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We study the correlations of the shear signal between triplets of sources inthe Canada-France-Hawaii Lensing Survey (CFHTLenS) to probe cosmologicalparameters via the matter bispectrum. In contrast to previous studies, weadopted a non-Gaussian model of the data likelihood which is supported by oursimulations of the survey. We find that for state-of-the-art surveys, similarto CFHTLenS, a Gaussian likelihood analysis is a reasonable approximation,albeit small differences in the parameter constraints are already visible. Forfuture surveys we expect that a Gaussian model becomes inaccurate. Ouralgorithm for a refined non-Gaussian analysis and data compression is then ofgreat utility especially because it is not much more elaborate if simulateddata are available. Applying this algorithm to the third-order correlations ofshear alone in a blind analysis, we find a good agreement with the standardcosmological model: $\Sigma_8$=$\sigma_8$ $(\Omega_{\rmm}/0.27)^{0.64}$=$0.79^{+0.08}_{-0.11}$ for a flat $\Lambda\rm CDM$ cosmologywith $h=0.7\pm0.04$ ($68\%$ credible interval). Nevertheless our models provideonly moderately good fits as indicated by $\chi^2/{\rm dof}=2.9$, including a$20\%$ r.m.s. uncertainty in the predicted signal amplitude. The models cannotexplain a signal drop on scales around 15 arcmin, which may be caused bysystematics. It is unclear whether the discrepancy can be fully explained byresidual PSF systematics of which we find evidence at least on scales of a fewarcmin. Therefore we need a better understanding of higher-order correlationsof cosmic shear and their systematics to confidently apply them as cosmologicalprobes.

CFHTLenS: A Gaussian likelihood is a sufficient approximation for a cosmological analysis of third-order cosmic shear statistics