Parameter estimation using quantized observations is of importance in manypractical applications. Under a symmetric $1$-bit setup, consisting of azero-threshold hard-limiter, it is well known that the large sample performanceloss for low signal-to-noise ratios (SNRs) is moderate ($\frac{2}{\pi}$ or$-1.96$dB). This makes low-complexity analog-to-digital converters (ADCs) with$1$-bit resolution a promising solution for future wireless communications andsignal processing devices. However, hardware imperfections and external effectsintroduce the quantizer with an unknown hard-limiting level different fromzero. In this paper, the performance loss associated with pilot-based channelestimation, subject to an asymmetric hard limiter with unknown offset, isstudied under two setups. The analysis is carried out via the Cram\'{e}r-Raolower bound (CRLB) and an expected CRLB for a setup with random parameter. Ourfindings show that the unknown threshold leads to an additional informationloss, which vanishes for low SNR values or when the offset is close to zero.
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