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

  • 2
    $\begingroup$ 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. $\endgroup$
    – Xi'an
    Commented Mar 26, 2015 at 14:04
  • 1
    $\begingroup$ The question at stats.stackexchange.com/questions/32430 answers this one in the affirmative. $\endgroup$
    – whuber
    Commented Mar 26, 2015 at 14:54
  • 2
    $\begingroup$ 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 $\sigma^2$? $\endgroup$
    – cardinal
    Commented Mar 26, 2015 at 16:34

2 Answers 2


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.

  • $\begingroup$ Nice and simple example. Furthermore, we might remind CRLB with many unbiased estimators. $\endgroup$
    – kurtkim
    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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.