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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"?

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    $\begingroup$ 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. $\endgroup$
    – Noah
    Sep 29 at 20:39
  • $\begingroup$ 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? $\endgroup$
    – moses.rivera100
    Oct 4 at 17:39
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    $\begingroup$ 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. $\endgroup$
    – Noah
    Oct 4 at 19:11
  • $\begingroup$ @Noah Thank you very much. $\endgroup$
    – moses.rivera100
    Oct 6 at 18:28

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You should say "bias of the estimator" because bias is defined in expectation for the estimator which is a random variable.

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

It is incorrect to talk about the "bias of an estimate."

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  • $\begingroup$ In that case, is it also true that we should say "bias of the standard error of an estimator" (rather than "of an estimate")? $\endgroup$
    – moses.rivera100
    Sep 29 at 20:18
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    $\begingroup$ @mosesrivera100 Standard errors are properties of the estimator rather than of the estimate. But what would you mean by "bias of the standard error of the estimator"? The standard error of an estimator is typically defined as the square root of the expected square of the error (which is the standard deviation of the estimator when the estimator is unbiased). $\endgroup$
    – Henry
    Sep 29 at 22:58
  • $\begingroup$ @Henry Thank you. By "bias of the standard error of the estimator" I meant the "standard error of a distribution of {estimates generated by the estimator}" (curly braces added to help with interpretation). For example, I want to conduct a Monte Carlo simulation in which an estimator's estimate is computed across many simulated samples. Then, I want to evaluate the standard error of that distribution of estimates. I'm not sure whether I should call that the "standard error of the estimates" (plural) or, alternatively, the "standard error of the estimator". $\endgroup$
    – moses.rivera100
    Oct 4 at 17:17

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