In this paper we assess the relationship of the population of Active Galactic Nuclei ( AGN ) selected by hard X-rays to the traditional population of AGN with strong optical emission lines . First , we study the emission-line properties of a new hard X-ray selected sample of 47 local AGN ( classified optically as both Type 1 and Type 2 AGN ) . We find that the hard X- ray ( 3-20 keV ) and [ OIII ] \lambda 5007 optical emission-line luminosities are well-correlated over a range of about four orders-of-magnitude in luminosity ( mean luminosity ratio 2.15 dex with a standard deviation of \sigma = 0.51 dex ) . Second , we study the hard X-ray properties of a sample of 55 local AGN selected from the literature on the basis of the flux in the [ OIII ] line . The correlation between the hard X-ray ( 2-10 keV ) and [ OIII ] luminosity for the Type 1 AGN is consistent with what is seen in the hard X-ray selected sample . However , the Type 2 AGN have a much larger range in the luminosity ratio , and many are very weak in hard X-rays ( as expected for heavily absorbed AGN ) . We then compare the hard X-ray ( 3-20 keV ) and [ OIII ] luminosity functions of AGN in the local universe . These have similar faint-end slopes with a luminosity ratio of 1.60 dex ( 0.55 dex smaller than the mean value for individual hard X-ray selected AGN ) . We conclude that at low redshift , selection by narrow optical emission- lines will recover most AGN selected by hard X-rays ( with the exception of BL Lac objects ) . However , selection by hard X-rays misses a significant fraction of the local AGN population with strong emission lines .