Inference for Density Families Using Functional Principal Component Analysis

Abstract

We consider t = 1,...,T samples of iid 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 numbersL, we draw conclusions on the dimensionality of mixture models for describing the densities.

KEY WORDS:

• Density evolution
• Density mixtures
• Family expenditure survey data
• Household head age
• Household income
• K-sample problems
• Kernel smoothing
• Nonparametric density estimation
• Prediction