In this letter, the performance of mismatched likelihood ratio detectors for binary Bayesian hypothesis testing problems is considered. Based on large deviation theory, a method for achieving the maximum Bayesian error exponent for a mismatched likelihood ratio detector is presented. It is shown that the maximum Bayesian error exponent is given by generalized Chernoff information, which is an extension of the Chernoff information to the case of two mismatched distributions and has similar properties to those of the original Chernoff information. As an application example, energy detection under the Gauss-Markov signal model, is considered. It is shown that the generalized Chernoff information of energy detection, which is achieved by optimally choosing the detection threshold, is close to the original Chernoff information for the considered signal model, and thus, the performance of suboptimal energy detection can be improved significantly simply by choosing the detection threshold judiciously.