Unbiased Estimators And Their Efficiency

In this discussion, we will denote by 0 a parameter to be estimated and by 6 the corresponding point estimator. As we saw in Chapter 6, it is sometimes possible to find the sampling distribution of the random variable 6. That estimator is said to be unbiased if the mean of its sampling distribution is the unknown parameter 6.

Definitions

The estimator 6 is said to be an unbiased estimator of the parameter 0 if the mean of the sampling distribution of 0 is 0; that is

We say that the corresponding point estimate is obtained through an unbiased estimation procedure.

It follows from the notion of expectation that if the sampling procedure is repeated many times, then on the average, the value obtained for an unbiased estimator will be equal to the population parameter. It seems reasonable to assert that, all other things being equal, unbiasedness is a desirable property in a point estimator. Figure 7.1 shows the sampling distributions of two estimators, one unbiased and one not.

For three of the estimators being considered, we saw in Chapter 6 that

Thus, we can say that the sample mean, variance, and proportion are unbiased estimators of the corresponding population parameters. It is for this reason that in defining the sample variance, we divided the sum of squared discrepancies from the sample mean by (« — 1) rather than n. The former produces an unbiased estimator; the latter does not. The mean of the sampling distribution of the sample standard deviation is not equal to the population standard deviation. Hence, the sample standard deviation is not an unbiased estimator of the population standard deviation.

FIGURE 7.1 Probability density functions for the estimators 0, and $i being an unbiased estimator of 6 and 62 not

FIGURE 7.1 Probability density functions for the estimators 0, and $i being an unbiased estimator of 6 and 62 not

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Unbiasedness of Some Estimators

(i) The sample mean, variance, and proportion are unbiased estimators of the corresponding population quantities.

(ii) In general, the sample standard deviation is not an unbiased estimator of the population standard deviation.

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