Multi-stage Minimum Risk Point Estimation Strategies for Comparing the Locations from Two Negative Exponential Models and Second-Order Asymptotics: Illustrations with Simulated Data and Bone Marrow Transplant Data

The negative exponential (NE) distributions have been used in the literature for studying the growth of certain kinds of tumors in cancer research in addition to its wider applicability in other areas including life testing, reliability analysis, soil science, survival analysis, and weed science. We consider two independent NE populations with unknown location parameters but unknown and unequal scale parameters. We develop minimum risk point estimation (MRPE) methodologies for comparing the location parameters. Under a set of assumptions, we have developed a general theory of multi-stage sampling strategies leading to important and practical properties such as (1) the asymptotic first-order (f.o.) risk efficiency and (2) the asymptotic second-order (s.o.) expansion of the regret. Then, we successively incorporate a wide range of asymptotics associated with specific multi-stage strategies such as (1) purely sequential, (2) accelerated sequential, (3) three-stage, and (4) two-stage sampling methods. The proposed theory and methodology are then validated with thorough analyses of data obtained from simulations. To wrap this up, we supplement with illustrations via bone marrow transplant (BMT) data, but we have also included a number of other potential real data sources for new additional illustrations including one from reliability studies.