  This paper proposes a new model-based approach to small area estimation of
general finite-population parameters based on grouped data or frequency data,
which is often available from sample surveys. Grouped data contains information
on frequencies of some pre-specified groups in each area, for example the
numbers of households in the income classes, and thus provides more detailed
insight about small areas than area-level aggregated data. A direct application
of the widely used small area methods, such as the Fay-Herriot model for
area-level data and nested error regression model for unit-level data, is not
appropriate since they are not designed for grouped data. The newly proposed
method adopts the multinomial likelihood function for the grouped data. In
order to connect the group probabilities of the multinomial likelihood and the
auxiliary variables within the framework of small area estimation, we introduce
the unobserved unit-level quantities of interest which follows the linear mixed
model with the random intercepts and dispersions after some transformation.
Then the probabilities that a unit belongs to the groups can be derived and are
used to construct the likelihood function for the grouped data given the random
effects. The unknown model parameters (hyperparameters) are estimated by a
newly developed Monte Carlo EM algorithm using an efficient importance
sampling. The empirical best predicts (empirical Bayes estimates) of small area
parameters can be calculated by a simple Gibbs sampling algorithm. The
numerical performance of the proposed method is illustrated based on the
model-based and design-based simulations. In the application to the city level
grouped income data of Japan, we complete the patchy maps of the Gini
coefficient as well as mean income across the country.
