Sufficient but not minimal statistic in $\mathbb{R}$?

Question

We know that sufficient statistic is not necessarily minimal. For example, consider a random sample $X_1,\cdots,X_n \sim \text{Bernoulli}(\theta) $. It can be shown easily that both $T'(X_1,\cdots,X_n) = (X_1,\cdots,X_n)$ and $T(X_1,\cdots,X_n) = \sum_{i=1}^n X_i$ are sufficient statistic. However, $T'$ is not minimal. This is not surprising because after all, $T'$ has "more dimensions" than $T$ , and thus achieve less data reduction. More specifically, range of $T'$ is in $\mathbb{R}^n$ and range of $T$ is in $\mathbb{R}$.

In general, we consider a random sample $X_1,\cdots,X_n \sim f_X (x\mid\theta)​$. Does there exist a sufficient statistic $S$ for $\theta$ such that it is sufficient, its range is in $\mathbb{R}$, but it is not minimal?

Answer 1

Consider a Normal sample $(x_1,\ldots,x_n)$ from ${\cal N}(0,\sigma^2)$. Then it is a standard result that the statistic$$S(x_1,\ldots,x_n)=\sum_{i=1}^n x_i^2$$is minimal sufficient. If I now define the statistic $$Z(x_1,\ldots,x_n)=\text{sign}(x_1)\sum_{i=1}^n x_i^2$$where$$\text{sign}(x_1)=\begin{cases} 1&\text{if }x_1\ge 0\\ -1&\text{if }x_1< 0\\\end{cases}$$ then $Z$ is also sufficient but not minimal sufficient (since the signs of the observations are ancillary).

Answer 2

Suppose $T$ and $T'$ are both sufficient statistics of dimension 1 in $\mathbb{R}$, and suppose without loss of generality, that $T$ is minimal sufficient but $T'$ is not.

Then there exists a function $g$ such that $T = g(T')$ but there is no function $f$ such that $T' = f(T)$ (otherwise $T'$ would be minimal sufficient). It follows that $g$ isn't invertible, so $g(x)$ is not unique for some $x \in \mathbb{R}$. Suppose now that $T'(X_1,...,X_n) = x$. Then $T = g(T')$ takes more than one value, and so (after a bit of hand-waving) $T$ cannot be a sufficient statistic. This is a contradiction, and so both $T$ and $T'$ are minimal.