Empirical Bayes testing for guarantee lifetime: Non identical components case

ABSTRACT

This article considers an empirical Bayes testing problem for the guarantee lifetime in the two-parameter exponential distributions with non identical components. We study a method of constructing empirical Bayes tests under a class of unknown prior distributions for the sequence of the component testing problems. The asymptotic optimality of the sequence of empirical Bayes tests is studied. Under certain regularity conditions on the prior distributions, it is shown that the sequence of the constructed empirical Bayes tests is asymptotically optimal, and the associated sequence of regrets converges to zero at a rate O(n^{− 1 + 1/[2(r + α) + 1]}) for some integer r ⩾ 0 and 0 ⩽ α ⩽ 1 depending on the unknown prior distributions, where n is the number of past data available when the (n + 1)st component testing problem is considered.

Keywords:

• Asymptotic optimality
• non identical component
• rate of convergence
• regret

MATHEMATICS SUBJECT CLASSIFICATION:

• 62C12
• 62F15