Invariance property

Question

I am a bit confused regarding what exactly is the invariance property of sufficient estimators, consistent estimators and maximum likelihood estimators. As far as I know,

1. Invariance property of consistent estimators is : If $T$ is a consistent estimator of $\theta$, and $f$ is a continuous function then $f(T)$ is a consistent estimator of $f(\theta)$.

2. Invariance property of sufficient estimators is : If $T$ is sufficient estimator of $\theta$ and $f$ is one-one, onto function then $f(T)$ is sufficient estimator of $f(\theta)$, also $f(T)$ is sufficient estimator of $\theta$, and $T$ is sufficient estimator of $f(\theta)$.

3. Invariance property of maximum likelihood estimators(MLE) is : If $T$ is a MLE of $\theta$, and $f$ is a continuous/ one-one, onto function then $f(T)$ is a MLE of $f(\theta)$.

Please correct me if I am wrong somewhere and please tell me the least I need to check for it as I am appearing for a competitive exam where time really matters.

Answer 1

These are somewhat different properties , and throwing them together under the same name mostly is confusing. But:

1. This is correct, but I have never before seen this named invariance.

2. Confused. We talk about sufficient statistics not estimators. Of course an estimator is a statistic, but $T$ being sufficient statistic (in a model parametrized by $\theta$, or for $\theta$) says in itself not that $T$ is a good estimator! If $T$ is sufficient, and also a good estimator for $\theta$, then still $1000000 T$ is sufficient, but maybe not a good estimator (for $\theta$.) Of your requirements for $f$ then one-to-one is essential, onto is unnecessary. Then we can reformulate: If $T$ is sufficient for $\theta$ and $f$ is one-one, then $f(T)$ is also sufficient for $\theta$.

3. Mostly right, but you don't need all those conditions on $f$. See Invariance property of maximum likelihood estimator?.