Parameter estimation using quantized observations is of importance in many practical applications. Under a symmetric 1-bit setup, consisting of a zero-threshold hard limiter, it is well known that the large sample performance loss for low signal-to-noise ratios (SNRs) is moderate (2/pi or -1.96dB). This makes low-complexity analog-to-digital converters (ADCs) with 1-bit resolution a promising solution for future wireless communications and signal processing devices. However, hardware imperfections and external effects introduce the quantizer with an unknown hard-limiting level different from zero. In this paper, the performance loss associated with pilot-based channel estimation, subject to an asymmetric hard limiter with unknown offset, is studied under two setups. The analysis is carried out via the Cramer-Rao lower bound (CRLB) and an expected CRLB for a setup with random parameter. Our findings show that the unknown threshold leads to an additional information loss, which vanishes for low SNR values or when the offset is close to zero.