Table 1.

Mortality models belonging to the location–scale (LS) and log–location–scale (LLS) families and models closely related to them, together with their parameterization in terms of the classic, LS and LLS force of mortality $\mu (x)$, ${\mu }_{\mathrm{LS}}(x)$, ${\mu }_{\mathrm{LLS}}(x)$, location u and scale c parameters

Models belonging to the LS family	$\mu (x)$	$\frac{1}{c}{\mu }_{\mathrm{LS}}\left(\frac{x-u}{c}\right)$	u	c
Logistic	$\frac{b\text{exp}(a+bx)}{1+\text{exp}(a+bx)}$	$\frac{1}{c}\frac{\text{exp}(\frac{x-u}{c})}{1+\text{exp}(\frac{x-u}{c})}$	$-\frac{a}{b}$	$\frac{1}{b}$
Normal	$\frac{1}{\sigma }\frac{\varphi \left(\frac{x-\lambda }{\sigma }\right)}{1-\mathrm{\Phi }\left(\frac{x-\lambda }{\sigma }\right)}$	$\frac{1}{c}\frac{\varphi \left(\frac{x-u}{c}\right)}{1-\mathrm{\Phi }\left(\frac{x-u}{c}\right)}$	$\lambda$	$\sigma$
Smallest Extreme–Value	$\frac{1}{\sigma }\text{exp}(\frac{x-\lambda }{\sigma })$	$\frac{1}{c}\text{exp}(\frac{x-u}{c})$	$\lambda$	$\sigma$
Largest Extreme–Value	$\frac{1}{\sigma }\frac{\text{exp}(-\frac{x-\lambda }{\sigma })}{\text{exp}\left[\text{exp},\left(-,\frac{x-\lambda }{\sigma }\right)\right]-1}$	$\frac{1}{c}\frac{\text{exp}(-\frac{x-u}{c})}{\text{exp}\left[\text{exp},\left(-,\frac{x-u}{c}\right)\right]-1}$	$\lambda$	$\sigma$

Models belonging to the LLS family	$\mu (x)$	$\frac{1}{cx}{\mu }_{\mathrm{LLS}}\left(\frac{ln(x)-u}{c}\right)$	u	c
Weibull	$ab{(ax)}^{b-1}$	$\frac{1}{cx}\text{exp}\left(\frac{ln(x)-u}{c}\right)$	$-ln(a)$	$\frac{1}{b}$
Log–Logistic	$\frac{b}{x}\frac{\text{exp}(a+bln(x))}{1+\text{exp}(a+bln(x))}$	$\frac{1}{cx}\frac{\text{exp}\left(\frac{ln(x)-u}{c}\right)}{1+\text{exp}\left(\frac{ln(x)-u}{c}\right)}$	$-\frac{a}{b}$	$\frac{1}{b}$
Log–Normal	$\frac{1}{\sigma x}\frac{\varphi \left(\frac{ln(x)-\lambda }{\sigma }\right)}{1-\mathrm{\Phi }\left(\frac{ln(x)-\lambda }{\sigma }\right)}$	$\frac{1}{cx}\frac{\varphi \left(\frac{ln(x)-u}{c}\right)}{1-\mathrm{\Phi }\left(\frac{ln(x)-u}{c}\right)}$	$\lambda$	$\sigma$

Models related to the LS family	$\mu (x)$	LS-like $\mu (x)$	u	c
Gompertz	$a\text{exp}(bx)$	$\frac{1}{c}\text{exp}\left(\frac{x-u}{c}\right)$	$\frac{1}{b}ln\left(\frac{b}{a}\right)$	$\frac{1}{b}$
Gamma–Gomp	$\frac{a\text{exp}(bx)}{1+\frac{a}{b}\gamma \left[\text{exp},(,b,x,),-,1\right]}$	$\frac{1}{c}\frac{\text{exp}\left(\frac{x-u}{c}\right)}{1+\gamma \left[\text{exp},\left(\frac{x-u}{c}\right),-,,\text{exp},\left(-,\frac{u}{c}\right)\right]}$	$\frac{1}{b}ln\left(\frac{b}{a}\right)$	$\frac{1}{b}$
Kannisto	$\frac{\text{exp}(a+bx)}{1+\text{exp}(a+bx)}$	$\frac{\text{exp}(\frac{x-u}{c})}{1+\text{exp}(\frac{x-u}{c})}$	$-\frac{a}{b}$	$\frac{1}{b}$
Minimal Generalized Extreme–Value	$\frac{1}{\sigma }{\left[1,+,\xi ,\left(-,\frac{x-\lambda }{\sigma }\right)\right]}^{-\frac{1}{\xi }-1}$	$\mu (x)$	$\lambda$	$\sigma$
Maximal Generalized Extreme–Value	$\frac{1}{\sigma }\frac{\text{exp}\left(-,s^{-\frac{1}{\xi }}\right)s^{-\frac{1}{\xi }-1}}{1-\text{exp}\left(-,s^{-\frac{1}{\xi }}\right)}$	$\mu (x)$	$\lambda$	$\sigma$
	where $s=1+\xi \left(\frac{x-\lambda }{\sigma }\right)$

$\varphi (z)=\frac{1}{\sqrt{2\pi }}\text{exp}\left(-,\frac{z^2}{2}\right)$ and $\mathrm{\Phi }(z)={\int }_{-\infty }^z\varphi (w)\mathrm{d}w$ denote the probability density function and the cumulative distribution function of the standardized normal distribution, respectively