# Invariance property

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.

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?.