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In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter θ 0 —having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to θ 0.
A consistent estimator is an estimator whose sequence of estimates converge in probability to the quantity being estimated as the index (usually the sample size) grows without bound. In other words, increasing the sample size increases the probability of the estimator being close to the population parameter.
The name weak consistency can be used in two senses. In the first sense, strict and more popular, weak consistency is one of the consistency models used in the domain of concurrent programming (e.g. in distributed shared memory, distributed transactions etc.). A protocol is said to support weak consistency if:
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 ...
We have two estimators for b: b 0 and b 1. Under the null hypothesis, both of these estimators are consistent, but b 1 is efficient (has the smallest asymptotic variance), at least in the class of estimators containing b 0. Under the alternative hypothesis, b 0 is consistent, whereas b 1 isn't. Then the Wu–Hausman statistic is: [6]
For example, an estimator is called consistent if it converges in probability to the quantity being estimated. Convergence in probability is also the type of convergence established by the weak law of large numbers .
The term consistency in statistics usually refers to an estimator that is asymptotically consistent. Fisher consistency and asymptotic consistency are distinct concepts, although both aim to define a desirable property of an estimator. While many estimators are consistent in both senses, neither definition encompasses the other.
Heteroskedasticity-consistent standard errors that differ from classical standard errors may indicate model misspecification. Substituting heteroskedasticity-consistent standard errors does not resolve this misspecification, which may lead to bias in the coefficients. In most situations, the problem should be found and fixed. [5]