The radiation pattern of an antenna array depends on the excitation weights and the geometry of the array. Due to wind and atmospheric conditions, outdoor millimeter wave antenna elements are subject to full or partial blockages from a plethora of particles like dirt, salt, ice, and water droplets. Handheld devices are also subject to blockages from random finger placement and/or finger prints. These blockages cause absorption and scattering to the signal incident on the array, modify the array geometry, and distort the far-field radiation pattern of the array. This paper studies the effects of blockages on the far-field radiation pattern of linear arrays and proposes several array diagnosis techniques for millimeter wave antenna arrays. The proposed techniques jointly estimate the locations of the blocked antennas and the induced attenuation and phase-shifts given knowledge of the angles of arrival/departure. Numerical results show that the proposed techniques provide satisfactory results in terms of fault detection with reduced number of measurements (diagnosis time) provided that the number of blockages is small compared to the array size.