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I've fitted my data to a generalized pareto distribution as to model the returns in the tails more accurately. The interior is fitted with kernel distributions.

I would like to now test whether the original returns conform to the hypothesized distribution (i.e. generalized pareto distribution). Can I do this with the Kolmogorov-Smirnov test? I've already QQ-plots. However, I would like to conduct a statistical significance test on top. Can some one help? Kind of struggling with implementing it in Matlab.

Best

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I don't know if there are any additional issues that arise with using goodness off fit with a piece-wise function. When I have fit generalized pareto distributions to series like financial market returns, I have noticed that it is common to differences between the estimated distribution and observed returns at the cutoff points. This is going to be the main difference between running the goodness of fit test on the entire GPD as opposed to the fits for the tails individually, since the kernel density fit will be good.

If you have an estimate of your hypothesized distribution, I would recommend using the Anderson-Darling test instead of the KS-test. The KS-test checks for the maximum distance between the empirical distribution function and the hypothesized function, and thus is not that sensitive to the tails which is what you care about. The Anderson-Darling test integrates over the squared difference between empirical distribution and the hypothesized, and places different weights on each part of the distribution. The weighting function effectively places greater weight in the tails.

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As an additional (simple) solution I would use the probability integral transform (PIT) of the returns with respect to the generalized pareto distribution. Under the null hypothesis that the distribution is correctly specified, outcomes of the PIT should be independent uniform U[0; 1] random variables. Then you can use traditional independence tests.

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  • $\begingroup$ Isn't that equivalent to qq-plot? $\endgroup$ – Kiwiakos Apr 1 '16 at 16:40

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