Semiparametric quantile regression imputation for a complex survey with application to the Conservation Effects Assessment Project
Section 6. Discussion

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QRI is developed for a complex survey setting. Alternative choices of weights are discussed, and a closed form variance estimator is provided based on a linear approximation. Consistency and asymptotic normality of the estimators are demonstrated under the framework of an infinite number of imputed values. In simulations designed to represent the CEAP data, the variance estimator based on the asymptotic distribution has a relative bias less than 6% in absolute value and leads to confidence intervals with coverage close to the nominal level for finite $J.$ Further, the estimator based on QRI is more efficient than an estimator based on PFI or NPI because QRI provides a reasonable compromise between bias and variance.

The quantile regression imputation procedure is applied to estimate mean erosion in seven states in the midwestern United States using data from the Conservation Effects Assessment Project. The analysis demonstrates that QRI presents a viable alternative to weighting adjustments currently used to account for nonresponse in CEAP.

Areas for improvement to QRI include the choice of ${\tau }_j,$ the choice of $b_i,$ refinements to estimation of the quantile curves, and variance estimation for non-differentiable $g(\cdot )$ functions. Development of automated methods to select the nuisance parameters, appropriate for selection of multiple quantiles in a complex survey setting, is an area for future research. Estimation of the quantile curves subject to a restriction that the estimated curves are non-overlapping, has potential to improve estimation of the derivatives needed for the variance estimator. Section E of the online supplement https://github.com/ emilyjb/Semiparametric-QRI-Supplement/blob/master/SupplementToQRI.pdf, (Berg and Yu, 2016) provides further discussion of areas for improvement.

Acknowledgements

This work was supported by Cooperative Agreement No. 68-3A75-4-122 between the USDA Natural Resources Conservation Service and the Center for Survey Statistics and Methodology at Iowa State University.

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