Means and variances for multivariate (MV) probability distributions can be tested against actual data in one step. The MV means test uses the two-sample Hotelling T2 test which tests whether two data matrices (mxn and pxn) have the same mean vectors, assuming normality and equality of covariance matrices of the data matrices. Assume historical data are arranged in an mxn matrix and the simulated data are in a pxn matrix, where p is the number of iterations, then the means can be tested with the Hotelling T2 test procedure. The Hotelling T2 test is analogous to a Student’s-t test of two means in a two-sample univariate case.

The equality of the covariance matrices for two data matrices with dimensions mxn and pxn, respectively, can be tested using a large sample likelihood ratio testing procedure. The Box’s M test of homogeneity of covariances can be used to test whether the covariance matrices of two or more data series, with n columns each, are equal. The assumptions under this test are that the data matrices are MV normal and that the sample is large enough for the asymptotic, or central Chi-Squared, distribution under the null hypothesis to be used.

The third MV test is the Complete Homogeneity test. This statistical test simultaneously tests the mean vectors and the covariance matrices for two distributions. The historical data’s mean vector and covariance matrix are test against the simulated sample’s mean vector and covariance matrix. If the test fails to reject that the means and covariance are equivalent then one can assume that the multivariate distribution in the historical series is being simulated appropriately. An example of this test is provided in Step 4 of DemoSimetar-Data.

The combination of the Hotelling T-Squared test, Box’s M-test, and the Complete Homogeneity test provides an additional tool for validating a multivariate probability distribution. These three MV distribution tests are performed by Simetar© when you specify an mxn matrix for the 1st Data Series and a pxn matrix for the 2nd Data Series after selecting the Compare Two Series tab (Figure 1) for the Hypothesis Testing dialog box opened by the Statistical tests for validating simulated random variables icon. The Compare Two Series tab invokes the MV distribution tests if the 1st and 2nd Data Series indicate two matrices and invokes the univariate student’s-t and F tests if the specified Data Series indicate two different columns.

Statistical tests for validating simulated random variables