The experiment is to generate a random sample \(\boldsymbol{X} = (X_1, X_2, \ldots, X_n)\) from the Pareto distribution with shape parameter \(a\) and scale parameter \(b\). The statistics of interest are \begin{align} U & = 1 + \sqrt{\frac{M^{(2)}}{M^{(2)} - M^2}} \\ V & = \frac{M^{(2)}}{M}\left(1 - \sqrt{\frac{M^{(2)} - M^2}{M^{(2)}}}\right) \\ W & = \frac{n}{\left(\sum_{i=1}^n \ln X_i - n \ln X_{(1)}\right)} \\ X_{(1)} & = \min\{X_1, X_2, \ldots, X_n\} \end{align} where \(M = \frac{1}{n} \sum_{i=1}^n X_i\) and \(M^{(2)} = \frac{1}{n} \sum_{i=1}^n X_i^2\)
Statistics \(U\) and \(V\) are the method of moments estimators of \(a\) and \(b\), respectively. Statistics \( W \) and \( X_{(1)} \) are the maximum likelihood estimators of \( a \) and \( b \) respectively. The parameters \(a\), \(b\), and \(n\) can be varied with the input controls.