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Simulate Menu

The Simulate Menu provides options to simulate the distribution and the MSE/Bias of the estimators provided in the menu system. The following options are available:

MAX domain: Distribution of Estimator
MSE/Bias of Estimator
POT domain: Distribution of Estimator
MSE/Bias of Estimator


Distribution of Estimator (POT)

Select a df with shape parameter gamma (or likewise, with shape parameter alpha) by entering its qf in the edit field on the right-hand side. Use the predefined function calls provided by the UserFormula facility and XPL. For example, enter paretoqf(2,x) to generate standard Pareto data under the shape parameter alpha=2. In the edit field below, enter the true parameter value for alpha or gamma. Xtremes needs this information to center the data.

Next, select an estimator on the left-hand side. Then enter values for sample size n, number of extremes k and number of simulations N. A set of estimates of gamma, say gamma(k,n,1), ..., gamma(k,n,N), is generated and written to a file of type Xtremes Univariate Data. The distribution of the estimator gamma[k,n] may be visualized by means of procedures in the Visualize menu or in the SUM domain (the same holds for sets of estimators for alpha as well):
Nonparametric:
Use kernel density, sample qf and sample df to get an approximation to the density, qf and df of the estimator.
Parametric:
Estimators of the shape parameter are asymptotically Gaussian, so that the SUM domain becomes applicable.

MSE/Bias of Estimator (POT)

A Monte Carlo Simulation is carried out to estimate the mean squared error (MSE) and the bias of estimators.

Select a df with shape parameter gamma (or likewise, with shape parameter alpha) by entering its qf in the edit field on the right-hand side. Use the
predefined function calls provided by the UserFormula facility and XPL. For example, enter paretoqf(2,x) to generate standard Pareto data under the shape parameter alpha=2. In the edit field below, enter the true parameter value for alpha or gamma. Xtremes requires this information to calculate MSE and bias.

Next, select an estimator on the left-hand side and enter values for sample size n, number k of extremes, and number N of simulations.

It is possible to specify either k or sample size as running parameters. According to from, to and step, K values of the parameter range are selected. For these K values N Monte-Carlo samples are generated. The computed values are written to a textfile. If running parameters are not wanted, one must select the corresponding radio button. The result of the simulation is stored as a trivariate data set with columns containing the running parameters, as well as the simulated MSE and bias.

Distribution of Estimator (MAX)

Select a df with shape parameter gamma (or likewise, with shape parameter alpha) by entering its qf in the edit field on the right-hand side. Use the predefined function calls provided by the UserFormula facility and XPL. For example, enter frechetqf(2,x) to generate standard Frechet data under the shape parameter alpha=2. In the edit field below, enter the true parameter value for alpha or gamma. Xtremes needs this information to center the data.

Next, select an estimator on the left-hand side. Then enter values for number n of maxima, block size m and number of simulations N. Note that the sample size equals the number of blocks multiplied by the block size. Maxima are computed for each block. A set of estimates of gamma, say gamma(k,n,1),..., gamma(k,n,N), is generated and written to a file of type Xtremes Univariate Data. Proceed in the same way as described in the POT domain (see also Distribution of estimator (GP)).

MSE/Bias of Estimator (MAX)

A Monte Carlo Simulation is carried out to estimate the mean squared error (MSE) and the bias of estimators.

Select a df with shape parameter gamma (or likewise, with shape parameter alpha) by entering its qf in the edit field on the right-hand side. Use the
predefined function calls provided by the UserFormula facility and XPL. For example, enter frechetqf(2,x) to generate standard Frechet data under the shape parameter alpha=2. In the edit field below, enter the true parameter value for alpha or gamma. Xtremes needs this information to calculate MSE and bias.

Next, select an estimator on the left-hand side and enter values for Number n of maxima, Block size m and Number N of simulations. Note that sample size is the number of maxima multiplied by the block size. Maxima are computed for each block.

It is possible to specify either n or m as running parameters. According to from, to and step, K values of the parameter range are selected. For these K values N Monte-Carlo samples are generated. The computed values are written to a textfile. If running parameters are not wanted, one must select the corresponding radio button. The result of the simulation is stored as a trivariate data set with columns containing the running parameters, as well as the simulated MSE and bias.

© 2005
Xtremes Group · updated Jun 21, 2005