I am wondering what "Minimizing the variance" in Statistics mean. As far as I understand that the variance is one of the parameters of a probability distribution, for example, a normal distribution and it is a measure of how spread the data points are from the mean. However, what is there between "Minimizing the variance" and the spread of data points?
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asked Apr 18, 2021 at 10:57
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3$\begingroup$ A statistical estimator is a random variable, hence is endowed with a variance of its own. $\endgroup$– Xi'anCommented Apr 18, 2021 at 12:41
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1 Answer
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A common task in statistics is to perform parameter estimation.
The parameter estimator is a function of the sample.
For example, consider the following two parameter estimator for the mean of normal distribution:
$\hat{\mu}_1=\frac{X_1+X_2}{2}$
$\hat{\mu}_2=X_1$
We have $Var(\hat{\mu}_1)=\frac{\sigma^2}{2}$ but $Var(\hat{\mu}_2)=\sigma^2$. Even though both are unbiased estimator of the mean, we would prefer the first estimators.
Given a few unbiased estimators, we would prefer the one with the smallest variance.
answered Apr 18, 2021 at 11:17