# Example of a consistent estimator that doesn't grow less variable with increased sample size?

I've had it asserted to me that any consistent estimator must necessarily also grow less variable with increased sample size.

I felt that this couldn't be correct, since there was nothing in the definition of a consistent estimator that forced this to be so, and the class of possible estimators is infinite.

What's an example of a consistent estimator that doesn't grow less variable with increased sample size?

I'd prefer the example be an estimator that's commonly used, rather than one contrived just to meet the demands of the question.

• If the limiting distribution is normal and the limiting variance is positive, it cannot happen. So an example will have to be fairly contrived. May 30, 2016 at 13:59
• As a thought experiment, I'd actually be very interested to see an example of when this might be the case -- no matter how contrived this example is. I came up with a few scenarios where variance would be expected to increase with additional observations, for some small samples, but all these examples had to do with how the population was sampled, not the actual estimator. May 30, 2016 at 15:27

The common meaning of "consistency" and its technical meaning are different. See this page for some discussion. Also, as noted by @hejseb in a comment on another answer here, lack of bias and consistency are not the same.

The estimator for the mean of a sequence proposed in another answer here: $$X_1 + \frac{1}{n}$$ thus is not consistent because it does not converge to the true value of the mean as the number of observations increases.