Bias, variance, MSE, and consistency

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Problem

For i.i.d. observations with mean $\mu$ and variance $\sigma^2$ , analyze the sample mean $\overline{X}_n$ as an estimator of $\mu$ : give its bias, variance, MSE, standard error, and whether it is consistent.

Reveal answer or reference solution

The bias is $0$ ; the variance and MSE are $\sigma^2/n$ ; and the standard error is $\sigma/\sqrt{n}$ . It is consistent because the variance tends to zero and the estimator is unbiased.

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Statistics Practice