Estimating the population variance

This article applies to the population. See also Estimating the Variance of an Estimator .

The maximum likelihood estimate is often used to estimate the variance of the population . The maximum likelihood estimate provides the uncorrected sample variance as an estimator of the unknown variance of the population , which, however, is only asymptotically true to expectations . An unbiased estimator, the corrected sample variance , is obtained by multiplying the uncorrected sample variance by the correction factor. ${\displaystyle 1/(n-1)}$

Variance estimation of a normally distributed population

Maximum likelihood estimation

Let be independently and identically distributed random variables from a normally distributed population with the unknown expected value and the unknown variance of the population . Let the realizations of the random variables be , then the likelihood function (also called plausibility function) is a sample with size ${\displaystyle X_{1},\ldots ,X_{n}}$ ${\displaystyle X_{i}\sim {\mathcal {N}}(\mu ,\sigma ^{2}),\;i=1,\ldots n}$ ${\displaystyle \mu }$ ${\displaystyle \sigma ^{2}}$ ${\displaystyle x_{1},\ldots ,x_{n}}$ ${\displaystyle n}$

${\displaystyle L(x_{1},\ldots ,x_{n}\mid \mu ,\sigma ^{2})=\prod _{i=1}^{n}{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\exp \left(-{\frac {(x_{i}-\mu )^{2}}{2\sigma ^{2}}}\right)=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left(-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}\right)}$

and the log likelihood function

${\displaystyle \log(L(x_{1},\ldots ,x_{n}\mid \mu ,\sigma ^{2}))=-{\frac {n}{2}}\log(2\pi \sigma ^{2})-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}}$.

In order to find an estimator for , the log-likelihood function is derived from ${\displaystyle {\hat {\sigma }}^{2}}$ ${\displaystyle \sigma ^{2}}$ ${\displaystyle \sigma ^{2}}$

${\displaystyle {\frac {\partial \log(L(x_{1},\ldots ,x_{n}\mid \mu ,\sigma ^{2}))}{\partial \sigma ^{2}}}=-{\frac {n}{2\sigma ^{2}}}+{\frac {1}{2\sigma ^{4}}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}}$

and set equal to zero to find a maximum

${\displaystyle 0=-{\frac {n}{2{\hat {\sigma }}^{2}}}+{\frac {1}{2{\hat {\sigma }}^{4}}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}\quad \Longrightarrow \quad {\hat {\sigma }}^{2}={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}}$

(for a derivation of the population variance in matrix notation, see classical linear model ). The second derivative results as

${\displaystyle {\frac {\partial ^{2}\log(L(x_{1},\ldots ,x_{n}\mid \mu ,\sigma ^{2}))}{\partial \sigma ^{2}\partial \sigma ^{2}}}={\frac {1}{\sigma ^{4}}}\left({\frac {n}{2}}-{\frac {\sum _{i=1}^{n}(x_{i}-\mu )^{2}}{\sigma ^{2}}}\right)}$

and at the point : ${\displaystyle \sigma ^{2}={\hat {\sigma }}^{2}}$

${\displaystyle {\frac {1}{{\hat {\sigma }}^{4}}}\left({\frac {n}{2}}-{\frac {\sum _{i=1}^{n}(x_{i}-\mu )^{2}}{{\hat {\sigma }}^{2}}}\right)={\frac {1}{{\hat {\sigma }}^{4}}}\left({\frac {n}{2}}-{\frac {n{\hat {\sigma }}^{2}}{{\hat {\sigma }}^{2}}}\right)=-{\frac {n}{2{\hat {\sigma }}^{4}}}<0}$,

d. H. it is a maximum if . ${\displaystyle {\hat {\sigma }}^{2}>0}$

Individual evidence

1. ↑ Jürgen Hedderich, Lothar Sachs : Applied Statistics: Collection of Methods with R. , p. 332.