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Using a sensitive statistical test we determine whether or not one can detectthe breakdown of linear response given observations of deterministic dynamicalsystems. A goodness-of-fit statistics is developed for a linear statisticalmodel of the observations, based on results on central limit theorems fordeterministic dynamical systems, and used to detect linear response breakdown.We apply the method to discrete maps which do not obey linear response and showthat the successful detection of breakdown depends on the length of the timeseries, the magnitude of the perturbation and on the choice of the observable.  We find that in order to reliably reject the assumption of linear responsefor typical observables sufficiently large data sets are needed. Even forsimple systems such as the logistic map, one needs of the order of $10^6$observations to reliably detect the breakdown, if less observations areavailable one may be falsely led to conclude that linear response theory isvalid. The amount of data required is larger the smaller the appliedperturbation. For judiciously chosen observables the necessary amount of datacan be drastically reduced, but requires detailed {\em{a priori}} knowledgeabout the invariant measure which is typically not available for complexdynamical systems.  Furthermore we explore the use of the fluctuation-dissipation theorem (FDT)in cases with limited data length or coarse-graining of observations. The FDT,if applied naively to a system without linear response, is shown to be verysensitive to the details of the sampling method, resulting in erroneouspredictions of the response.

On spurious detection of linear response and misuse of the fluctuation-dissipation theorem in finite time series