Non-parametric hypothesis testing to address fundamental life testing issues in reliability analysis with some real applications

Life categories and probability distributions are part of a new field in reliability that has emerged as a result of the daily generation of data that has become more complex across practical fields. This study demonstrated how well the U-statistics technique can be applied to real-world testing problems, producing more efficient processes that are on par with or even more successful than conventional approaches. Furthermore, there was room for improvement in the performance of these methods. An approach tending toward normalcy was supported by comparing a unique U-statistic test with the used better than age in moment generating ordering (UBAmgf) test statistic, In this manuscript, a novel nonparametric technique has been developed to test the belonging of a dataset to a distribution of a new statistical class survival function, the moment generating function for used better than aged (UBAmgf). This type of test was crucial in practical life, such as implementing a specific strategy of proposed therapy for a particular disease, deeming it futile if the survival data was exponential (accepting ${\mathrm{H}}_0$) (the suggested therapeutic approach does not exhibit positive or negative effects on the patients). Once the survival data was UBAmgf, the treatment or device or system employed yields an expected overall current value better or higher than the older device governed by the asymptotic survival function (discussed in the Applications section). The appropriateness of the proposed statistical test's application range was properly determined by calculating its test efficiency and critical values and comparing them with other tests, whether in complete or censored data. Finally, we applied this proposed test technique in the manuscript to a different set of real data in both cases.