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In typical parameter estimation problems, the signal observation is a function of the parameter set to be estimated as well as some background (environmental/system) parameters assumed known. The assumed background could differ from the true one, leading to biased estimates even at high signal-to-noise ratio (SNR). To analyze this background mismatch problem, a Ziv-Zakai-type lower bound on the mean-square error (MSE) is developed based on the mismatched likelihood ratio test (MLRT). At high SNR, the bound incorporates the increase in MSE due to estimation bias; at low SNR, it includes the threshold effect due to estimation ambiguity. The kernel of the bound's evaluation is the error probability associated with the MLRT. A closed-form expression for this error probability is derived under a random signal model typical of the bearing estimation/passive source localization problem. The mismatch is then analyzed in terms of the related ambiguity functions. Examples of bearing estimation with system (array shape) mismatch demonstrate that the developed bound describes the simulations of the maximum-likelihood estimate well, including the sidelobe-introduced threshold behavior and the bias at high SNR.