In this paper, moment inequalities for a new class of life distributions called new renewal better than used in Laplace
transform order (NRBUL) are proposed. For the new class NRBUL, the preservation under convolution and mixture are
studied. A new test statistic for testing exponentiality versus NRBUL is investigated based on these moment inequalities.
Pitman asymptotic eciencies of the test are proposed. The critical values of this test are tabulated. Some examples for
censored and non-censored data are applied to the new test. Finally a new test for censored data is proposed