Using the Bright Ultra-Hard XMM-Newton Survey to define an IR selection
  of luminous AGN based on WISE colours
  We present a highly complete and reliable mid-infrared (MIR) colour selection
of luminous AGN candidates using the 3.4, 4.6, and 12 um bands of the WISE
survey. The MIR colour wedge was defined using the wide-angle Bright Ultra-Hard
XMM-Newton Survey (BUXS), one of the largest complete flux-limited samples of
bright (f(4.5-10 keV)>6x10^{-14} erg cm^-2 s^-1) "ultra-hard" (4.5-10 keV)
X-ray selected AGN to date. BUXS includes 258 objects detected over a total sky
area of 44.43 deg^2 of which 251 are spectroscopically identified and
classified, with 145 being type-1 AGN and 106 type-2 AGN. Our technique is
designed to select objects with red MIR power-law spectral energy distributions
(SED) in the three shortest bands of WISE and properly accounts for the errors
in the photometry and deviations of the MIR SEDs from a pure power-law. The
completeness of the MIR selection is a strong function of luminosity. At L(2-10
keV)>10^{44} erg s^-1, where the AGN is expected to dominate the MIR emission,
97.1_{-4.8}^{+2.2}% and 76.5_{-18.4}^{+13.3}% of the BUXS type-1 and type-2 AGN
meet the selection. Our technique shows one of the highest reliability and
efficiency of detection of the X-ray selected luminous AGN population with WISE
amongst those in the literature. In the area covered by the BUXS survey our
selection identifies 2755 AGN candidates detected with SNR>5 in the three
shorter wavelength bands of WISE with 38.5% having a detection at 2-10 keV
X-ray energies. We also analyzed the possibility of including the 22um WISE
band to select AGN candidates, but neither the completeness nor the reliability
of the selection improves. This is likely due to both the significantly
shallower depth at 22um compared with the first three bands of WISE and
star-formation contributing to the 22um emission at the WISE 22um sensitivity.
