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 a\in\mathbb{R},\\ T(F_{aX})&=aT(F_X),&&\forall a\neq0;\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 b\in\mathbb{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?