Wikipedia says
... consider the map $f:p_{\theta }\mapsto p_{T\,\mid\, \theta }$ which takes each distribution on model parameter $\theta$ to its induced distribution on statistic $𝑇$. The statistic $T$ is said to be complete when $f$ is surjective, and sufficient when $f$ is injective.
(emphasis mine)
Is this claim true? ie does $f:p_{\theta }\mapsto p_{T\,\mid\, \theta }$ being injective imply statistic $T$ is sufficient?
My research so far:
I think wikipedia is incorrect, as I can prove by counterexample. ie I can provide an example where $f:p_{\theta }\mapsto p_{T\,\mid\, \theta }$ is injective but $T$ is not a sufficient statistic.
Consider this conditional probability distribution of samples $X$ given parameters $\theta$, ie $p_{X\,\mid \,\theta}$:
(table 1)
| $\theta_1$ | $\theta_2 $ | |
|---|---|---|
| $x_1$ | $0.1 $ | $0.2 $ |
| $x_2$ | $ 0.2$ | $0.2 $ |
| $x_3$ | $0.3$ | $0.3 $ |
| $x_4$ | $ 0.4 $ | $0.3 $ |
and here is the map of samples $ X$ to statistic $T$, meaning that statistic $T$ calculated for sample $x_i$ (column 1) has value equal to $t_j$ (column 2) ie $T(x_i)=t_j$:
(table 2)
| sample | statistic |
|---|---|
| $x_1$ | $t_1$ |
| $x_2$ | $t_1$ |
| $x_3$ | $t_2$ |
| $x_4$ | $t_2$ |
which leads to the following conditional probability distribution of statistic $T$, given parameters $\theta$, ie $p_{T\,\mid\,\theta}$:
(table 3)
| $\theta_1$ | $\theta_2$ | |
|---|---|---|
| $t_1$ | $0.3$ | $0.4$ |
| $t_2$ | $0.7$ | $0.6 $ |
In this case $f:p_{\theta }\mapsto p_{T\,\mid\, \theta }$ is injective (can be deduced from table 3), but the statistic $ T$ is not sufficient, as for a given $T $ the conditional probability of $X $ is a function of $\theta$ (can be deduced form table 1 and table 2).
