# More than one unbiased estimator for a single unknown parameter?

Is it possible to have more than one unbiased estimator for a single unknown parameter?If "Yes" then how and if "No" then why?

• Add an unbiased estimator of zero to a first unbiased estimator and you will get another unbiased estimator. Hence the lack of unicity of unbiased estimators. Commented Mar 26, 2015 at 14:04
• The question at stats.stackexchange.com/questions/32430 answers this one in the affirmative.
– whuber
Commented Mar 26, 2015 at 14:54
• Somewhat more concretely: Consider a random sample $X_1,\ldots,X_n$ such that each $X_i \sim \mathcal N(\mu,1)$. What is the expected value of the estimator $\hat \mu_1 = X_1$? What about the alternative estimator $\hat \mu_2 = X_2$? Can you construct any other estimators of $\mu$ from the sample? How many can you find? What if the sample were instead of $\mathcal N(0,\sigma^2)$ random variables and you wanted an unbiased estimator of $\sigma^2$? Commented Mar 26, 2015 at 16:34

As an example, from a i.i.d. sample of (finite) size $n$, where the common mean is $\mu \neq 0$ we can have an infinite (and not even countably) number of unbiased estimators of the form

$$\hat \mu(a) = aX_i + (1-a)X_{j}, \;i\neq j, \;a \in \mathbb R$$

The number of estimators is uncountably infinite because $\mathbb R$ has the cardinality of the continuum.

And that's just one way to obtain so many unbiased estimators.

• Nice and simple example. Furthermore, we might remind CRLB with many unbiased estimators. Commented Feb 24, 2023 at 9:22

To use an even simpler example, take an i.i.d. sample of size $n$ with mean $\mu$. Take the value of the first observation $X_1$ as an estimator for the mean. We have: $$\mathbb{E} X_1 = \mu$$ So the estimator is unbiased.

Now take the second observation $X_2$ as an estimator: $$\mathbb{E} X_2 = \mu$$ So this estimator is unbiased as well. You can check that unbiasedness for the mean holds for all convex combinations between observations.