2

In my textbook for my statistics class, it says that $s^2$, sample variance is a "unbiased estimator" for population variance, $\sigma^2$. Does this mean that when we use $s^2$ as a point estimator for $\sigma^2$, it precisely equals $\sigma^2$? So $s^2$ is not even an estimation/approximation for $\sigma^2$? What does unbiased estimator mean?

Thank You!

  • 1
    No, it may not be precisely equal to $\sigma^2$. $s^2$ is a random variable whose value depends on the sample. "Bias" of a statistic $X$ that is used to estimate some parameter $\theta$ is given $\Bbb E[X-\theta]$. When bias is zero, we say the estimator is unbiased. "Unbiased" means that the expectation of $s^2$ over all possible samples is equal to $\sigma^2$. – Shubham Johri Feb 17 '21 at 15:27

1 Answers1

2

$S^2$ is unbiased estimator for the population variance $\sigma^2$ because, as per definition

$$\mathbb{E}[S^2]=\sigma^2$$

there are other and most important properties of an estimator, i.e. consistency, sufficiency, efficiency, etc etc.

tommik
  • 32,733
  • 4
  • 15
  • 34
  • Ah, so an unbiased estimator is good because if there's bias then the expected value of the point estimate will be an approximation of the parameter, I.E. there's more error to account for, correct? Thus, it's better to have unbiased estimators when possible, correct? –  Feb 17 '21 at 15:41
  • 1
    @BillBillwater : yes of course but there are other important properties to say that an estimator is a good one. The most important property is Sufficiency. As an example, an estimator can be biased but with a correctable bias. Consider that most of the ML estimators are biased but they are very good estimators because they have other important properties – tommik Feb 17 '21 at 15:44
  • ok, I see, I guess I will learn more about properties like that in grad school or maybe further into my current course. Thank You! –  Feb 17 '21 at 15:46