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This paper investigates the sensitivity of the achievables rates for the full-duplex relay channel to small additive disturbances on channel links. The focus is on two relaying strategies - the decode-and-forward (DF) mode and the compress-and-forward (CF) mode. We use Fisher Information and De-Bruijn's identity to assess the decrease in the corresponding rates due to small additive contaminating noise. Analysis sheds light on the respective sensitivity levels of these schemes and hence, provides insights onto the choice of appropriate relaying strategies in the situations where some trade-off between transmission rate and sensitivity is needed. Next, we show that these results can be used to emphasize the effect of channel estimation error on relaying transmissions. An important (somehow intuitive) observation at this stage is that transmission through the direct link (i.e., relay is off) may improve upon both decode-and-forward and compress-and-forward schemes, when the channel is "bad enough". Finally, a lower bound on the capacity of a relay channel under channel estimation error is obtained by combining well known relaying strategies, each over the appropriate SNR range. Analysis is supported by some examples.