Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Probability distributions ▷ Location parameter

Definition: Let $\mathcal{D}(\mu, \lambda)$ denote a family of probability distributions with statistical parameters $\mu$ and $\lambda$. Then, the parameter $\mu$ is referred to as a “location parameter”, if and only if the following holds: When a random variable $X$ is following the probability distribution $\mathcal{D}(\mu^{*}, \lambda^{*})$, then the random variable $Y = X-\mu^{*}$ is following the probability distribution $\mathcal{D}(0, \lambda^{*})$.

\[\label{eq:para-loc} X \sim \mathcal{D}(\mu^{*}, \lambda^{*}) \quad \Rightarrow \quad Y = X-\mu^{*} \sim \mathcal{D}(0, \lambda^{*}) \; .\]

This implies an identity of the cumulative distribution functions:

\[\label{eq:para-loc-cdf} \begin{split} F_Y(y) &= F_X(y+\mu^{*}) \\ F_\mathcal{D}(y; \, 0, \lambda^{*}) &= F_\mathcal{D}(y+\mu^{*}; \, \mu^{*}, \lambda^{*}) \; . \end{split}\]

The parameter $\mu$ can be scalar-, vector- or matrix-valued for random variables, vectors or matrices $X$.

 
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Metadata: ID: D217 | shortcut: para-loc | author: JoramSoch | date: 2025-03-14, 14:56.