Let $(X_1,\ldots,X_n)$ be a random vector, whose distribution involves unknown: location parameter $\mu$ and a scale parameter $\sigma>0$. It follows, that any measurable function $f(X_1,\ldots,X_n)$, satisfying condition: $$ f(\frac{X_1-a}{b},\ldots,\frac{X_n-a}{b})=f(X_1,\ldots,X_n) \quad \forall a\in \mathbb{R}\quad\land b>0, $$ is ancillary statistic for the unknown parameter $(\mu,\sigma)$. How to prove the above implication? This statement appears in: "On statistics independent of a complete sufficient statistics", D. Basu.
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You say that $X$ has some parametric distribution, which is a location-scale family. Then aspect of the distribution which is not influenced by the location or scale is said to measure some aspect of the distributions shape. By transforming $$ X_i \mapsto \frac{X_i-a}{b} $$ you changes $X$ to have some distribution from the same family, only with location and scale changed. Your requirement is that this should not change the value of $f$. Then compare to the definition of ancillary statistic.
answered Jul 5 at 15:23
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$\begingroup$ Ok, so starting from $X$, which has some distribution $\mathbb{P_{\mu,\sigma}}$, or starting with $Y=\frac{X-a}{b}$, which has some distribution $\mathbb{P_{\mu',\sigma'}}$, does not change value of function $f$. But i further can't see, that the distribution of this function will be the same. Could you give me some hint about how i can show it analytically? $\endgroup$ – Mentossinho Jul 5 at 21:17
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$\begingroup$ @Mentossinho: If its value does not change by this transformation, and that is true for all $x$, how can its distribution change? $\endgroup$ – kjetil b halvorsen Jul 21 at 20:27
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1$\begingroup$ i think i've got it: if $X$ has distribution $\mathbb{P_{\mu,\sigma}}$, and $Y=b^{-1}X-a \mathbb{1}$ has distribution $\mathbb{P_{\mu',\sigma'}}$, then $\mathbb{P_{\mu',\sigma'}}(\{f(Y)\in B\})=\mathbb{P_{\mu,\sigma}}(\{f(b^{-1}X-a)\}\in B)=\mathbb{P_{\mu,\sigma}}(\{f(X)\in B\})$ for every measurable $B$ from range of $f$. $\endgroup$ – Mentossinho Jul 21 at 22:49
