A random variable $Y$ belongs to the location scale family if $F(Y)$ can be expressed as $$P(Y\le y)=F(y|\mu, \sigma)=\Phi\left(\frac{y-\mu}{\sigma}\right)$$
where
$\mu$ represents the location parameter, $\; \mu \in (-\infty,\infty)$
$\sigma$ represents the scale parameter, $\; \sigma \in [0,\infty)$
$\Phi$ represents the "standard" form of the cdf i.e. $\; (\mu, \sigma)=(0,1)$
Many distributions belong to the location-scale family
Normal
Smallest extreme value
Largest extreme value
Logistic
The statistical theory for the location-scale family is relatively simple
Statistical methods developed for the location-scale family can be applied to any member distribution
The more comfortable you become with location scale distributions, the better you will be at statistics
$T$ belongs to the log-location scale family if the cdf of $Y= \log(T)$ is a member of the location scale family $$P(T\le t)=F(t|\mu, \sigma)=\Phi\left(\frac{\log(y)-\mu}{\sigma}\right)$$
where
$\mu$ represents the location parameter, $\; \mu \in (-\infty,\infty)$
$\sigma$ represents the scale parameter, $\; \sigma \in [0,\infty)$
$\Phi$ represents the "standard" form of the cdf i.e. $\; (\mu, \sigma)=(0,1)$
Many distributions can belong to the log-location-scale family
Lognormal
Weibull
Loglogistic
Distributions in the log-location-scale family can be easily transformed to the location scale family using
$$Y = \log[T]$$
where
$T$ is a member of the log-location-scale family
$Y$ is a member of the location-scale family
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