A general result concerning noncompleteness of location families of probability measures on Euclidean space is pointed out. Examples include boundedly complete families, such as those generated by certain scale mixtures of the standard Gaussian distribution. These examples illuminate completeness criteria for location families and compare favourably in simplicity with previously known examples of incomplete boundedly complete (nonlocation) families.
"Some Incomplete But Boundedly Complete Location Families." Ann. Statist. 21 (4) 2158 - 2162, December, 1993. https://doi.org/10.1214/aos/1176349416