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I'm trying to think of an example of a consistent estimator with var=0 but I'm having a hard time. I pretty sure it exists but I'm unable to find one.

Any help\hints would be much appreciated.

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anything with a var=0 is a constant and is not a random variable. An estimator will always have a variance larger than zero, because you don't know the real parameter, and you only have a limited number ob observation, which you use to estimate the parameter. If you knew the real parameter, it would be a constant and have a variance 0.

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