Variance Estimation for Regression Imputed Quantiles and Inequality Indicators

This work is done in the framework of sampling theory. It is based on a scenario in which a sample survey has been carried out and only a sub-part of the selected sample has accepted to answer (total non-response). Moreover, some respondents did not answer all questions (partial non-response). This is common scenario in practice. We are particularly interested in income type variables. Generally, total non-response is treated by re-weighting and partial non-response through imputation. One further supposes here that the imputation is carried out by a regression model adjusted on the respondents. We then resume the idea presented by \citet{dev:sar:94} and tested afterwards by \citet{LeeRancSar:1994} which consists in constructing an unbiased estimator of the variance of a total based solely on the information at our disposal: the information known on the selected sample and the subset of respondents. While the two cited articles dealt with the exercise for a conventional total of an interest variable $y$, we reproduce here a similar development in the case where the considered total is one of the linearized variable of quantiles or of inequality indicators, and that, furthermore, it is computed from the imputed variable $y$. We show by means of simulations that regression imputation can have an important impact on the bias estimation as well as on the variance estimation of inequality indicators. This leads us to a method capable of taking into account the variance due to imputation in addition to the one due to the sampling design in the cases of quantiles. This method could be generalized to some of the inequality indicators.