A data scientist is fitting a parametric survival model to right-censored component lifetimes. She wants a distribution whose hazard can be written
h(t)=λ k t^{k−1} (λ>0, k>0)
so that covariate effects may be interpreted in either a proportional-hazards or an accelerated-failure-time (AFT) framework. She also needs the model to simplify to a constant-hazard process when k = 1. Which distribution and interpretation meet these requirements?
Weibull distribution; k = 1 indicates a constant hazard, reducing the model to an exponential distribution.
Gompertz distribution; k = 1 implies the hazard grows exponentially with time rather than remaining constant.
Log-logistic distribution; k = 1 produces a unimodal (peak-then-decline) hazard, not a constant one.
Log-normal distribution; k = 1 makes the hazard constant and turns the model into a Gaussian AFT model.
The hazard form h(t)=λ k t^{k−1} is specific to the Weibull distribution. The Weibull family is unique among common parametric survival models in that it supports both proportional-hazards and AFT parameterisations, letting analysts switch interpretive frameworks without changing the likelihood. When the shape parameter k equals 1, the t^{k−1} term cancels, the hazard becomes the constant λ, and the Weibull collapses to an exponential distribution.
The log-normal and log-logistic distributions do not yield the given hazard form and can only be used in the AFT (not proportional-hazards) framework; their hazard is non-monotonic and does not become constant when any single parameter equals one. The Gompertz model has hazard h(t)=λ e^{βt}; setting β or any other single parameter to one does not produce a constant hazard. Therefore, only the Weibull distribution with k = 1 satisfies all stated requirements.
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