📂Mathematical Statistics

Location Family

Definition

For a given cumulative distribution function $F$, suppose $F_{\theta}$ satisfies $F_{\theta} (x) = F \left( x - \theta \right)$ for all $x$.

$\left\{ F_{\theta} : \theta \in \mathbb{R} \right\}$ is referred to as a Location Family.

Example ¹

Considering a random sample $X_{1} , \cdots , X_{n}$ with parameter $\theta$ that possesses a cumulative distribution function $F_{0} (x) = F (x - 0) = F(x)$, the sample $Z_{1} , \cdots , Z_{n}$ can be expressed as $$X_{i} = Z_{i} + \theta$$. The length of the range as a statistical measure, $R = X_{n} - X_{(1)}$, should indeed be constant regardless of $\theta$. This is because $\theta$ merely increases or decreases the magnitude of values, not affecting their dispersion. In fact, the joint cumulative distribution function of $R$ is $$\begin{align*} F_{R} \left( r ; \theta \right) =& P_{\theta} \left( R \le r \right) \\ =& P_{\theta} \left( X_{(n)} - X_{(1)} \le r \right) \\ =& P_{\theta} \left( \max_{k} X_{k} - \min_{k} X_{k} \le r \right) \\ =& P_{\theta} \left( \max_{k} \left( Z_{k} + \theta \right) - \min_{k} \left( Z_{k} + \theta \right) \le r \right) \\ =& P_{\theta} \left( \max_{k} \left( Z_{k} \right) + \theta - \min_{k} \left( Z_{k} \right) - \theta \le r \right) \\ =& P_{\theta} \left( Z_{(n)} - Z_{(1)} \le r \right) \end{align*}$$. In other words, $R$ acts as an auxiliary statistic for $\theta$.

See Also

• Exponential Family
• Scale Family