Biased estimator

A statistical estimator whose expectation does not coincide with the value being estimated.

Let be a random variable taking values in a sampling space , , and let be a statistical point estimator of a function defined on the parameter set . It is assumed that the mathematical expectation of exists. If the function

is not identically equal to zero, that is, , then is called a biased estimator of and is called the bias or systematic error of .

Example. Let be mutually-independent random variables with the same normal distribution , and let

Then the statistic

is a biased estimator of the variance since

that is, the estimator has bias . The mean-square error of this biased estimator is

The best unbiased estimator of is the statistic

with mean-square error

When , the mean-square error of the biased estimator is less than that of the best unbiased estimator .

There are situations when unbiased estimators do not exist. For example, there is no unbiased estimator for the absolute value of the mathematical expectation of the normal law , that is, it is only possible to construct biased estimators for .

References

[1]	H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)