Summary: The new better than used failure rate (NBUFR) and new better than average failure rate (NBAFR) classes of the distributions, which are generalizations of the new better than used (NBU) class, and their dual classes the new worse than used failure rate (NWUFR) and the new worse than average failure rate (NWAFR), have been considered in the literature as natural weakenings of the NBU (NWU) property. This paper considers testing exponentiality against strictly NBUFR (NBAFR) alternatives, or their duals, based on kernel methods. The percentiles of these test statistics are tabulated for sample size \(n=5(1)50\). Pitman’s asymptotic efficiency of the tests are calculated for both classes and compared. An example of 40 patients suffering from blood cancer disease (Leukemia) demonstrates a practical application of the proposed tests in the Medical Sciences.