## Consistent Estimators

Given a sample from apopulation, we can estimate the mean of the population from

As the sample sizeincreases, the variance ofdecreases.This property makesa useful estimator for the population mean,sinceby increasing the sample size n, we can reduce the variance ofIfisalso an unbiased estimator for the population meanthenisa consistent estimator for

If an estimatorfora population parameterhasthe propertiesandasthe sample size for calculatingtendsto infinity, thenisa consistent estimator for

The sample meanisan unbiased estimator for the population meansinceSince also, if the variance of the population isandthe mean is found from a sample of size n usingthenandisa consistent estimator for

Similarly, for a binomialdistribution with proportionfromwhich a sample of sizeistaken, and we record the number of success asandthe proportion astheproportion of 'successes' is expected to be

The variance ofis

The variance oftendsto zero as n tends to infinity soisa consistent estimator for