Gamma Estimation Experiment

Distribution graph

Description

The experiment is to generate a random sample \((X_1, X_2, \ldots, X_n)\) of size \(n\) from the gamma distribution with shape parameter \(k\) and scale parameter \(b\). The probability density function is shown in blue in the graph, and on each update, the sample density function is shown in red. On each update, the following statistics are recorded: \[ U = \frac{M^2}{T^2}, \quad V = \frac{T^2}{M}, \quad U_b = \frac{M}{b}, \quad V_k = \frac{M}{k} \] where \[ M = \frac{1}{n} \sum_{i=1}^n X_i, \quad T^2 = \frac{1}{n} \sum_{i=1}^n (X_i - M)^2 \] Statistic \(U\) is an estimator of \(k\) and and statistic \(V\) is an estimator of \(b\). Statistic \(U_b\) is an estimator of \(k\) assuming that \(b\) is known, and \(V_k\) is an estimator of \(b\) with \(k\) known. On each update, the empirical bias and mean square error of each estimator are recorded in the table for that estimator. The parameters \(k\), \(b\), and \(n\) can be varied with the input controls..