📂Mathematical Statistics

Scale Families

Definition

The cumulative distribution function $F$ is said to satisfy $F_{\sigma}$ for all $x$ if $F_{\sigma} (x) = F \left( x / \sigma \right)$ holds.

$\left\{ F_{\sigma} : \sigma > 0 \right\}$ is called a Scale Family.

Example ¹

Consider a random sample $X_{1} , \cdots , X_{n}$ with parameter $\sigma$ having a cumulative distribution function $F_{1} (x) = F ( x / 1) = F(x)$, then for the random sample $Z_{1} , \cdots , Z_{n}$ we can express $$X_{i} = \sigma Z_{i}$$ in this manner. If a statistic of this sample is a function of only $${{ X_{1} } \over { X_{n} }} , \cdots , {{ X_{n-1} } \over { X_{n} }}$$ then it’s an auxiliary statistic. It necessarily follows, because regardless of the scale parameter $\sigma$, the ratios of that random sample will cancel each other out in numerator-denominator. Indeed, the joint cumulative distribution of these ratios $$\begin{align*} F \left( y_{1} , \cdots , y_{n} ; \sigma \right) =& P_{\sigma} \left( {{ X_{1} } \over { X_{n} }} \le y_{1} , \cdots , {{ X_{n-1} } \over { X_{n} }} \le y_{n-1} \right) \\ =& P_{\sigma} \left( {{ \sigma Z_{1} } \over { \sigma Z_{n} }} \le y_{1} , \cdots , {{ \sigma Z_{n-1} } \over { \sigma Z_{n} }} \le y_{n-1} \right) \\ =& P_{\sigma} \left( {{ Z_{1} } \over { Z_{n} }} \le y_{1} , \cdots , {{ Z_{n-1} } \over { Z_{n} }} \le y_{n-1} \right) \end{align*}$$ does not depend on $\sigma$.

See Also

• Exponential Family
• Location Family