If we have a minimal sufficient statistic for a parameter, is this also a minimal sufficient statistic for any function of the parameter?

Question

If we know that $\theta^{*}$ is a minimal sufficient statistic for a certain parameter $\theta$ , is it true that $\theta^{*}$ will also be a minimal sufficient statistic for any one-to-one function of $\theta$, $f(\theta)$?

Answer 1

Denote $\eta = f(\theta)$. $f$ is one-one.

For any sufficient statistic $T$ of $\eta$ we have

\begin{align*} T(x)=T(y)\implies\delta_\eta(x)&=\delta_\eta(y) \\ \implies\delta_{f(\theta)}(x)&=\delta_{f(\theta)}(y)\\ \implies\delta_\theta(x)&=\delta_\theta(y)\\ \end{align*}

(The last $\implies$ holds because we can treat $\delta$ a function of $\theta$)

Thus $T$ is sufficient for $\theta$. Similarly, $\theta^*$ is sufficient for $\eta$.

Since $\theta^*$ is a minimal sufficient statistic of $\theta$ and $T$ sufficient for $\theta$, $\theta^*$ is a function of $T$.

This means for any $T$ sufficient for $\eta$, and $\theta^*$, being sufficient for $\eta$, is a function of $T$, thus $\theta^*$ is minimal sufficient for $\eta$.