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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.

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    $\begingroup$ If the limiting distribution is normal and the limiting variance is positive, it cannot happen. So an example will have to be fairly contrived. $\endgroup$
    – kjetil b halvorsen ♦
    Commented May 30, 2016 at 13:59
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    $\begingroup$ 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. $\endgroup$
    – 5ayat
    Commented May 30, 2016 at 15:27

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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.

This quote from the Wikipedia page may help remove some confusion:

Bias is related to consistency as follows: a sequence of estimators is consistent if and only if it converges to a value and the bias converges to zero. Consistent estimators are convergent and asymptotically unbiased (hence converge to the correct value): individual estimators in the sequence may be biased, but the overall sequence still consistent, if the bias converges to zero. Conversely, if the sequence does not converge to a value, then it is not consistent, regardless of whether the estimators in the sequence are biased or not.

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.

The requirement for convergence means that the estimator must get arbitrarily close to to the true value as sample size increases. That would seem to require that the estimator "grow less variable with increased sample size," for any reasonable definition of "less variable."

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  • $\begingroup$ Where is the example the OP was asking for? Does it not exist? I conjecture so based on your last sentence. $\endgroup$
    – Richard Hardy
    Commented Nov 2, 2017 at 9:41
  • $\begingroup$ @RichardHardy I'm not aware of "an example of a consistent estimator that doesn't grow less variable with increased sample size," but I haven't proven that such an example is impossible. $\endgroup$
    – EdM
    Commented Nov 2, 2017 at 13:40

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