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Testing Menu

Univariate Mode

Discrete domain: Poisson, Goodness-of-Fit
SUM domain: Gaussian
MAX domain: LR(EV0/EV)
POT domain: LR(GP0/GP)
Hasofer-Wang(GP0/GP)

Multivariate Mode*

SUM domain: Gaussian*
Wilcoxon*


Poisson, Goodness-of-Fit

The Poisson model can be tested by means of chi-square- and likelihood-ratio-tests (see Statistical Analysis, pages 106-107). First, fix cell boundaries in the left-hand edit field. Equally sized cells can be obtained by means of the From, To and Step-Width options. Pressing OK, the pertaining frequencies within the cells (in the right-hand field) and the p-value are displayed.

Recall that the Poisson model is rejected for small p-values.

Gaussian testing, one-sample

Let mu denote the mean and sigma the standard deviation of a Gaussian distribution. Within the Gaussian model one is testing

one-sided: H0: mu < mu0 against H1: mu > mu0,

two-sided*: H0: mu = mu0 against H1: mu <> mu0,

with

In the dialog box, specify

Press OK to obtain the p-value. Recall that for small p-values the null hypothesis H0 is rejected (Statistical Analysis, Section 3.3).

LR(EV0/EV)

Within the EV model, the null hypothesis gamma = 0 is tested against the alternative gamma <> 0 with unknown location and scale parameters by means of a Likelihood-Ratio-Test. Thus, the Gumbel distributions are tested against other EV distributions. The output consists again of the p-value.

The significance level is attained with a higher accuracy by employing a Bartlett correction; in this case a modified p-value is used.

Recall that the Gumbel model is rejected for small p-values.

LR(GP0/GP)

Within the GP model, the null hypothesis gamma = 0 is tested against the alternative gamma <> 0 with unknown scale parameters by means of a likelihood-ratio-test. More precisely, exponential-like upper tails are tested against other GP-like upper tails (see Statistical Analysis, page 154).

Choose the number of upper extremes on which the test should be based on. Alternatively, if exceedances over a threshold u are dealt with, prepare the data set in the data editor: Take the exceedances over u and include u in the data set. Then apply the LR option with number of extremes equal to the full sample size (which is the default value).

The significance level is attained with a higher accuracy by employing a Bartlett correction; in this case a modified p-value is used.

The output consists again of the p-value. The hypothesis of an exponential-like upper tail is rejected for small p-values.

Hasofer-Wang(GP0/GP)

This is another test besides the likelihood-ratio-test (see LR(GP0/GP) ) for testing exponential-like upper tails against other GP-like upper tails. The Hashofer-Wang test is based on the reciprocal squared coefficient of variation (see Statistical Analysis, page 154). Make sure that the shape parameter gamma is smaller than 1/2.

The output consists again of the p-value. Recall that the hypothesis of an exponential-like upper tail must be rejected for small p-values.

Gaussian testing, two-sample*

The test procedures are applied to bivariate data of type Xtremes Multivariate Data. The data x[1], ... , x[n] and y[1], ... , y[m] in the 1st and 2nd column may be of a different length whereby empty entries are filled with a full stop. Notice that two univariate data sets may be combined to a bivariate one by using the Data... Convert to... Multivariate Data... option.

Let mux, sigmax and muy, sigmay denote the means and standard deviations of the Gaussian distributions in the 1st and 2nd component. Within the Gaussian model one is testing

one-sided: H0: mux <= muy against H1: mux > muy,

two-sided: H0: mux = muy against H1: mux <> muy,

with

In the dialog box, specify

Press OK to obtain the p-value. Recall that for small p-values the null hypothesis H0 is rejected (Statistical Analysis, Section 3.3).

Wilcoxon test, two-sample*

This test procedures is applied to bivariate data of type Xtremes Multivariate Data. The data x[1], ... , x[n] and y[1], ... , y[m] in the 1st and 2nd column may be of a different length whereby empty entries are filled with a full stop. Notice that two univariate data sets may be combined to a bivariate one by using the Data... Convert to... Multivariate Data option.

The Wilcoxon test (also known under the names Mann-Whitney test and U-test) is a nonparametric tests based on samples x[1], ... , x[n] and y[1], ... , y[m] governed by dfs F and, respectively, G. One is testing

H0: F = G against H1: F <> G.

Execute the option to obtain the p-value. Recall that for small p-values the null hypothesis H0 is rejected (Statistical Analysis, Section 3.3).

© 2005
Xtremes Group · updated Jun 21, 2005