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.

It is often desired for practical reasons to cease a life test at a predetermined period ${𝓽}_0$. In this study, we provide sequential inspection sampling plan (SISP) for amputated life tests underlying the Burr Ⅻ (BTXII) distribution. We considered the ${\rho }^{\mathrm{t}\mathrm{h}}$ percentile lifetime of a product as the quality parameter. The sequential sampling plan is a dynamic and efficient technique to quality control and statistical decision-making. Unlike fixed sample plans, which require a predetermined number of samples before deciding, sequential plan enables ongoing analysis as a batch is examined. This flexible plan enables decisions- such as rejecting, accepting or continuing the sampling process - to be made after each sample, which is cumulative data for the number of nonconforming items. Acceptance, rejection limit lines, and optimal sample size at levels of manufacturer and customer errors were analyzed. A technique is provided for calculating the operation characteristic (OC) function and average sample number (ASN) in the suggested SISP. The effectiveness of the SISP was compared to the single, double, and repeating sampling strategies. In comparison to the single, double, and repetitive acceptance sampling plans (RASPs) the proposed sequential sampling acceptance strategy requires fewer sample resources on average for amputated life testing. The suggested method's uses are demonstrated with illustrated instances. In industrial applications, two actual sets of data are employed to demonstrate the SISP's flexibility.