Minimum risk sequential point estimation of the stress-strength reliability parameter for exponential distribution

Abstract

In this article, using purely and two-stage sequential procedures, the problem of minimum risk point estimation of the reliability parameter (R) under the stress–strength model, in case the loss function is squared error plus sampling cost, is considered when the random stress (X) and the random strength (Y) are independent and both have exponential distributions with different scale parameters. The exact distribution of the total sample size and explicit formulas for the expected value and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress–strength model are provided under the two-stage sequential procedure. Using the law of large numbers and Monte Carlo integration, the exact distribution of the stopping rule under the purely sequential procedure is approximated. Moreover, it is shown that both proposed sequential procedures are finite and for special cases the exact distribution of stopping times has a degenerate distribution at the initial sample size. The performances of the proposed methodologies are investigated with the help of simulations. Finally, using a real data set, the procedures are clearly illustrated.

Keywords:

• Law of large numbers
• purely sequential sampling
• stopping rule
• two-stage sequential sampling

SUBJECT CLASSIFICATIONS:

• 62L15
• 60G40
• 62F12
• 60K10

Acknowledgments

The authors are grateful to the Editor-in-Chief, Professor Nitis Mukhopadhyay, an Associate Editor, and anonymous referee(s) for their valuable comments, which led to improvement of the article. We thank Springer publisher for giving us permission to use data for our illustration(s).

Additional information

Funding

Eisa Mahmoudi and Ashkan Khalifeh are also indebted to Yazd University for supporting this research.