Total Time on Test Transform via Quantile Function

Esfahani, Mojtaba; Amini, Mohammad; Mohtashami Borzadaran, Gholam Reza

doi:10.22055/jamm.2026.49182.2359

Total Time on Test Transform via Quantile Function

Document Type : Original Paper

Authors

Mojtaba Esfahani ¹

Mohammad Amini ²

Gholam Reza Mohtashami Borzadaran ²

¹ Drpartment of Statistics, Velayat University of Iranshahr, Iranshahr, Iran

² Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad

10.22055/jamm.2026.49182.2359

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.

Keywords

Total Time on Test Transform‎

‎Quantile Function‎‎

‎Ageing Concepts‎

‏‎Economic Inequality

Subjects

Statistics

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