In this paper, moment inequalities for a new class of life distributions called new renewal better than used in Laplace transform order (N RBU L) are proposed. For the new class N RBU L, the preservation under convolution and mixture are studied. A new test statistic for testing exponentiality versus N RBU L is investigated based on these moment inequalities. Pitman asymptotic efficiencies 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.

We hope to assess the different processing methodologies or the effectiveness of the devices or systems applied in this nonparametric statistical test by observing the failure behavior of the recorded survival data. The proposed second-order approach of new better than used (NBU 2) will be employed, which requires that the test data behaves either like NBU 2 property or exponentially. If the survival data is NBU 2, the proposed treatment method is likely to be beneficial. If the data are exponential, on the other hand, the recommended treatment method has no positive or negative influence on patients, as shown in the application section. To establish the validity of the test, we calculated the power of the proposed test and efficiency on both complete and censored data, compared the results to those of existing tests, and then applied the test to a range of real-world data.

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In this article we introduce new classes of life distributions namely new better (worse) than used in expectation at specific age t 0 NBUE-t 0 (NWUE-t 0 ) and harmonic new better (worse) than used in expectation at specific age t 0 HNBUE-t 0 (HNWUE-t 0 ). The closure properties under various reliability operations such as convolution, mixture, mixing and the homogeneous Poisson shock model of these classes are studied. Furthermore, nonparametric tests are proposed to test exponentiality versus the NBUE-t 0 and HNBUE-t 0 classes. The critical values and the powers of this tests are calculated to assess the performance of the tests. It is shown that the proposed tests have high efficiencies for some commonly used distributions in reliability. Sets of real data are used as examples to elucidate the use of the proposed tests for practical problems.