Does κ = 0 recover the ordinary Weibull?

Yes, by taking the continuous limit of the stated survival function.

Is β the usual Weibull scale?

No. With scale λ in this convention, β = λ^−α.

Is every κ-deformed Weibull-shaped density this distribution?

No. The survival definition and derivative factor identify the construction used here.

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Kaniadakis Weibull distribution

Definition

The Kaniadakis Weibull distribution, or κ-Weibull, is a positive-valued probability model commonly defined by the survival function S(x) = expκ(−βx^α), with α and β positive. It generalizes the ordinary Weibull through the κ-exponential. As κ approaches zero, the survival becomes exp(−βx^α). For positive κ, the far right tail follows a power law. This survival-based construction must be distinguished from other κ-deformed densities that have a Weibull limit but different formulas away from κ = 0.

What it means

The survival function gives the probability that the random variable exceeds a threshold. The ordinary Weibull can describe several shapes of duration behavior through α, while κ-deformation modifies its tail. Here β is an inverse scale raised to the power α: if a conventional scale λ is desired, β = λ^−α. It is therefore not generally correct to call β the inverse of the mean or the inverse of a scale without stating the convention. The parameter κ changes both tail behavior and the relationship between parameters and summary statistics.

How it works

Use expκ(u) = exp[asinh(κu)/κ], with the ordinary exponential as its κ = 0 limit. For x ≥ 0, set F(x) = 1 − S(x). Differentiating gives f(x) = αβx^(α − 1) expκ(−βx^α)/√(1 + κ²β²x^(2α)). The hazard is the same expression without its survival factor: h(x) = αβx^(α − 1)/√(1 + κ²β²x^(2α)). In the common domain 0 < κ < 1, the survival tail is proportional to x^(−α/κ), so a positive moment of order r is finite only when r < α/κ. Fit and evaluate the complete distribution, including the observation mechanism.

Three practical examples

Duration analysis: explore a κ-Weibull model when positive durations have a heavier tail than a conventional Weibull describes.
Reliability research: compare hazard patterns while checking whether the proposed shape matches the physical failure process.
Simulation: generate positive scenarios using the model's quantile function to examine sensitivity to long-tail assumptions. These are possible analytical uses; observed fit and domain evidence determine whether the model is appropriate.

Think of it this way

Think of a curve showing how much of a population remains beyond each duration. An ordinary Weibull and its κ-deformed counterpart may look similar over part of the range, yet the latter can retain more probability far out. The survival curve itself must decrease, but the hazard asks a different question: the instantaneous rate conditional on having reached that point. Those two curves should not be interpreted interchangeably.

Worked example

Choose α = 2, β = 0.25 and κ = 0.5. At x = 2, βx^α = 1, so S(2) = exp₀.₅(−1) = (√1.25 − 0.5)², approximately 0.381966. Thus F(2) is approximately 0.618034. The hazard at 2 is [2 × 0.25 × 2]/√1.25, approximately 0.894427 per unit of x. To find the median, use lnκ(u) = (u^κ − u^−κ)/(2κ). Solving S(m) = 0.5 gives m = [lnκ(2)/β]^(1/α), approximately 1.681793. Substitution returns survival 0.5. At these parameters the survival tail exponent is α/κ = 4. The second moment exists, but the fourth moment diverges at the boundary r = 4. Neither the median calculation nor normalization implies all moments are finite.

Illustrative case study

Fictional case study: Pine Service Research compares disruption-duration models using observations that include ongoing, unfinished disruptions. The analysts fit a κ-Weibull with a likelihood that accounts for right censoring, rather than treating unfinished durations as completed events. They compare held-out survival predictions and investigate whether the estimated hazard shape is plausible. A model with a long tail changes scenario estimates, but the team reports uncertainty and sensitivity to the assumed observation process. It avoids presenting one fitted curve as a guaranteed forecast of rare future disruptions.

Common misconceptions and mistakes

Common mistakes include replacing the exponential in a Weibull density without adding the derivative factor required by this survival construction, using the wrong scale convention, and assuming the classical Weibull hazard rules transfer unchanged. For α > 1 and κ > 0, this hazard rises from zero but eventually declines, behaving like α/(κx) in the far tail. A classical Weibull with α > 1 instead has an increasing hazard. A heavier tail does not establish a specific physical mechanism. Sparse extremes and censoring can make tail parameters especially uncertain.

Business use

Businesses should evaluate this as one candidate among plausible duration or positive-outcome models. Clarify the units and whether data are censored, truncated or selected, and use validation that reflects the intended prediction horizon. Report the exact survival function, parameter convention and uncertainty around relevant quantiles or exceedance probabilities. Check moment existence before supplying average or variance-based summaries. When model choices lead to materially different decisions, present that sensitivity and seek additional evidence instead of selecting the most convenient tail estimate.

Check your understanding

Why is the example's fourth moment infinite when its variance is finite?

Answer: Positive moments require r < α/κ = 4. The second moment satisfies the strict inequality, but the fourth is on the divergent boundary.

Related terms

Weibull distribution
Survival function
Hazard rate
κ-exponential
right censoring
quantile function
power-law tail

Learn next

derive the density and hazard from the survival function, then compare them with the classical Weibull limit. Practice using the inverse κ-logarithm expression for quantiles, and study likelihood contributions from censored observations before fitting duration data.

Survival function

Sources

Sources: G. Kaniadakis, New power-law tailed distributions emerging in κ-statistics, https://arxiv.org/abs/2203.01743, section III (survival, density, hazard and quantile). D. T. Hristopulos et al., Finite-Size Effects on Return Interval Distributions for Weakest-Link-Scaling Systems, https://arxiv.org/abs/1308.1881 (κ-Weibull construction). Numerical example and fictional case are original illustrations.

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