Inference for Density Families Using Functional Principal Component Analysis

We consider t = 1,. . . , T samples of iid observations X1t, …, Xntt from unknown population densities ft. To characterize differences and similarities of ft, we assume their expansions into the first L principal components. From the given observations Xit, 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 numbersL, we draw conclusions on the dimensionality of mixture models for describing the densities.