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Therefore, the sample variance S 2 is an unbiased estimator of the population variance
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Although S 2 is unbiased for 2, S is a biased estimator of . For large samples, the bias is very small. However, there are good reasons for using S as an estimator of in samples from normal distributions, as we will see in the next three chapters when are discuss con dence intervals and hypothesis testing.
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Sometimes there are several unbiased estimators of the sample population parameter. For example, suppose we take a random sample of size n 10 from a normal population and obtain the data x1 12.8, x2 9.4, x3 8.7, x4 11.6, x5 13.1, x6 9.8, x7 14.1, x8 8.5, x9 12.1, x10 10.3. Now the sample mean is x 12.8 11.04 the sample median is ~ x 10.3 2 11.6 10.95 9.4 8.7 11.6 13.1 10 9.8 14.1 8.5 12.1 10.3
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and a 10% trimmed mean (obtained by discarding the smallest and largest 10% of the sample before averaging) is xtr1102 8.7 10.98 We can show that all of these are unbiased estimates of . Since there is not a unique unbiased estimator, we cannot rely on the property of unbiasedness alone to select our estimator. We need a method to select among unbiased estimators. We suggest a method in Section 7-2.3. 9.4 9.8 10.3 11.6 8 12.1 12.8 13.1
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Proof That S is a Biased Estimator of
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7-2.3 Variance of a Point Estimator
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Suppose that 1 and 2 are unbiased estimators of . This indicates that the distribution of each estimator is centered at the true value of . However, the variance of these distributions may be different. Figure 7-1 illustrates the situation. Since 1 has a smaller variance than 2, the estimator 1 is more likely to produce an estimate close to the true value . A logical principle of estimation, when selecting among several estimators, is to choose the estimator that has minimum variance. De nition If we consider all unbiased estimators of , the one with the smallest variance is called the minimum variance unbiased estimator (MVUE).
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Figure 7-1 The sampling distributions of two unbiased estimators 1 and 2 .
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In a sense, the MVUE is most likely among all unbiased estimators to produce an estimate that is close to the true value of . It has been possible to develop methodology to identify the MVUE in many practical situations. While this methodology is beyond the scope of this book, we give one very important result concerning the normal distribution.
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Theorem 7-1
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If X1, X2, p , Xn is a random sample of size n from a normal distribution with mean and variance 2 , the sample mean X is the MVUE for .
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In situations in which we do not know whether an MVUE exists, we could still use a minimum variance principle to choose among competing estimators. Suppose, for example, we wish to estimate the mean of a population (not necessarily a normal population). We have a random sample of n observations X1, X2, p , Xn and we wish to compare two possible estimators for : the sample mean X and a single observation from the sample, say, Xi. Note that both X and Xi are unbi2 n from Equation 5-40b and the ased estimators of ; for the sample mean, we have V1X 2 2 variance of any observation is V1Xi 2 . Since V1X 2 V1Xi 2 for sample sizes n 2, we would conclude that the sample mean is a better estimator of than a single observation Xi.
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7-2.4 Standard Error: Reporting a Point Estimate
When the numerical value or point estimate of a parameter is reported, it is usually desirable to give some idea of the precision of estimation. The measure of precision usually employed is the standard error of the estimator that has been used. De nition The standard error of an estimator is its standard deviation, given by 2V1 2 . If the standard error involves unknown parameters that can be estimated, substitution of those values into produces an estimated standard error, denoted by .
Sometimes the estimated standard error is denoted by s or se1 2 . Suppose we are sampling from a normal distribution with mean and variance 2 . Now the distribution of X is normal with mean and variance 2 n, so the standard error of X is 1n
If we did not know but substituted the sample standard deviation S into the above equation, the estimated standard error of X would be