A major outstanding question regarding the formation of planetary systems is whether wide-orbit giant planets form differently than close-in giant planets .
We aim to establish constraints on two key parameters that are relevant for understanding the formation of wide-orbit planets : 1 ) the relative mass function and 2 ) the fraction of systems hosting multiple companions .
In this study , we focus on systems with directly imaged substellar companions , and the detection limits on lower-mass bodies within these systems .
First , we uniformly derive the mass probability distributions of known companions .
We then combine the information contained within the detections and detection limits into a survival analysis statistical framework to estimate the underlying mass function of the parent distribution .
Finally , we calculate the probability that each system may host multiple substellar companions .
We find that 1 ) the companion mass distribution is rising steeply toward smaller masses , with a functional form of N \propto M ^ { -1.3 \pm 0.3 } , and consequently , 2 ) many of these systems likely host additional undetected sub-stellar companions .
Combined , these results strongly support the notion that wide-orbit giant planets are formed predominantly via core accretion , similar to the better studied close-in giant planets .
Finally , given the steep rise in the relative mass function with decreasing mass , these results suggest that future deep observations should unveil a greater number of directly imaged planets .