Identification from corrupted input-output measurements of systems that do not necessarily belong to the model class used is investigated. This leads to a nonstandard set membership (SM) identification problem. The ’goodness’ of different model classes is measured by the conditional radius of information, a generalization of the radius in standard SM identification theory, giving a measure of the minimal worst-case modeling error. Upper and lower bounds on the radius are derived for linearly parameterized model classes. Specific formulas for the upper and lower bounds are given for the case of H-2 identification of exponentially stable systems in the presence of power-bounded noise. The bounds are shown to coincide with the conditional radius when the model space dimension is equal to the number of output measurements. An almost-optimal identification algorithm is derived, giving identification error within the range of the derived bounds.