We present a spectroscopically complete sample of 147 infrared-color-selected AGN down to a 22 $mu$m flux limit of 20 mJy over the $sim$270 deg$^2$ of the SDSS Stripe 82 region. Most of these sources are in the QSO luminosity regime ($L_{rm bol} gtrsim 10^{12} L_odot$) and are found out to $zsimeq3$. We classify the AGN into three types, finding: 57 blue, unobscured Type-1 (broad-lined) sources; 69 obscured, Type-2 (narrow-lined) sources; and 21 moderately-reddened Type-1 sources (broad-lined and $E(B-V) > 0.25$). We study a subset of this sample in X-rays and analyze their obscuration to find that our spectroscopic classifications are in broad agreement with low, moderate, and large amounts of absorption for Type-1, red Type-1 and Type-2 AGN, respectively. We also investigate how their X-ray luminosities correlate with other known bolometric luminosity indicators such as [O III] line luminosity ($L_{rm [OIII]}$) and infrared luminosity ($L_{6 mu{rm m}}$). While the X-ray correlation with $L_{rm [OIII]}$ is consistent with previous findings, the most infrared-luminous sources appear to deviate from established relations such that they are either under-luminous in X-rays or over-luminous in the infrared. Finally, we examine the luminosity function (LF) evolution of our sample, and by AGN type, in combination with the complementary, infrared-selected, AGN sample of Lacy et al. (2013), spanning over two orders of magnitude in luminosity. We find that the two obscured populations evolve differently, with reddened Type-1 AGN dominating the obscured AGN fraction ($sim$30%) for $L_{5 mu{rm m}} > 10^{45}$ erg s$^{-1}$, while the fraction of Type-2 AGN with $L_{5 mu{rm m}} < 10^{45}$ erg s$^{-1}$ rises sharply from 40% to 80% of the overall AGN population.
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