Algorithm 1.

Fast Imputation Algorithm Step

Step 1: Quantile regression with the complete data on a fine grid of quantile levels $0<{\tau }_1<\cdots <{\tau }_k<\cdots <{\tau }_{k_n}<1$.

Step 2: Model the conditional density f (x|z) parametrically as f (x | z, η), and estimate η based on the complete data.

Step 3: Simulate M x from the estimated $f(x|z,\widehat{\eta })$ for each missing xi, 1 ≤ i ≤ n1.

Step 4: Calculate the weights using the model induced density from Step 1, and assemble the weighted estimating function as in (5) to get the final estimator.