How to model censored regressor (not censored dependent variable)


How to model a censored regressor? Imagine:
Y= b_1*X_1 +b_2*X_2 + b_0
I observe X_1 and X_2 and Y and I want to obtain b_1 and b_2 and b_0. However, X_1 is truncated and whenever it is greater than a number (300 here) we just observe 300. Other values including dependent variable are not censored.
I think one way of doing it is to treat it like a missing value, but in this way I am not using the information that this missed value is greater than 300.



Seems like you would need a nested model

Y|X_1,X_2 = b_1 X_1 + b_2 X_2 + b_0 + \epsilon
X_1|U_1 = \left\{\begin{array}{cc} 300 & U_1 > 300 \\\ U_1 & U_1 \leq 300\end{array}\right.

You would then need a prior \pi(U_1, b_0, b_1, b_2) for inference.



You would need to assume some distribution over X_1 and follow the imputation strategy described here