The experiment is to generate a random sample \(\boldsymbol{X} = (X_1, X_2, \ldots, X_n)\) from the beta distribution with left parameter \(a\) and right parameter \(b\). The statistics of interest are \begin{align} U & = \frac{M(M - M^{(2)})}{M^{(2)} - M^2} \\ V & = \frac{(1 - M)(M - M^{(2)})}{M^{(2)} - M^2} \\ U_b & = b \frac{M}{1 - M} \\ V_a & = a \frac{1 - M}{M} \\ W & = - \frac{n}{\sum_{i = 1}^n \ln(X_i)} \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 estimators of \(a\) and \(b\), respectively. Statistic \(U_b\) is an estimator of \(a\) assuming that \(b\) is known, while \(V_a\) is an estimator of \(b\) assuming that \(a\) is known. Statistics \(W\) is an estimator of \(a\), assumping that \(b = 1\). The parameters \(a\), \(b\), and \(n\) can be varied with the input controls.