In many practical signal detection problems, the detectors have to be designed from training data. Due to limited training data, which is usually the case, it is imperative to exploit some inherent signal structure for reliable detector design. The signals of interest in a variety of applications manifest such structure in the form of nuisance parameters. However, data-driven design of detectors by exploiting nuisance parameters is virtually impossible in general due to two major difficulties: identifying the appropriate nuisance parameters, and estimating the corresponding detector statistics. We address this problem by using recent results that relate joint signal representations (JSRs), such as time-frequency and time-scale representations, to quadratic detectors for a wide variety of nuisance parameters. We propose a general data-driven framework that: (1) identifies the appropriate nuisance parameters from an arbitrarily chosen finite set, and (2) estimates the second-order statistics that characterize the corresponding JSR-based detectors. Simulation results demonstrate that for limited training data, exploiting the structure of nuisance parameters via our framework can deliver substantial gains in performance as compared to empirical detectors which ignore such structure.