# Is it more correct to say "bias of the standard error of the estimator" or "bias of the standard error of the estimate"

I understand an estimator is a "rule" (e.g., a function, say $$g$$) that produces an estimate ($$\hat\theta$$) of an estimand (say, a population parameter, $$\theta$$).

My question: is it technically correct to say "bias of the estimate" and "bias of the standard error of the estimate", or should we instead be saying "bias of the estimator" and "bias of the standard error of the estimator"?

• Note that if you are talking about the performance of an estimator of the standard error of an estimate, you would talk about the bias of the estimator of the standard error of the estimate, not the bias of the standard error of the estimate/estimator. Standard errors of estimators simply are what they are, but an estimator of them might be biased. For example, with heteroscedasticity, the usual OLS estimator of the standard error of the model coefficient estimates is biased, but the robust standard error is not.
– Noah
Sep 29 at 20:39
• Ah, I think I'm starting to understand. To confirm my understanding: I want to estimate the model across many simulated samples, and, in each sample, I obtain (1) the estimate of a model coefficient, and (2) the estimate of the standard error (SE) of the estimate of the coefficient. Then, I want to compare two things to each other: (1) the standard deviation of the distribution (i.e., distribution across samples) of the estimates of the coefficient, and (2) the average estimated SE across the samples. Thus, I guess this would be the bias of the estimator of the standard error of the estimate? Oct 4 at 17:39
• Yes exactly. I agree its confusing. Typically we use an estimator's estimated variance rather than standard error, as standard deviations in general are biased even when consistent.
– Noah
Oct 4 at 19:11
• @Noah Thank you very much. Oct 6 at 18:28

$$Bias(X) = E(\bar{X}) - \mu_X$$