A technique known as digital predistortion (DPD) is the subject of much current research and is being contemplated for the next generation RF power amplifiers. DPD has the capability of linearizing the overall amplifier response and is a cost effective technique for applications requiring wideband operation with low out-of-band spurious emissions, also known as spectral regrowth. Estimation of the DPD parameters is hampered by the presence of measurement noise at the output of the amplifier, even though this noise is not actually transmitted on air. The end effect of the noise is to degrade the DPD performance, leading to higher spectral growth. This paper shows that a major cause of this degradation can be attributed to coefficient bias induced by measurement noise in a nonlinear model. Two techniques are proposed for mitigating these effects. One uses a specially designed periodic training signal with frame averaging, and the other uses forward estimation followed by offline noiseless inverse estimation. Simulations show that spectral regrowth reductions in excess of 10 dB are possible using these techniques.