The Weibull Distribution

Alpha decays. A quant strategy that works today might stop working next month as other traders discover the same edge, or it might keep working steadily for years. If you're trying to model when a strategy will stop being profitable — its "time to failure" — you need a distribution where the risk of failure isn't necessarily constant over time. The exponential distribution assumes a constant failure rate no matter how long something has already survived; real strategies (and real machines) usually don't behave that way. The Weibull distribution is built specifically to let the failure rate rise, fall, or stay flat, controlled by a single parameter.

The analogy: a lightbulb versus a bearing

A lightbulb has roughly constant risk of burning out each day it's already survived — day 500 isn't meaningfully riskier than day 50. A mechanical bearing wears out: the longer it's run, the higher its risk of failing soon, because of accumulated wear. The Weibull distribution captures both stories (and the opposite one — "infant mortality," where new components fail early but survivors get more reliable) with one shape knob: below 1 means decreasing risk over time, exactly 1 means constant risk (the exponential case), above 1 means increasing risk over time.

Writing it down

The Weibull distribution with shape $k>0$ and scale $\lambda>0$ has CDF

$$F(x; k, \lambda) = 1 - e^{-(x/\lambda)^k}, \qquad x \ge 0.$$

In words: the probability of having failed by time $x$ rises toward 1 as an exponential of $x$ raised to the power $k$, rescaled by $\lambda$. The instantaneous hazard rate — the risk of failing right now, given survival so far — is

$$h(x) = \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}.$$

In words: the hazard rate itself is a power function of time, and its exponent, $k-1$, is what determines whether risk is climbing ($k>1$), falling ($k<1$), or flat ($k=1$, which collapses the whole formula to the exponential distribution's constant hazard $1/\lambda$). This is the single most useful fact about Weibull: fit $k$ from data and you directly learn whether the thing you're modeling is "wearing in," "wearing out," or memoryless.

Try exponents below 1, at 1, and above 1 here — that's a direct stand-in for the Weibull hazard shape itself: a sub-1 power curve falls steeply then flattens (decreasing hazard — strategies that survive their fragile early months tend to keep going), while a super-1 power curve accelerates upward (increasing hazard — strategies that inevitably get arbitraged away by crowding).

Worked example 1: strategy survival with increasing hazard

A backtested signal's live "time to failure" (months until it stops being profitable) is modeled as $\text{Weibull}(k=2, \lambda=18)$ — increasing hazard, consistent with crowding risk. What's the probability it survives past 24 months? $P(X>24) = e^{-(24/18)^2} = e^{-1.778} \approx 0.169$ — under a 17% chance of lasting two years. Compare to the probability of surviving past 6 months: $e^{-(6/18)^2} = e^{-0.111} \approx 0.895$ — the strategy is likely fine in its first half-year but the rising hazard makes the two-year mark a real long shot, exactly the "gets more dangerous with age" story $k>1$ encodes.

Worked example 2: infant mortality in a new execution algorithm

A newly deployed execution algorithm's time-to-critical-bug is modeled as $\text{Weibull}(k=0.5, \lambda=100)$ (days) — decreasing hazard, typical of software with early bugs that get shaken out. Probability of failing within the first 10 days: $F(10) = 1-e^{-(10/100)^{0.5}} = 1-e^{-0.316}=1-0.729=0.271$ — a substantial 27% chance of an early failure. But conditional on surviving 60 days bug-free, the hazard has already dropped sharply (falling like $x^{-0.5}$), so the algorithm is now comparatively much safer — precisely the justification for a "burn-in" monitoring period before trusting new code with size.

Set the rate to zero and this curve is flat — a constant hazard, exactly the $k=1$ Weibull case (plain exponential decay). Push the rate away from zero and the curve's steepness change mimics what varying $k$ does to the Weibull hazard: the further from flat, the more the risk of failure is trending with age rather than staying constant.

What this means in practice

• Strategy decay modeling. Weibull's shape parameter directly answers "does this edge get riskier the longer it runs (crowding) or safer (proven robustness)" — a question a plain exponential "constant decay rate" model can't even ask.
• Hardware and infrastructure risk. Server and network hardware failure modeling in low-latency trading infrastructure is a classic reliability-engineering Weibull application.
• Extreme value connections. The Weibull distribution is one of the three possible limiting shapes (alongside Gumbel and Fréchet) for the maximum of many random variables — directly relevant to tail risk modeling. See Extreme Value Theory for Tail Risk.