Before revision:  The noise-robust estimators are constructed as a plug-in version of their counterparts, where we replace the efficient price, which is non-observable in our framework , by an estimator based on the raw price and the limit order book data.
After revision:  The noise-robust estimators are constructed as a plug-in version of their counterparts, where we replace the efficient price, which is non-observable in our framework , by an estimator based on the raw price and limit order book data.