# Difference between revisions of "TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Bura)/Übungen 2019W/5.4"

Line 7: | Line 7: | ||

What would have been the expectation if in <math>S^2</math> we had scaled with n instead of n − 1? | What would have been the expectation if in <math>S^2</math> we had scaled with n instead of n − 1? | ||

− | == Lösungsvorschlag von [[ | + | == Lösungsvorschlag von [[Benutzer:Ikaly|Ikaly]] == |

Copied from: https://en.wikipedia.org/wiki/Bias_of_an_estimator#Examples | Copied from: https://en.wikipedia.org/wiki/Bias_of_an_estimator#Examples |

## Revision as of 21:42, 11 November 2019

- Unbiasedness of the empirical variance

Let n ≥ 2 and be i.i.d. (independent and identically distributed) random varia- bles, with . Calculate the expectation of the empirical variance

- .

What would have been the expectation if in we had scaled with n instead of n − 1?

## Lösungsvorschlag von Ikaly

Copied from: https://en.wikipedia.org/wiki/Bias_of_an_estimator#Examples

### Sample variance

Vorlage:Main
The sample variance of a random variable demonstrates two aspects of estimator bias: firstly, the naive estimator is biased, which can be corrected by a scale factor; second, the unbiased estimator is not optimal in terms of mean squared error (MSE), which can be minimized by using a different scale factor, resulting in a biased estimator with lower MSE than the unbiased estimator. Concretely, the naive estimator sums the squared deviations and divides by *n,* which is biased. Dividing instead by *n* − 1 yields an unbiased estimator. Conversely, MSE can be minimized by dividing by a different number (depending on distribution), but this results in a biased estimator. This number is always larger than *n* − 1, so this is known as a shrinkage estimator, as it "shrinks" the unbiased estimator towards zero; for the normal distribution the optimal value is *n* + 1.

Suppose *X*_{1}, ..., *X*_{n} are independent and identically distributed (i.i.d.) random variables with expectation *μ* and variance *σ*^{2}. If the sample mean and uncorrected sample variance are defined as

then *S*^{2} is a biased estimator of *σ*^{2}, because

To continue, we note that by subtracting from both sides of , we get

Meaning, (by cross-multiplication) . Then, the previous becomes: