Archimedean Copulas Theodore Charitos MSc. Student CROSS

Task-Goal Examination of relations between Archimedean copulas and diagonal band or minimum information copulas given correlation constraints Computation of relative information with respect to uniform distribution for each family.

Accomplishment of tasks Use of the algorithm provided in the paper of Christian Genest and Louis-Paul Rivest “Statistical Inference Procedures for Bivariate Archimedean Copulas”. Use of small program in Matlab for calculating numerically the relative information with respect to uniform distribution.

Structure of presentation Theoretical Background Explanation and description of the whole procedure proposed by Genest and Rivest. Analysis of datasets sampled from Unicorn software. Results Conclusions-Discussions

Theoretical Background Definition: A bivariate distribution function with marginals and is said to be generated by an Archimedean copula if it can be expressed in the form for some convex, decreasing function on in such a way that

Proposition: Let and be uniform random variates whose dependence function is of the form for some convex decreasing function defined on with the property that. Set, and. Then is uniformly distributed on, is distributed as and, are independent random variables.

Proposition: Let X and Y be uniform random variables with dependence function. For let and define The function is convex,decreasing and satisfies if and only if for all. It is obvious from the above propositions that is determined as long as can be determined from the dataset. This will be done in our case via a nonparametric estimation of the distribution of V based on a decomposition of Kendall’s tau.

A pair of random variables is concordant if large values of one tend to be associated with large values of the other and vice versa. More precisely, if we have two observations and from a vector of continuous random variables we say that and are concordant if and. Similarly, and are discordant if and or vice versa.

Definition The Kendall’s tau for the sample is defined as where c is the number of concordant pairs, d is the number of discordant pairs from n observations of a vector and is the number of distinct pairs of observations in the sample.

For Archimedean copulas the Kendall’s tau statistic can be conveniently computed via the identity Apparently, the problem now of estimating the bivariate dependence function relies on the estimation of. Genest and Rivest provide a nonparametric procedure for estimating and also.

Analysis of various datasets The algorithm proposed by Genest and Rivest uses the variables where the symbol # stands for the cardinality of a set. If denotes the distribution function of a point mass at the origin, then a nonparametric estimator of is given by Knowing that, a sample equivalent for the estimation of is

Family Clayton Frank Gumbel -

The next step of the analysis concerns the performance of a Pearson chi-squared goodness of fit test statistic for each family in order to assess the fit of the various models. This means that a classification of the dataset is made each time constituting the observed frequencies. However, since the chi-squared test requires predicted values for its computation, it is necessary to generate random variates whose joint distribution belongs to one of the mentioned Archimedean families

Algorithm for sampling from Archimedean families 1.Generate two independent uniform variates u and t. 2. Set 3. Set v = 4. The desired pair is

Archimedean Family Clayton Frank Gumbel

Clayton’s joint density with a=1.514 Frank’s joint density with a=4.604 Gumbel’s joint density with a=0.757

In general, the relative information with respect to uniform distribution for the bivariate case is computed as where is the joint density of and An approximation of the real solution in each case will be provided, which however is enough to indicate what should someone expect from each Archimedean family.

To illustrate the above procedure six datasets (n=1000) were at first sampled and thoroughly analyzed. The correlations were 0.2, 0.65 and 0.9 for both the diagonal band and the minimum information copulas. A classification of the frequencies was also decided. For the sake of completeness, six more datasets with similar correlations constraints but different size (n=5000) and classification were also analyzed in order to compare results.

Recapitulation-Steps Sample from diagonal band and minimum information copulas. Estimate Kendal’s tau and the empirical lambda function. Estimate the parameters for each family according to the previous results. Estimate the lambda functions for each family. Classify the dataset in categories and simulate values from each family according to their estimated parameters. Perform chi-square goodness of fit test and compare the resulting fits. Compute the relative information with respect to uniform distribution. Repeat the whole procedure for different correlations and size of the dataset.

Examples of classifications from diag.band with 0.2 Cross-Classification of X and Y (Observed values) X\Y Cross-Classification of X and Y (Observed values) X\Y

= Statistic df Clayton Frank Gumbel = Statistic df Clayton Frank Gumbel

= Statistic df Clayton Frank Gumbel = Statistic df Clayton Frank Gumbel

= Statistic df Clayton Frank Gumbel = Statistic df Clayton Frank Gumbel

= Statistic df Clayton Frank Gumbel = Statistic df Clayton Frank Gumbel

= Statistic df Clayton Frank Gumbel = Statistic df Clayton Frank Gumbel

= Statistic df Clayton Frank Gumbel = Statistic df Clayton Frank Gumbel

Relative Information with respect to uniform distribution Clayton Frank Gumbel Diag.band 0.2 (n=1000) Min.inf.0.2 (n=1000) Diag.band 065 (n=1000) Min.inf.0.65 (n=1000) Diag.band 0.9 (n=1000) Min.inf.0.9 (n=1000) Diag.band0.2 (n=5000) Min.inf.0.2 (n=5000) Diag.band 0.65 (n=5000) Min.inf.0.65 (n=5000) Diag.band0.9 (n=5000) Min.inf.0.9 (n=5000)

Conclusions-Comments for correlation 0.2 all the three families seem to fit reasonably well when n=1000,but when n=5000 only Frank’s and also Gumbel’s for the min.information. for correlation 0.65 the results are quite promising only when n=1000 for correlation 0.9 Frank’s and Gumbel’s family seem to fit the data when n=1000 and only Frank’s family when n=5000. the results are more promising in the cases of minimum information copula and this is actually a fact that holds for all the datasets no matter what the correlation is. the results are much better when the size of the dataset is smaller. It is obvious that the chi-square test statistic is sensitive to the number of cells.For greater size n and 6x6 cross-classification the results in almost all cases are disappointing. A performance of another goodness of fit test might result in more encouraging conclusions. for correlation 0.2 Clayton’s family has the smallest values of relative information with respect to uniform distribution. Nonetheless, for greater correlations, Frank’s family has the smallest values.