Ordering probability distributions by tail behavior

We introduce, what we call, "the exponent function of a probability distribution" which can be used for ordering probability distributions according to their tail behavior. Some properties of this function as well as the related "failure rate quantile function" are discussed. An ordering among probability distributions based on the right tail exponent of Parzen (1979) is introduced and called "Parzen ordering". Some properties of this ordering along with examples are given.