Definition
Suppose we have a statistical model parameterized by θ giving rise to a probability distribution for observed data, and a statistic θ^ which serves as an estimator of θ based on any observed data . That is, we assume that our data follow some unknown distribution (where is a fixed constant that is part of this distribution, but is unknown), and then we construct some estimator that maps observed data to values that we hope are close to . Then the bias of this estimator is defined to be
where E denotes expected value over the distribution, i.e. averaging over all possible observations .
An estimator is said to be unbiased if its bias is equal to zero for all values of parameter θ.
There are more general notions of bias and unbiasedness. What this article calls "bias" is called "mean-bias", to distinguish mean-bias from the other notions, with the notable ones being "median-unbiased" estimators. For more details, the general theory of unbiased estimators is briefly discussed near the end of this article.
In a simulation experiment concerning the properties of an estimator, the bias of the estimator may be assessed using the mean signed difference.
Read more about this topic: Bias Of An Estimator
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