total time on test transform via quantile function

This paper investigates the total time on test (ttt) transform throughthe framework of the quantile function and explores its theoretical properties. the studyestablishes the connection between the ttt transform and several concepts in reliabilitytheory, including aging and hazard rate functions, expressed in terms of quantile functions.furthermore, the paper highlights the relationship between the ttt transformand economic indices based on quantile functions, and demonstrates its effectiveness as atool for measuring income inequality. the main motivation of this study was to establisha unified framework linking reliability theory and income inequality measurement, showingthat the ttt transform can not only enhance the analysis of lifetime distributionsbut also serve as a more sensitive and reliable index of economic inequality. in addition torestating and extending known results, we present new findings that strengthen the roleof the ttt transform as both a reliability measure and an economic inequality index.numerical examples and graphical illustrations are provided to clarify the applicabilityof the proposed results. this paper investigates the total time on test (ttt) transform throughthe framework of the quantile function and explores its theoretical properties. the studyestablishes the connection between the ttt transform and several concepts in reliabilitytheory, including aging and hazard rate functions, expressed in terms of quantile functions.furthermore, the paper highlights the relationship between the ttt transformand economic indices based on quantile functions, and demonstrates its effectiveness as atool for measuring income inequality. the main motivation of this study was to establisha unified framework linking reliability theory and income inequality measurement, showingthat the ttt transform can not only enhance the analysis of lifetime distributionsbut also serve as a more sensitive and reliable index of economic inequality. in addition torestating and extending known results, we present new findings that strengthen the roleof the ttt transform as both a reliability measure and an economic inequality index.numerical examples and graphical illustrations are provided to clarify the applicabilityof the proposed results.