Likelihood Maximization of Lifetime Distributions With Bathtub-Shaped Failure Rate

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Equipment lifetime distributions with bathtub-shaped failure rate can be fitted to data by the maximum likelihood criterion. In the literature, a commonly used method is to find a point in the parameter space where the partial derivatives of the log-likelihood function are zero. As the log-likelihood function is typically nonconvex, this approach may yield a suboptimal fit. In this work, we maximize the log-likelihood function, using a multistart of 100 optimization procedures, by three nonlinear optimization algorithms: 1) Nelder–Mead with adaptive parameters; 2) sequential least squares quadratic programming (SLSQP); 3) limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm with box constraints (L-BFGS-B). We perform a systematic study of refitting ten key lifetime distributions with bathtub-shaped failure rate from the literature to two widely studied datasets. The multistart nonlinear optimization yields better fits than those reported in the literature in 14 out of 19 distribution-dataset pairs, for which reference parameters are available. Based on the results, if gradient information of the log-likelihood function is available, our recommended optimization algorithm for the purpose is SLSQP.
Original languageEnglish
Pages (from-to)759-773
Number of pages15
Issue number2
Early online date11 Aug 2022
Publication statusPublished - Jun 2023
MoE publication typeA1 Journal article-refereed


  • Additives
  • Data analysis
  • Data models
  • equipment lifetime modeling
  • Integrated circuits
  • maximum likelihood estimation
  • optimization
  • reliability
  • Symbols
  • Systematics
  • Terminology
  • Weibull distribution


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