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In statistics, the bias of an estimator (or bias function) 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. In statistics, "bias" is an objective property of an estimator.
One way of seeing that this is a biased estimator of the standard deviation of the population is to start from the result that s 2 is an unbiased estimator for the variance σ 2 of the underlying population if that variance exists and the sample values are drawn independently with replacement. The square root is a nonlinear function, and only ...
The bias of an estimator is the difference between an estimator's expected value and the true value of the parameter being estimated. Although an unbiased estimator is theoretically preferable to a biased estimator, in practice, biased estimators with small biases are frequently used. A biased estimator may be more useful for several reasons.
We say that the estimator is a finite-sample efficient estimator (in the class of unbiased estimators) if it reaches the lower bound in the Cramér–Rao inequality above, for all θ ∈ Θ. Efficient estimators are always minimum variance unbiased estimators. However the converse is false: There exist point-estimation problems for which the ...
Difference between estimators: an unbiased estimator is centered around vs. a biased estimator . A desired property for estimators is the unbiased trait where an estimator is shown to have no systematic tendency to produce estimates larger or smaller than the true parameter.
Among unbiased estimators, minimizing the MSE is equivalent to minimizing the variance, and the estimator that does this is the minimum variance unbiased estimator. However, a biased estimator may have lower MSE; see estimator bias.
This sequence is consistent: the estimators are getting more and more concentrated near the true value θ 0; at the same time, these estimators are biased. The limiting distribution of the sequence is a degenerate random variable which equals θ 0 with probability 1.
This may occur either if for any unbiased estimator, there exists another with a strictly smaller variance, or if an MVU estimator exists, but its variance is strictly greater than the inverse of the Fisher information. The Cramér–Rao bound can also be used to bound the variance of biased estimators of given bias.