Mean squared error

In statisticsmean squared erroran estimator Tan unobservable parameter θ is

i.e.,isexpected value ofsquare of"error". The "error" isamount by whichestimator differs fromquantitybe estimated. The mean squared error satisfiesidentity

where

i.e.,bias isamount by whichexpected value ofestimator differs fromunobservable quantitybe estimated.

Here isconcrete example. Suppose

i.e., this israndom samplesize n fromnormally distributed population. Two estimatorsσ²sometimes used (asothers):

where

is"sample mean". The firstthese estimators ismaximum likelihood estimator,is biased, i.e., its biasnot zero, but hassmaller variance thansecond, whichunbiased. The smaller variance compensates somewhat forbias, so thatmean squared error ofbiased estimatorslightly smaller than that ofunbiased estimator.