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The detection of small exoplanets with the radial velocity (RV) technique is limited by various poorly known noise sources of instrumental and stellar origin. As a consequence, current detection techniques often fail to provide reliable estimates of the significance levels of detection tests (p-values). We designed an RV detection procedure that provides reliable p-value estimates while accounting for the various noise sources. The method can incorporate ancillary information about the noise (e.g., stellar activity indicators) and specific data- or context-driven data (e.g., instrumental measurements, simulations of stellar variability) . The detection part of the procedure uses a detection test that is applied to a standardized periodogram. Standardization allows an autocalibration of the noise sources with partially unknown statistics. The estimation of the p-value of the test output is based on dedicated Monte Carlo simulations that allow handling unknown parameters. The procedure is versatile in the sense that the specific pair (periodogram and test) is chosen by the user. We demonstrate by extensive numerical experiments on synthetic and real RV data from the Sun and aCenB that the proposed method reliably allows estimating the p-values. The method also provides a way to evaluate the dependence of the estimated p-values that are attributed to a reported detection on modeling errors. It is a critical point for RV planet detection at low signal-to-noise ratio to evaluate this dependence. The python algorithms are available on GitHub. Accurate estimation of p-values when unknown parameters are involved is an important but only recently addressed question in the field of RV detection. Although this work presents a method to do this, the statistical literature discussed in this paper may trigger the development of other strategies.