Refer to the results in Example 7.1. We can now state that the estimates of the population mean, variance, and proportion of stocks for which the price-earnings ratio exceeded 8.5

are obtained through unbiased estimation procedures. However, the estimate of the population standard deviation, sx — 3.97, is not obtained through an unbiased estimation procedure.

An estimator that is not unbiased is said to be biased. The extent of the bias is the difference between the mean of the estimator and the true parameter.

Definition

Let 6 be an estimator of $. The bias in 6 is defined as the difference between its mean and 0; that is

Bias(0) = E(0) - Q It follows that the bias of an unbiased estimator is 0.

In many practical problems, different unbiased estimators can be obtained, and some method of choosing among them needs to be found. In this situation, it is natural to prefer the estimator whose distribution is most closely concentrated about the population parameter being estimated. Values of such an estimator are less likely to differ, by any fixed amount, from the parameter being estimated than are those of its competitors. Using variance as a measure of concentration, we introduce the concept of efficiency of an estimator as a criterion for preferring one estimator over another.

Definitions

Let ft and d2 be two unbiased estimators of 0, based on the same number of sample observations. Then

(i) $i is said to be more efficient than 62 if

(ii) The relative efficiency of one estimator with respect to the other is the ratio of their variances; that is

Relative efficiency =

Figure 7.2 shows the sampling distributions of two unbiased estimators. Clearly, 0, is the more efficient.

EXAMPLE X], X2,... ,Xnbe a. random sample from a normal distribution with mean fix

7 ^ and variance cr*2. The sample mean, X, is an unbiased estimator of the population mean, with variance

As an alternative estimator, we could use the median of the sample observations. It can be shown that this estimator is also unbiased for ¡xx and that its variance is

The sample mean is more efficient than the median, the relative efficiency of the mean with respect to the median being

The variance of the sample median is 57% higher than that of the sample mean. Here, in order for the sample median to have as small a variance as the sample mean, it would have to be based on 57% more observations. In Chapter 2, we noted that one advantage of the median over the mean is that it gives far less weight to extreme observations. We now see, in terms of its relative inefficiency, a potential disadvantage in using the sample median as a measure of central location.

FIGURE 7.2 Probability density functions of two unbiased estimators, 61 and d2\ 0i is the more efficient

FIGURE 7.2 Probability density functions of two unbiased estimators, 61 and d2\ 0i is the more efficient

In some estimation problems,1 the point estimator with the smallest variance among a group of unbiased estimators is sought. In a few relatively simple cases, it is possible to find the most efficient of all unbiased estimators for a parameter.

If d is an unbiased estimator of 6, and no other unbiased estimator has smaller variance, then 6 is said to be the most efficient or minimum variance unbiased estimator of $.

Specific examples of minimum variance unbiased estimators include

1. The sample mean when sampling from a normal distribution

2. The sample variance when sampling from a normal distribution

3. The binomial sample proportion

The use of minimum variance unbiased estimators is appealing. However, it is not always possible to find such estimators.

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