A new model is proposed for the characterization of failure distributions which allows a monotone characterization for distributions with nonmonotone hazard or failure rates. The new model is based on the log odds rate and can be seen to characterize a failure time distribution in terms of increasing log odds rate (ILOR) and decreasing log odds rate (DLOR) in a manner similar to the IFR and DFR concepts. The model is described and developed sufficiently in this paper to allow a reliability engineer to use the characterizations based on it. Examples are provided which illustrate how one can use log odds plots to indicate whether a distribution has ILOR or DLOR in terms of failure time, log failure time or log-log failure time. It is believed that this new model will give reliability engineers more alternative choices, more intuitive choices and more organized choices in the failure time distributions when the failure rate is nonmonotone or the IFR concept appears to be too stringent.