A U-statistic is derived for testing exponentiality against new renewal better (worse) than used. For this class of life distributions, a nonparametric procedure (U-statistic) is presented in this investigation. Selected critical values are tabulated for sample size n = 5(1)39. The Pitman asymptotic efficiency relative to tests of other classes such as "decreasing mean residual life and new better than used in expectation" are discussed. Our procedure is simple to implement and is asymptotically normal both under the null hypothesis and alternative hypothesis with good efficiency against other procedures for some commonly used alternatives.