Bathtub-Shaped Distribution

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What is a Bathtub-Shaped Distribution?

The bathtub-shaped distribution is used in survival analysis and reliability engineering to describe a product or system’s’s failure rate over its lifespan. The term “bathtub” comes from the shape of the hazard function when plotted over time. The distribution is high at the beginning, fairly low and constant in the middle, and increasing towards the end.

The Bathtub curve can also be described as U-shaped distribution.

Bathtub-Shaped Distributions

The following distributions have bathtub-shaped hazard rates:

• Muth distribution. It’s hazard function has one vertical asymptote [1].
• Alpha distribution: This distribution has a bathtub-shaped hazard rate function corresponding to one version of the probability density function (Johnson’s) [2].
• The Exponentiated Weibull can model bathtub-shaped hazard rates as well as increasing, decreasing, and unimodal hazard rates.
• Wilson-Hilferty distribution (also called the WH distribution)
  A two-parameter special case of the Amoroso distribution. It is ideal for modelling data with increasing, decreasing and bathtub shape hazard rates.
• The GG distribution is well-suited for modeling a diverse range of hazard rate functions, including increasing, decreasing, bathtub, and arc-shaped functions.
• The exponential power distribution, a generalization of the normal distribution, has a bathtub-shaped hazard function.
• The generalized Gompertz distribution (GGD) differs from the “regular” distribution in that it can have a bathtub curve failure rate depending upon the shape parameter [3].

Bathtub-shaped distributions are not exclusively connected to hazard functions. While the bathtub is most commonly associated with the hazard function, other distributions and functions can also have a bathtub shape. For example:
Other Distributions with Bathtub Shapes:

• Probability Density Functions (PDFs): Some distributions have PDFs that can show a bathtub shape. For example, the Beta distribution with parameters that emphasize both ends of the range can produce a PDF that is high at the extremes and low in the middle.
• Income Distribution: In some models, income distribution curves can be bathtub-shaped, showing high frequencies of low and high-income earners with fewer individuals in the middle-income range.

Bathtub Curve Phases

The two ends of the bathtub and the central low point mark three distinct phases for the distribution:

• Infant Mortality Phase: This initial phase has a high failure rate that decreases over time. Early failures often result from manufacturing defects, poor-quality components, or installation errors. Products that survive this phase without failing are likely to perform reliably for a significant period, leading to the useful life phase.
• Useful Life Phase: In this middle phase, the failure rate stabilizes and remains relatively constant. This period represents the product’s optimal performance window, where failures occur randomly and are usually due to unforeseen stresses or accidental damage rather than inherent flaws.
• Wear-Out Phase: The final phase sees an increasing failure rate as the product ages. Components begin to wear out due to fatigue, corrosion, or other degradation processes. Maintenance becomes more critical, and the likelihood of failure grows the longer the product remains in use.

References

1. Kosznik-Biernacka, S. (2007). Makeham’s Generalised Distribution. Computational Methods in Science and Technology 13(2), 113-120 from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.552.4465&rep=rep1&type=pdf
2. Corderiro et al. The Beta Alpha Distribution. Online: http://www.est.ufmg.br/portal/arquivos/rts/Beta_Alpha_Distribution_RT_UFMG.pdf
3. El-Gohary, A et. al. (2013). The generalized Gompertz distribution. Applied Mathematical Modelling. Volume 37, Issues 1–2, January 2013, Pages 13–24