Survival Models based on Log-Lifetime

Definition

Assume that the log survival time $T$ can be modeled with Log-Linear Model $Y=\ln T=\sigma W+\alpha$ where $\sigma$ is the Scale Parameter, $\alpha$ is the Location Parameter, and $W$ is some well-known distribution.

Kinds

Standard Gumbel Distribution

If $W$ follows the standard Gumbel distribution (extreme value distribution), then $f(y)=\frac{1}{\sigma }\exp \left[(\frac{y-\alpha }{\sigma })-\exp (\frac{y-\alpha }{\sigma })\right]$ $f(t)=\frac{1}{\sigma t}\exp \left[(\frac{\ln t-\alpha }{\sigma })-\exp (\frac{\ln t-\alpha }{\sigma })\right]$ If $\alpha =-\ln \lambda$ and $\sigma =\frac{1}{\rho }$, then $f(t)=\rho \lambda (\lambda t{)}^{\rho -1}\exp [-(\lambda t{)}^{\rho }]$ follows a Weibull Distribution.

Normal Distribution

If $W$ follows the standard normal distribution, then $T$ follows log-normal distribution $f(t)=\frac{1}{\sqrt{2\pi }}\frac{\rho }{t}\exp \left[-\frac{{\rho }^2}{2}(\ln \lambda t{)}^2\right]$ where $\alpha =-\log \lambda$ and $\sigma =\frac{1}{\rho }$

Logistic Distribution

If $W$ follows a Logistic Distribution, then $T$ follows Log-Logistic Distribution $f(y)=\frac{1}{\sigma }\frac{\exp (\frac{y-\alpha }{\sigma })}{[1+\exp (\frac{y-\alpha }{\sigma }){]}^2}$ $f(t)=\lambda \rho \frac{(\lambda t{)}^{\rho -1}}{[1+(\lambda t{)}^{\rho }{]}^2}$ where $\alpha =-\ln \lambda$ and $\sigma =\frac{1}{\rho }$.

Gumbel Distribution

If $W$ follows a Gumbel Distribution (generalized extreme value distribution), then $f(t)=\lambda \rho (\lambda t{)}^{\rho k-1}\frac{\exp [-(\lambda t{)}^{\rho }]}{\Gamma (k)}$ where $\alpha =-\ln \lambda$ and $\sigma =\frac{1}{\rho }$.

Special Cases

If $\rho =1$ then $T\sim \Gamma (k,\lambda )$ If $k=1$ then $T\sim \mathrm{Weibull}(\lambda ,k)$ If $\rho =k=1$ then $T\sim \mathrm{Exp}(\lambda )$ If $k\to \infty$ then $T\sim \mathrm{Lognormal}(\mu ,{\sigma }^2)$

Exponential-F Distribution

If $e^W$ follows a F-Distribution $F(2m,2n)$, then $T$ follows a generalized F-distribution.

Special Cases

If $m=n=1$ then $T$ follows Log-Logistic Distribution If $n\to \infty$ then $T$ follows generalized gamma distribution If $m,n\to \infty$ then $T$ follows Log-Normal Distribution