New method is described to investigate the damping factor \barγ which represents the exponential decay of the magnetophonon resonance amplitude in the inverse magnetic field. The method is based on measurement of the second derivative signals of the magnetoresistance utilizing magnetic field modulation technique and analysis using the empirical formula obtained by Stradling and Wood. Measurements are performed between 58 and 235 K in n-InSb with carrier concentrations 3.9 ×10 13 ∼8.3 ×10 14 cm -3 . The observed temperature, impurity and carrier concentration dependence of \barγ ∝( T n s 2 / n ) 1/4 at T <160 K indicates that the dominant process is the band tailing effect. A shift of the extremal position to higher magnetic field is observed. The shift is found to be larger for samples with larger damping factor \barγ. A tentative interpretation for the shift is made in terms of Landau level shift induced by the real part of the electron self-energy.