Gompertzlike

Gompertzlike describes models, curves, or distributions that resemble the Gompertz function, especially in the sense that a rate or hazard increases approximately exponentially with time. In the canonical Gompertz form, the hazard rate is h(t) = eta exp(b t). The corresponding survival function is S(t) = exp(-H(t)) with the cumulative hazard H(t) = (eta/b)(exp(b t) - 1) when b ≠ 0, and the cumulative distribution function is F(t) = 1 - S(t).

Gompertzlike models are used when empirical data show rapidly increasing risk with age or time, but may include extensions such as the Gompertz-Makeham model, which adds a constant baseline hazard to account for time-invariant risk. The term is often used descriptively rather than to denote a single official distribution.

Applications span demography, actuarial science, toxicology, oncology, and reliability engineering. In longevity research, Gompertz-like hazards capture mortality acceleration with age; in product reliability, they reflect rising failure rates over time. Estimation typically proceeds via maximum likelihood, with parameters interpreted in terms of initial risk and the rate of acceleration.

Model selection involves comparing Gompertzlike fits to alternatives such as the Weibull or log-normal, and diagnostics examine fit to the hazard and survivor curves. Limitations include sensitivity to sample size and potential identifiability issues when data do not strongly support exponential-time acceleration.