|
In statistics, the bias (or bias function) of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called unbiased. Otherwise the estimator is said to be biased. In statistics, "bias" is an objective statement about a function, and while not a desired property, it is not pejorative, unlike the ordinary English use of the term "bias". Bias can also be measured with respect to the median, rather than the mean (expected value), in which case one distinguishes ''median''-unbiased from the usual ''mean''-unbiasedness property. Bias is related to consistency in that consistent estimators are convergent and ''asymptotically'' unbiased (hence converge to the correct value), though individual estimators in a consistent sequence may be biased (so long as the bias converges to zero); see bias versus consistency. All else equal, an unbiased estimator is preferable to a biased estimator, but in practice all else is not equal, and biased estimators are frequently used, generally with small bias. When a biased estimator is used, the bias is also estimated. A biased estimator may be used for various reasons: because an unbiased estimator does not exist without further assumptions about a population or is difficult to compute (as in unbiased estimation of standard deviation); because an estimator is median-unbiased but not mean-unbiased (or the reverse); because a biased estimator reduces some loss function (particularly mean squared error) compared with unbiased estimators (notably in shrinkage estimators); or because in some cases being unbiased is too strong a condition, and the only unbiased estimators are not useful. Further, mean-unbiasedness is not preserved under non-linear transformations, though median-unbiasedness is (see effect of transformations); for example, the sample variance is an unbiased estimator for the population variance, but its square root, the sample standard deviation, is a biased estimator for the population standard deviation. These are all illustrated below. ==Definition== Suppose we have a statistical model, parameterized by a real number ''θ'', giving rise to a probability distribution for observed data, , and a statistic which serves as an estimator of ''θ'' based on any observed data . That is, we assume that our data follows some unknown distribution (where ''θ'' 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 ''θ''. Then the bias of this estimator (relative to the parameter ''θ'') is defined to be : where denotes expected value over the distribution , i.e. averaging over all possible observations . The second equation follows since ''θ'' is measurable with respect to the conditional distribution . 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「bias of an estimator」の詳細全文を読む スポンサード リンク
|