Consider two estimators of
𝜃
.

𝜃
^
1
has expected value
𝜃
and variance
𝜃
𝑛
.

𝜃
^
2
has expected value
𝜃
and variance
𝜃
𝑛
2
.

Which estimator has smaller bias?

Question

Consider two estimators of 
𝜃
.

𝜃
^
1
has expected value 
𝜃
 and variance 
𝜃
𝑛
.

𝜃
^
2
has expected value 
𝜃
 and variance 
𝜃
𝑛
2
.

Which estimator has smaller bias?

BlackTom AI · Accepted Answer

To determine the bias of an estimator, we examine the expected value of the estimator relative to the true parameter value. An estimator is unbiased if its expected value equals the true parameter value. In this problem, both estimators $ \hat{\theta}_1 $ and $ \hat{\theta}_2 $ have expected values equal to $ \theta $, which is the true parameter value. Therefore, both estimators are unbiased, as their expected values do not deviate from $ \theta $. Bias is defined as the difference between the expected value of the estimator and the true parameter value, which is zero for both estimators in this case. Thus, they have the same bias, which is zero.

The variance of the estimators, $ \frac{\theta}{n} $ for $ \hat{\theta}_1 $ and $ \frac{\theta}{n^2} $ for $ \hat{\theta}_2 $, does not affect the bias but rather the precision of the estimators. A smaller variance indicates a more precise estimator, but it does not influence the bias. Therefore, the correct answer is 'They have the same bias.' The other options are incorrect because they either suggest a difference in bias that does not exist or imply insufficient information when the expected values clearly indicate unbiasedness.

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