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Inference for density families using functional principal component analysis. (With comments). (English) Zbl 1019.62060

Summary: We consider \(t=1,\dots,T\) samples of iid observations \(\{X_{1t}, \dots, X_{n_tt}\}\) 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.

MSC:

62H25 Factor analysis and principal components; correspondence analysis
62P20 Applications of statistics to economics
62G07 Density estimation

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