generalized total time on test transform for weighted variables, properties and applications

In this article, the generalized  total  time on test transform and some related transforms for weighted variables are stated. their characteristics and relationship with each other have been considered and also these transforms have been investigated in the weighted mode from the point of view of stochastic orders. also, by presenting graphs of generalized total time on test transform   for some common weight functions, its behavior with respect to the weighted function is studied. then the relationship of this transform with its initial state is expressed. in the following, the topic under discussion is explained with some practical examples. then providing a comprehensive exploration of the applications of the studied transforms within the domains of insurance and reliability. by delving into these practical contexts, we gain valuable insights into how these mathematical tools can be effectively utilized to address complex challenges in risk assessment, decision-making, and resource allocation. additionally, the examination of the nbu class of distributions offers a deeper understanding of their behavior, shedding light on their relevance and applicability in various statistical analyses.    finally, the article concludes with a detailed discussion of a specific real dataset, offering a concrete demonstration of how the topic under study can be applied in practice.