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Total time on test-based goodness-of-fit statistics for the reciprocal property in fatigue-life models

Author

Listed:
  • Cecilia Castro

    (Centre of Mathematics, Universidade do Minho)

  • Marta Azevedo

    (Centre of Mathematics, Universidade do Minho)

  • Víctor Leiva

    (Escuela de Ingenieria Industrial, Pontificia Universidad Catolica de Valparaiso)

  • Luís Meira-Machado

    (Centre of Mathematics, Universidade do Minho)

Abstract

We propose a new goodness-of-fit procedure designed to verify the reciprocal property that characterizes the fatigue-life or Birnbaum–Saunders (BS) distribution. Under this property, scaling a random variable that takes positive values by its median results in the same distribution as its reciprocal, a feature frequently encountered in reliability and survival studies. Our procedure employs total time on test (TTT) curves to compare the behavior of the observed data and its reciprocal counterpart, capturing both local and global discrepancies through supremum- and area-based statistics. We establish the theoretical validity of these statistics under mild assumptions, showing that they deliver accurate inference for moderate to large samples. Simulation evidence indicates that our TTT-based procedures are sensitive to subtle departures from log-symmetry, particularly when the distribution underlying the data has heavier or lighter tails than the assumed one. Illustrative real data examples further reveal how overlooking deviations from the reciprocal property can distort reliability estimates and predictions of failure times, showing the practical importance of the new goodness-of-fit procedure. Overall, our findings strengthen the BS framework and provide robust tools for model validation and selection when log-symmetric modeling assumptions are in place.

Suggested Citation

  • Cecilia Castro & Marta Azevedo & Víctor Leiva & Luís Meira-Machado, 2025. "Total time on test-based goodness-of-fit statistics for the reciprocal property in fatigue-life models," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 31(2), pages 422-441, April.
    • Handle: RePEc:spr:lifeda:v:31:y:2025:i:2:d:10.1007_s10985-025-09653-2
    DOI: 10.1007/s10985-025-09653-2
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    References listed on IDEAS

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    1. Kundu, Debasis & Kannan, Nandini & Balakrishnan, N., 2008. "On the hazard function of Birnbaum-Saunders distribution and associated inference," Computational Statistics & Data Analysis, Elsevier, vol. 52(5), pages 2692-2702, January.
    2. Ng, H. K. T. & Kundu, D. & Balakrishnan, N., 2003. "Modified moment estimation for the two-parameter Birnbaum-Saunders distribution," Computational Statistics & Data Analysis, Elsevier, vol. 43(3), pages 283-298, July.
    3. Cui-Juan Kong & Han-Ying Liang, 2024. "Quantile difference estimation with censoring indicators missing at random," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 30(2), pages 345-382, April.
    4. Lucia Santana & Filidor Vilca & V�ctor Leiva, 2011. "Influence analysis in skew-Birnbaum--Saunders regression models and applications," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(8), pages 1633-1649, July.
    5. Azevedo, Cecilia & Leiva, Víctor & Athayde, Emilia & Balakrishnan, N., 2012. "Shape and change point analyses of the Birnbaum–Saunders-t hazard rate and associated estimation," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 3887-3897.
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