# Solved – the mathematical definition of location / scale / shape parameters

I am trying to understand the exact definition of the location / scale / shape parameters (e.g. \$a\$ is called the shape parameter and \$c\$ is scale parameter in Pareto Type I). But the books I referred to (The Cambridge Dictionary of Statistics, HMC's Introduction to Mathematical Statistics, Feller's An Introduction to Probability Theory and its Applications, etc) only (seemingly) provided descriptive definitions for these parameters (location parameter are called centering parameter in Feller's). Wikipedia provided definitions in terms of cdf and pdf but without any sources given.

Based on the concepts in non-parametric statistics (say Ch.10 of HMC) I suspect the location / scale / shape parameters can be defined as the following:

Let \$X\$ be a random variable with cdf \$F_X\$. A parameter \$theta=T(F_X)\$, where \$T\$ is a functional, is a location
parameter if begin{align*}T(F_{X+a})&=T(F_X)+a,&&forall ainmathbb{R},\
T(F_{aX})&=aT(F_X),&&forall aneq0;end{align*} and it is a scale parameter
if begin{align*}T(F_{aX})&=aT(F_X),&&forall a>0,\
T(F_{X+b})&=T(F_X),&&forall binmathbb{R},\ T(F_{-X})&=T(F_X);end{align*} and it is a shape parameter if it is neither
location nor scale.

Am I correct? Or did I confused some unrelated concepts?

Contents

It is often true that these correspond to (some function of) the first, second and third moment as noted by @GuðmundurEinarsson. However, there are exceptions: For example for a Cauchy distribution Evans, Hastings, and Peacock (2000) call the first parameter a location parameter, but it represents the median instead of the mean. The mean is not even defined for a Cauchy distribution.

A more encompasing but less precise description would be:

• the location parameter shifts the entire distribution left or right
• The scale parameter compresses or stretches the entire distribution
• the shape parameter changes the shape of the distribution in some other way.

Merran Evans, Nicholas Hastings, and Brian Peacock (2000) Statistical Distributions, third edition. Wiley.

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