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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 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.
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 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.
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]
In Julia, the CovarianceMatrices.jl package [11] supports several types of heteroskedasticity and autocorrelation consistent covariance matrix estimation including Newey–West, White, and Arellano. In R , the packages sandwich [ 6 ] and plm [ 12 ] include a function for the Newey–West estimator.
In statistics, efficiency is a measure of quality of an estimator, of an experimental design, [1] or of a hypothesis testing procedure. [2] Essentially, a more efficient estimator needs fewer input data or observations than a less efficient one to achieve the Cramér–Rao bound.
Horowitz in a recent review [1] defines consistency as: the bootstrap estimator (,) is consistent [for a statistic ] if, for each , | (,) (,) | converges in probability to 0 as , where is the distribution of the statistic of interest in the original sample, is the true but unknown distribution of the statistic, (,) is the asymptotic ...