Abstract. This paper investigates the Total Time on Test (TTT) transform through the framework of the quantile function and explores its theoretical properties. The study establishes the connection between the TTT transform and several concepts in reliability theory, including aging and hazard rate functions, expressed in terms of quantile functions. Furthermore, the paper highlights the relationship between the TTT transform and economic indices based on quantile functions, and demonstrates its effectiveness as a tool for measuring income inequality. The main motivation of this study was to establish a unified framework linking reliability theory and income inequality measurement, showing that the TTT transform can not only enhance the analysis of lifetime distributions but also serve as a more sensitive and reliable index of economic inequality. In addition to restating and extending known results, we present new findings that strengthen the role of the TTT transform as both a reliability measure and an economic inequality index. Numerical examples and graphical illustrations are provided to clarify the applicability of the proposed results.

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