Which type of estimator has a standard deviation that does not converge to 0 as the sample size increases, meaning more data does not make estimates more precise?

Which type of estimator has a standard deviation that does not converge to 0 as the sample size increases, meaning more data does not make estimates more precise?

• A. Unbiased
• B. Biased
• C. Consistent
• D. Inconsistent

Answer: (D)

This question targets the idea of estimator consistency and how precision improves with more data. An estimator is consistent when, as the sample size grows, its distribution concentrates around the true parameter value. That means both the bias must go to zero and the variance (hence the standard deviation) must shrink to zero. If the standard deviation does not approach zero as n increases, the spread of the estimates stays roughly the same regardless of how much data you collect, so the estimates do not converge to the true parameter. In other words, more data won’t make them more precise in the long run. This describes an inconsistent estimator. Note that unbiasedness alone doesn’t guarantee consistency—an estimator can be unbiased but still have variance that doesn’t vanish, and thus not be consistent.

This question targets the idea of estimator consistency and how precision improves with more data. An estimator is consistent when, as the sample size grows, its distribution concentrates around the true parameter value. That means both the bias must go to zero and the variance (hence the standard deviation) must shrink to zero. If the standard deviation does not approach zero as n increases, the spread of the estimates stays roughly the same regardless of how much data you collect, so the estimates do not converge to the true parameter. In other words, more data won’t make them more precise in the long run. This describes an inconsistent estimator. Note that unbiasedness alone doesn’t guarantee consistency—an estimator can be unbiased but still have variance that doesn’t vanish, and thus not be consistent.