple
and efficient estimation of the target distribution. In this paper we introduce
for the first time formal testing procedures for the null hypothesis of
ignorable sampling bias with doubly truncated data. The asymptotic properties
of the proposed test statistic are investigated. A bootstrap algorithm to
approximate the null distribution of the test in practice is introduced. The
finite sample performance of the method is studied in simulated scenarios.
Finally, applications to data on onset for childhood cancer and Parkinson's
disease are given. Variance improvements in estimation are discussed and
illustrated.
