On Two-Stage Methodologies for Estimating the Mean Time to Failure with Applications
by
Nitis Mukhopadhyay
University of Connecticut, Department of Statistics, CLAS Building-U4120, 215 Glenbrook Road, Storrs, CT 06269-4120, U.S.A.

One may model time to failure data with an exponential distribution. Now, suppose that we have random samples from an exponential distribution with its mean time to failure (MTTF) parameter unknown. It is known that sequential estimation of MTTF is an important problem and it has attracted many researchers over the years.

Our goal is to estimate MTTF in such a way that the associated mean squared error (MSE) does not exceed a preset risk-bound c (> 0). First, we will discuss a genuine two-stage estimation methodology due to Mukhopadhyay and Pepe (2006, Sequential Analysis, 25, no. 1, 85-101) that was developed in the spirit of Stein (1945, Annals of Mathematical Statistics, 16, 243-258). What was remarkable about the Mukhopadhyay-Pepe estimator of MTTF was that its associated MSE did not exceed c even though their terminal estimator and the sample size were dependent on each other. More recently, Zacks and Mukhopadhyay (2006, Sequential Analysis, 25, no. 4, 437-452) improved upon Mukhopadhyay-Pepe’s two-stage estimator significantly reducing any over-sampling to a near minimum. Zacks and Mukhopadhyay (2006) relied upon exact calculations of a number of characteristics including the MSE of the terminal estimator of MTTF. The performances of these methodologies will be summarized with the help of simulations and exact calculations. Illustrations will also be included with data from a multicenter clinical trial involving patients with acute myeloctic leukemia (AML) prepared for transplantation with radiation-free conditioning regimen. In the end, a number of generalizations and extensions of the proposed two-stage estimation methodology will be explored with an exact upper risk-bound.

Date received: March 5, 2007