On renewal failure rate classes of life distributions

Comparisons between the failure rate of a new unit with that of the residual life of a used unit are used to introduce new classes of life distributions. These are the new better (worse) than used renewal failure rate and the new better (worse) than average used renewal in failure rate. Their relationship with the existing classes are established and closure properties of them under mixture and convolution are studied. Some test statistics are proposed for testing exponentiality versus the two classes of renewal failure rate alternatives