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Hills1234
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Copula - Goodness of Fit Testing

March 1st, 2012, 9:58 am

Hello,I have a high dimensional dataset...it is a 250 by 58 matrix of uniform observations. I have fitted various copula models to it including Student T, Gaussian, Archimedean, Vine copulas, etc. Now I would like to assess the fit. I have obtained the classical measures of fit such as Log-Likelihood, AIC and BIC...but these do not directly tell me whether to accept or reject the model. I am looking now to apply formal goodness of fit tests on the fitted models. I have tried implementing test statistics based on the distance between the empirical copula and the fitted copula. For example the Kolmogorov-Smirnov (KS), Cram´er-von Mises (CvM), Anderson-Darling (AD), etc. But these all rely on computing the CDF of the fitted copula. This has been a real problem for me since calculating the CDF of a T copula, for example, at a grid of points (100 by 58) is very time consuming. In fact, when I try to perform this within Matlab, I get the following error message Number of dimensions must be less than or equal to 25.So my question is...is there a more appropriate method (given my high dimensional dataset) of accepting or rejecting a copula model?Any help/recommendations will be greatly appreciated!
 
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Hills1234
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Copula - Goodness of Fit Testing

March 2nd, 2012, 9:03 am

Thanks for the reply but not quite sure what you mean....could you be a bit more specific?I have been looking around. There does exist the Probability Integral Transform (PIT), which is often used in multivariate goodness of fit tests. The idea is to transform the data first using the PIT under my null hypothesis (e.g. the data comes from a student t copula) and then perform a test of multivariate independance. If it passes this test then null hypothesis is accepted. PIT is basically the inverse of simulation. But the definition of PIT is not so straightforward. It relies on calculatingP(X_d < x_d | X_1 = x_1, ........X_(d-1) = x_(d-1)) (i.e. copula conditioned on many variables)There does exist explicit formulas when conditioning on a single variable. For example P(X_2 < x_2 | X_1 = x_1) can be easily calculated under a Gaussian, Student T, Gumbel, etc. But what if I wanted to know P(X_4 < x_4 | X_1 = x_1, X_2 = x_2, X_3 = x_3) ?PIT is better explained on page 3 at http://www.danielberg.no/publications/C ... .pdfThanks
 
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Hills1234
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Copula - Goodness of Fit Testing

March 2nd, 2012, 1:07 pm

Thanks...that makes things clearer. Did you think of this, off the top of your head?...Or is this something which has been used extensively in literature?
 
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Hills1234
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Copula - Goodness of Fit Testing

March 5th, 2012, 8:25 am

Ok Thanks. Although, what you have recommended is not a recognized GoF test...I think I'll go about it by conducting simulation studies and then assessing statistics such as Type I error and Type II error. This would help me determine whether in fact this dimension reduction method of hypothesis testing does in fact accurately tell me whether certain data comes from a s specific copula. Another thing...you mentioned how once the multidimensional data is converted to the univariate scale, then apply a Chi-Square Test to it. I'm not sure how a Chi-Square test is relevant here. Chi-Square tests are only appropriate for determining whether a specific sample comes from a specific distribution. Something like a 2 sample Kolmogorov Smirnov test to see if the two samples (sample 1: random numbers from hypothesized copula and sample 2: observations) would be more appropriate..???
 
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ClosetChartist
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Copula - Goodness of Fit Testing

March 7th, 2012, 6:37 pm

Hills1234:Your originally wrote QuoteI have a high dimensional dataset...it is a 250 by 58 matrix of uniform observations. I have fitted various copula models to it including Student T, Gaussian, Archimedean, Vine copulas, etc. Now I would like to assess the fit. I have obtained the classical measures of fit such as Log-Likelihood, AIC and BIC...but these do not directly tell me whether to accept or reject the model. This is a perfect setup for simulation-based hypothesis testing. For exampleDraw many sets of 250x58 from the Student T Copula.Compute the Log-Likelihood statistic for each set.Now you know the distribution of the Log-Likelihood statistic when the data is ACTUALLY FROM the Student T Copula.If your observed sample Log-Likelihood statistic lies in the tails, then you can reject the Student T Copula
 
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Hills1234
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Copula - Goodness of Fit Testing

March 11th, 2012, 8:06 pm

Good idea...thanks
 
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Yenlowang
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Copula - Goodness of Fit Testing

October 3rd, 2014, 3:43 pm

Hello Hills1234;I have more a question than a recommendation, I am new in the Matlab world, so I am a bit lost... I have a big dataset as well and I obtain my Uniform data with the cdf of my marginal. Afterwards, I did the copulafit and obtain the parameters.I would like to know which is the copula that fits better, can you help me in order to apply BIC and AIC. As I told you, I am really new into this.Thank you very much