- Valid interval estimation typically requires a method of estimating
the variance of the point estimator or .
- Let
represent
the true variance of a point estimator , and let
represent an estimator of
based on the data
.
- Suppose that

where*B*is called the bias in the variance estimator. - It is desirable to have

in which case is said to be an unbiased estimator of variance, . - If it is an unbiased estimator, the statistic

is approximatly*t*distribution with some degree*f*of freedom. - An approximate
confidence interval for
is given by

- This relation involves three parameters, estimator for mean,
estimator for variance, and the degree of freedom. How to determine
these values?
- Estimator for mean is calculated as above as a point estimator

- Estimator for the variance and for the degree of freedom
has to consider two separate cases
- If s are statistically independent observations
then use

to calculate

with the degree of freedom*f = n - 1*. - If s are not statistically independent, then the
above estimator for variance is biased. s is an autocorrelated
sequence, sometimes called a time series.
In this case,

i.e. one needs to calculate co-variance for every possible pair of observations. Too expensive.If the simulation is long eough to have passed the transient phase, the output is approximately

*covariance stationary*. That is depends on in the same way as depends onFor a covariance stationary time series s, define the lag

*k*autocovariance by

For k = 0, becomes the population variance

The lag

*k*autocorrelation is the correlation between any two observations*k*apart.

and has the property

If a time series is covariance stationary, then the calculation of sample variance can be substantially simplified.

So all we need is to calculate covariance between one sample and every other samples, but not every sample with every other samples.

Some discussions about why autocorrelation make it difficult to estimate are skipped

- If s are statistically independent observations
then use

- Estimator for mean is calculated as above as a point estimator