Pareto Estimation Experiment

Distribution graph

Description

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 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: \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. On each update, the empirical bias and mean square error of each estimator are recorded in the table for that estimator. The parameters $$a$$, $$b$$, and $$n$$ can be varied with the input controls.