Consistency (statistics)

A consistent estimator is one for which, when the estimate is considered as a random variable indexed by the number n of items in the data set, as n increases the estimates converge in probability to the value that the estimator is designed to estimate. An estimator that has Fisher consistency is one for which, if the estimator were applied to the entire population rather than a sample, the true value of the estimated parameter would be obtained. ==Tests==

A statistical hypothesis test is consistent if and only if, under any alternative hypothesis, the probability of rejecting the null hypothesis increases to 1 as the number of data items increases. In other words, for any alternative hypothesis, the statistical power \to 1 as N \to \infty. ==Classification==

In statistical classification, a consistent classifier is one for which the probability of correct classification, given a training set, approaches, as the size of the training set increases, the best probability theoretically possible if the population distributions were fully known. == Relationship to unbiasedness ==

An estimator or test may be consistent without being unbiased. A classic example is the sample standard deviation which is a biased estimator, but converges to the expected standard deviation almost surely by the law of large numbers. Phrased otherwise, unbiasedness is not a requirement for consistency, so biased estimators and tests may be used in practice with the expectation that the outcomes are reliable, especially when the sample size is large (recall the definition of consistency). In contrast, an estimator or test which is not consistent may be difficult to justify in practice, since gathering additional data does not have the asymptotic guarantee of improving the quality of the outcome. ==See also==