Estimation of quantiles based on Fay-Herriot models


Robust methods

Ann-Kristin Kreutzmann (Freie Universität Berlin) (Speaker)
Nicola Salvati (Università di Pisa)
Timo Schmid (Freie Universität Berlin, Germany)

Central banks and politics often use robust measures like quantiles in order to describe the distribution of income or wealth in a country. However, estimates on a disaggregated level are rarely reported due to small sample sizes and following large variances. Small area estimation is one way to handle this issue but standard approaches are mainly suited for the estimation of means and totals. Recently proposed methodologies like the Empirical Best Prediction (EBP) (Molina and Rao 2010) and the World Bank method (Elbers et al. 2003) enable the estimation of complex indicators like poverty measures as well as the estimation of quantiles. However, these methods need covariate information on the unit- or rather household-level (micro-level). Due to confidentiality issues, this requirement is often not given for data related to income or wealth in a country. What are the alternatives for the data analyst? Recently, Fabrizi and Trivisano (2016) discussed the estimation of the Gini coefficient based on aggregated models that can be used without the availability of census data at the micro-level. Similarly, we discuss the estimation of quantiles based on aggregated data.
The objective of the present work is twofold. First, as Fay-Herriot type models (Fay and Herriot 1979) rely on the assumption of approximated unbiased direct estimators, we assess the properties of direct estimators with regard to the bias in small samples. In particular, we propose modified direct estimators of quantiles characterized by a smaller bias in small samples. For instance, we incorporate sampling weights to the Harrel-Davis estimator (Harrel and Davis 1982) and to the weighted average at order statistic X[np+0.5] proposed in Parrish (1990) and develop suitable variance estimation. Second, the modified direct estimators are used for the estimation of quantiles based on Fay-Herriot models. To link the direct estimates to the area level information, we consider linear mixed models based on normality. The introduced methodology is assessed in large-scaled empirical evaluations. Finally, we illustrate the proposed methods to analyse the net wealth in 96 spatial planning regions in Germany using data from the Household Finance and Consumption Survey (HFCS).

Elbers, C., Lanjouw, J. O., & Lanjouw, P. (2003) Micro−level estimation of poverty and inequality, Econometrica, 71(1), 355–364. Fabrizi, E. & Trivisano, C. (2016) Small area estimation of the Gini concentration coefficient, Computational Statistics and Data Analysis, 99, 223–234. Fay, R.E. & Herriot, R.A. (1979) Estimates of income for small places: an application of James-Stein procedures to census data, Journal of the American Statistical Association, 74(366), 269–277. Harrel, F. E. & Davis, E. C. (1982) A new distribution-free quantile estimator, Biometrika, 69(3), 635–640. Molina, I., & Rao, J.N.K. (2010) Small area estimation of poverty indicators, Canadian Journal of Statistics, 38(3), 369–385. Parrish, R.S. (1990) Comparison of quantile estimators in normal sampling, Biometrics, 46(1), 247–257.