068. Properties of Good Estimators

A distinction should be drawn between an Estimator and an Estimate. An Estimator is the rule or procedure, usually expressed as a formula, that is used to derive the estimate. For example,

Is the estimator for the population mean.

If the value of the estimator is found to be, say, 10, then 10 is the estimate of the population mean.

To perform reliably, estimators must be

(1) Unbiased. An estimator is unbiased if the mean of the sampling distribution equals the corresponding parameter. To cite a specific example, is an unbiased estimator of p because the mean of the sampling distribution of sample means, , equals. Thus, E () = =

(2) Efficient. Given any unbiased estimators, the most efficient estimator is the one with the smallest variance.

(3) Consistent. An estimate is consistent if, as N increases, the value of the statistic approaches the parameter. For an estimate to be consistent, it must be unbiased and its variance must approach zero as N increases. The variance of the sampling distribution of the sample means, , is 2/n. As N gets larger, will approach zero. Therefore, it can be said that Is a consistent estimator of µ.

(4) Sufficient. An estimator is sufficient if no other estimator could provide more information about the parameter.

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