Consistent Estimator

Let

Notes:

• Consistent estimators are appealing since estimation errors associated with such point estimators are close to zero with high probability when sample sizes are sufficiently large.
• An estimator is consistent if its mean squared error tends to zero as n tends to infinity. For example, when based on a random sample, the sample mean is a consistent estimator of a population mean since
  
  and this tends to zero as n tends to infinity. The consistency of the sample mean is also a consequence of the Weak Law of Large Numbers. The two routes used to arrive at the same conclusion are of course connected.