  This paper describes a novel approach based on "proportional imputation" when
identical units produced in a batch have random but independent installation
and failure times. The current problem is motivated by a real life industrial
production-delivery supply chain where identical units are shipped after
production to a third party warehouse and then sold at a future date for
possible installation. Due to practical limitations, at any given time point,
the exact installation as well as the failure times are known for only those
units which have failed within that time frame after the installation. Hence,
in-house reliability engineers are presented with a very limited, as well as
partial, data to estimate different model parameters related to installation
and failure distributions. In reality, other units in the batch are generally
not utilized due to lack of proper statistical methodology, leading to gross
misspecification. In this paper we have introduced a likelihood based
parametric and computationally efficient solution to overcome this problem.
