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Characteristics of Estimators

Module by: David Lane

This section discusses two important characteristics of point estimates: bias and precision. Bias refers to whether an estimator tends to either over or underestimate the parameter. Precision refers to how close the estimator comes to the parameter.

Have you ever noticed that some bathroom scales give you very different weights each time you weigh yourself? With this in mind, lets compare two scales. Scale 1 is a very high-tech digital scale and gives essentially the same weight each time you weigh yourself; it varies by at most 0.02 pounds from weighing to weighing. Although this scale has the potential to be very accurate, it is calibrated incorrectly and, on average, overstates your weight by one pound. Scale 2 is a cheap scale and gives very different results from weighing to weighing. However, it is just as likely to underestimate as overestimate your weight. Sometimes it vastly overestimates it and sometimes it vastly underestimates it. However, the average of a large number of measurements would be your actual weight. Scale 1 is biased since, on average, its measurements are one pound higher than your actual weight. Scale 2, by contrast, gives unbiased estimates of your weight. However, Scale 2 is not at all precise. Its measurements are often very far from your true weight. Scale 1, in spite of being biased, is fairly precise. Its measurements are never more than 1.02 pounds from your actual weight.

We now turn to more formal definitions of bias and precision. However, the basic ideas are the same as in the bathroom scale example.

Bias

A statistic is biased if the long-term average value of the statistic is not the parameter it is estimating. More formally, a statistic is biased if the mean of the sampling distribution of the statistic is not equal to the parameter. The mean of the sampling distribution of a statistic is sometimes referred to as the expected value of the statistic.

As we saw in the section on the sampling distribution of the mean, the mean of the sampling distribution of the (sample) mean is the population mean ( m m ). Therefore the sampling distribution of the mean is an unbiased estimate of m m . Any given sample mean may underestimate or overestimate m m, but, there is no systematic tendency for sample means to either under or overestimate m m

In the section on variability, we saw that the formula for the variance in a population is

σ2=∑X-μ2N σ 2 X μ 2 N (1)
whereas the formula to estimate the variance from a sample is
s2=∑X-M2N-1 s 2 X M 2 N 1 (2)
Notice that the denominators of the formulas are different: NN for the population and N-1 N 1 for the sample. We saw in the "Estimating Variance Simulation" that if NN is used in the formula for s2 s 2 , then the estimates tend to be too low and therefore biased. The formula with N-1 N 1 in the denominator gives an unbiased estimate of the population variance.

Precision

The bathroom-scale example shows that it is possible to be unbiased yet imprecise. The precision of a statistic refers to how accurately the statistic estimates the parameter. A statistic's precision is usually measured by its standard error; the smaller the standard error, the more precise the estimate. For example, the standard error of the mean is a measure of the precision of the mean. Recall that the formula for the standard error of the mean is

σ M =σN σ M σ N (3)
The larger the sample size ( NN), the smaller the standard error of the mean and thereforfe the more precise the sample mean is as an estimate of the population mean.

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