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In
statistics, an '''estimator''' is a
function of the known data that is used to estimate an unknown
parameter; an ''estimate'' is the result from the actual application of the function to a particular set of data. Many different estimators are possible for any given parameter. Some criterion is used to choose between the estimators, although it is often the case that a criterion cannot be used to clearly pick one estimator over another.
There are two types of estimators:
point estimators and
interval estimators.
== Point estimators ==
For a point estimator '''θ''' of parameter θ:
# The ''
bias'' of '''θ''' is defined as B('''θ''') = E['''θ'''] − θ
# '''θ''' is an ''
unbiased estimator'' of θ iff B('''θ''') = 0 for all θ
# The ''mean square error'' of '''θ''' is defined as MSE('''θ''') = E[('''θ''' − θ)
2]
# MSE('''θ''') = V('''θ''') + (B('''θ'''))
2
where V(X) is the
variance of X and E is the
expected value operator.
The
standard deviation of '''θ''' (the square root of the variance) is also called the ''standard error'' of '''θ'''.
Occasionally one chooses the unbiased estimator with the lowest variance. Sometimes it is preferable not to limit oneself to unbiased estimators; see
bias (statistics). Concerning such "best unbiased estimators", see also
Gauss-Markov theorem,
Lehmann-Scheff theorem,
Rao-Blackwell theorem.
:''See also:''
maximum likelihood
