Method of moments estimate
A simple way to obtain an estimate of a single unknown parameter from a random sample is the method of moments. This is defined in the same way as when estimating a parameter in a discrete distribution — it is the parameter value that makes the distribution's mean equal to that of the random sample and is therefore the solution to the equation
\[ E[X] \;\; = \; \; \overline{X} \]German tank problem
Consider a rectangular distribution,
\[ X \;\; \sim \; \; \RectDistn(0, \beta) \]where the upper limit, \(\beta\), is an unknown parameter.
This is often called the German tank problem since it is related to Allied estimates of German tank production in the second war from the serial numbers of captured tanks. The serial number of the captured tanks, \(\{x_1, x_2, \dots, x_n\}\), can be treated as a random sample from this kind of rectangular distribution with \(\beta\) being the total number of tanks produced. (The distribution is actually discrete uniform distribution but when \(\beta\) is large, such as in this application, it can be approximated by a rectangular distribution.)
Since the mean and variance of this rectangular distribution are
\[ E[X] \;\; = \; \; \frac {\beta} 2 \spaced{and} \Var(X) = \frac {\beta^2} {12} \]the method of moments estimator is
\[ \hat{\beta} \;\;=\;\; 2\overline{X}\]It is an unbiased estimator and has standard error
\[ \se(\hat{\beta}) \;\;=\;\; \sqrt{\Var(2\overline{X})} \;\;=\;\; \sqrt{ \frac {4\Var(X)} n } \;\;=\;\; \frac {\beta} {\sqrt{3n}} \]Although this method of moments estimator is unbiased, it has one major problem. Consider the random sample
12,17, 42, 97
The resulting estimate of \(\beta\) would be
\[ \hat{\beta} \;\;=\;\; 2\overline{X} \;\;=\;\; 84\]yet the maximum of the distribution cannot be 84 since we have already observed one value greater than this.
The method of moments usually gives reasonable parameter estimates, but can sometimes result in estimates that are not feasible.