What makes the Weibull distribution useful for modeling time-to-failure?

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AcadiFi · Accepted Answer

The Weibull distribution generalizes the exponential by allowing a non-constant hazard rate. Its CDF is F(t) = 1 - e^(-(t/eta)^beta), where beta is the shape parameter and eta is scale. When beta = 1, Weibull reduces to exponential. For beta > 1 the hazard increases with time (wear-out), beta < 1 means decreasing hazard (infant mortality).

Quillcrest Systems models payment processor failures using Weibull with beta = 2.3, eta = 18 months. Hazard at t = 12 months is (2.3/18)*(12/18)^1.3 = 0.075 per month; at t = 24 months it rises to 0.17 per month. This captures aging infrastructure risk that exponential cannot. The mean time to failure is eta * Gamma(1 + 1/beta) = 18 * 0.887 = 15.96 months.

In finance, Weibull fits loan prepayment timing, software system failures in trading platforms, and time-to-default when hazard changes with seasoning. Estimation uses MLE on observed failure/censoring data. The distribution connects to extreme value theory: maxima of Weibull samples converge to a Gumbel distribution. Diagnose misfit with Weibull probability plots; straight line suggests good fit, curvature indicates shape misspecification.

What makes the Weibull distribution useful for modeling time-to-failure?

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