Which statement defines a consistent estimator?

Which statement defines a consistent estimator?

• A. The estimator is unbiased for all n.
• B. The estimator's variance goes to 0 as n increases.
• C. The standard deviation of the estimator converges to 0 as n increases.
• D. The estimator's bias remains fixed as n grows.

Answer: (C)

Consistency means the estimator converges in probability to the true parameter as the sample size grows. A clear way this happens is that the estimator’s sampling distribution becomes more tightly packed around the parameter value as n increases. If the standard deviation of the estimator goes to zero, the spread of its distribution shrinks to a point. When the estimator is unbiased (its mean equals the true parameter) this shrinking spread centers right on the parameter, so the estimator concentrates at the true value as n grows, i.e., it is consistent. This makes the statement about the standard deviation going to zero the best fit among the options: it directly reflects the increasing precision with larger samples, which is core to consistency. The other ideas don’t guarantee consistency on their own: being unbiased for all n doesn’t ensure the variability vanishes; variance going to zero helps but doesn’t guarantee the center aligns with the true parameter; and having a fixed bias means the estimator continues to miss the true value regardless of how tight the spread becomes.

Consistency means the estimator converges in probability to the true parameter as the sample size grows. A clear way this happens is that the estimator’s sampling distribution becomes more tightly packed around the parameter value as n increases. If the standard deviation of the estimator goes to zero, the spread of its distribution shrinks to a point. When the estimator is unbiased (its mean equals the true parameter) this shrinking spread centers right on the parameter, so the estimator concentrates at the true value as n grows, i.e., it is consistent.

This makes the statement about the standard deviation going to zero the best fit among the options: it directly reflects the increasing precision with larger samples, which is core to consistency. The other ideas don’t guarantee consistency on their own: being unbiased for all n doesn’t ensure the variability vanishes; variance going to zero helps but doesn’t guarantee the center aligns with the true parameter; and having a fixed bias means the estimator continues to miss the true value regardless of how tight the spread becomes.