Inference for density families using functional principal component analysis

We consider t = 1,..., T samples of lid observations {X-1t,...,X-ntt} from unknown population densities {f(t)}. To characterize differences and similarities of {f(t)}, we assume their expansions into the first L principal components. From the given observations {X-it}, we study inference on the components and on their required number L. A detailed asymptotic theory is presented. Our method is applied in the analysis of yearly cross-sectional samples of British households. Interpretation of the estimated principal components and their scores provides new insights into the evolution and interplay of household income and age distributions from 1968-1988. From estimating their required numbers L, we draw conclusions on the dimensionality of mixture models for describing the densities.