I was wondering what the exact definition of this is. Apparently the exponential distribution is an example of such a family of densities. But what is the precise definition of a scale family? Also, I've heard of location and location-scale families in the same context and I'm not sure what these are either.


1 Answer 1


Definition 3.5.4 from Casella & Berger:

Let $f(x)$ be any pdf. Then for any $\sigma >0$, the family of pdfs $(1/\sigma) f(x/\sigma)$, indexed by the parameter $\sigma$, is called the scale family with standard pdf $f(x)$ and $\sigma$ is called the scale parameter of the family.

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    $\begingroup$ +1. I am surprised, though, that C&B chose such a narrow definition. We could let $F$ be any distribution, continuous or not, and define the associated scale family as the set of distributions given by $F(x/\sigma), \sigma\gt 0$. $\endgroup$
    – whuber
    Jan 4, 2017 at 17:20

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