Location Model

Definition

Location Functional

Let $X$ be continuous Random Sample with CDF $F_X$. If $T(F_X)$ satisfies the following, then it is called location functional

• $\forall a\in \mathbb{R},Y=X+a\Rightarrow T(F_Y)=T(F_X)+a$
• $\forall a\ne 0,Y=aX\Rightarrow T(F_Y)=aT(F_X)$

Location Model

Let ${\theta }_X=T(F_X)$ be a location functional, then $X_1,X_2,\dots ,X_n$ is called to follow a location model with ${\theta }_X=T(F_X)$ if

• $X_i={\theta }_X+{ϵ}_i$, where ${ϵ}_i$‘s are i.i.d. with PDF $f(x)$ and $T(F_{ϵ})=0$

Facts

Mean functional is a locational functional

Let $X$ be Random Variable with CDF $F_X$ and PDF $f_x$ which is symmetric about $a$. If $T(F_X)$ is a location functional, then $T(F_X)=a$