The problem is to detect a multi-dimensional source transmitting an unknown sequence of complex-valued symbols to a multi-sensor array. In some cases the channel subspace is known, and in others only its dimension is known. Should the unknown transmissions be treated as unknowns in a first-order statistical model, or should they be assigned a prior distribution that is then used to marginalize a first-order model for a second-order statistical model? This question motivates the derivation of subspace detectors for cases where the subspace is known, and for cases where only the dimension of the subspace is known. For three of these four models the GLR detectors are known, and they have been reported in the literature. But the GLR detector for the case of a known subspace and a second-order model for the measurements is derived for the first time in this paper. When the subspace is known, second-order generalized likelihood ratio (GLR) tests outperform first-order GLR tests when the spread of subspace eigenvalues is large, while first-order GLR tests outperform second-order GLR tests when the spread is small. When only the dimension of the subspace is known, second-order GLR tests outperform first-order GLR tests, regardless of the spread of signal subspace eigenvalues. For a dimension-1 source, first-order and second-order statistical models lead to equivalent GLR tests. This is a new finding. The work by I. Santamaria was supported by the Ministerio de Ciencia e Innovación of Spain, and AEI/FEDER funds of the E.U., under grants TEC2016-75067-C4-4-R (CARMEN) and PID2019-104958RB-C43 (ADELE). The work by L. Scharf is supported by the Air Force Office of Scientific Research under contract FA9550-18-1-0087, and by the National Science Foundation (NSF) under contract CCF-1712788. The work of D. Ramírez was supported by the Ministerio de Ciencia, Innovación y Universidades under grant TEC2017-92552-EXP (aMBITION), by the Ministerio de Ciencia, Innovación y Universidades, jointly with the European Commission (ERDF), under grant TEC2017-86921-C2-2-R (CAIMAN), and by The Comunidad de Madrid under grant Y2018/TCS-4705 (PRACTICO-CM). Publicado