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.

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?.

• But for the sufficient Statistic, it holds both ways right? i.e, If T is sufficient statistic of theta and f is a one-one function then f(T) is a sufficient statistic of theta, and also f(T) is sufficient statistic of f(theta) , and T is also a sufficient statistic of f(theta). Commented Feb 6, 2019 at 3:35
• Yes, its right. Commented Feb 6, 2019 at 7:39