Error distributions from equations

For many common parameter estimates, statistical theory can provide the error distribution (or an approximation).

There is a formula for the standard error of most common parameter estimates.

Error distributions from simulations

However there are also situations where statistical theory (or easily accessible statistical software to implement the theory) does not provide the error distribution for the estimator of interest. In these situations, a simulation can often be used to find properties of the error distribution numerically.

(We used simulations in earlier in this chapter to illustrate some of the concepts in estimation, but not as a practical tool.)

This methodology will first be illustrated with a simulation to find the standard error of a sample proportion. For this application, we gave formulae earlier in this section to provide the standard error directly,

standard error  =  

A simulation is therefore unnecessary and would never be used in practice. However the application does provide a simple and easily understood illustration of how simulation can be used.

Rice survey

In the rice survey, a proportion p = 17/36 = 0.472 of the n = 36 farmers used 'Old' varieties. If this is a random sample of farmers in the region, the number using 'Old' varieties should have a binomial distribution,

X  ~  binomial (n = 36,  π)

If we know the value of π, we could take repeated samples from this binomial distribution, find the estimation error, (p - π) for each sample, and build up the error distribution.

Of course, we do not know the value of π, but we could perform a similar simulation by replacing π with our best estimate, p = 0.472. The diagram below does such a simulation.

Click Accumulate and take about 100 samples (each of 36 farmers) to build up the error distribution.

Observe that the estimates could be in error by as much as 0.2.

The proportion of farmers in the whole region using 'Old' varieties could therefore be quite different from our estimate of 0.472.

Click Estimate s.e. and bias to find the standard deviation and mean of the errors. These should be fairly close to the values that could be obtained from formulae for this example.

bias  =  0

standard error  =    =  0.0832

We again stress that a simulation would not be used to find the standard error in this type of example — we should use the formula for the standard error of a proportion.