Characterizations from Relationships Between Failure Rate Functions and Conditional Moments

Abstract

We obtain general results on characterization of distributions from relationships between failure rate functions and conditional moments. Specifically, we study characterizations from linear relationships between the usual failure rate function and the left truncated conditional moment m _h (x) = E(h(X) | X > x), extending some characterizations of mixtures given recently. Moreover, we also study characterizations from relationships between the generalized failure rate functions and the doubly truncated conditional moment function m _h (x, y) = E(h(X)|x < X < y). Our results extend some particular characterizations based on mean residual life and failure rate. As a consequence, we obtain characterizations based on reversed failure rate and reversed mean residual life (expected inactivity time). Finally, we apply our theoretical results to obtain some new characterizations for usual models.

Key Words:

• Characterization
• Failure rate
• Hazard function
• Mean residual life
• Conditional moments

Mathematics Subject Classification:

• Primary 62E10
• Secondary 62N05

Acknowledgment

We wish to thank the anonymous referees for several helpful comments. This work was partially supported by Ministerio de Ciencia y Tecnologia under grant BFM2003-02947.