Wavelet-based estimation in aggregated functional data with positive and correlated errors

Abstract

We consider the statistical problem of estimating constituent curves from observations of their aggregated curves, referred to as \textit{aggregated functional data}, in models with strictly positive random errors following a Gamma distribution and correlated errors structured through AR(1) and ARFIMA processes. This problem arises in several areas of knowledge, such as chemometrics, for example, when absorbance curves of the constituents of a given substance must be estimated from its aggregated absorbance curve according to the Beer--Lambert law. In this context, we propose Bayesian wavelet-based methods to estimate the component functions within a functional data analysis framework. This approach has the advantage of accurately estimating curves with important local features, such as discontinuities, peaks, and oscillations, due to the representation properties of functions in wavelet bases. We further evaluate the performance of the proposed method through computational simulations, as well as applications to real data.

Figures (2)

• Figure 1: Donoho and Johnstone test functions.
• Figure 2: Boxplots of the MSE of the proposed method in the model under iid gamma errors for $L=2$ component functions (Bumps and Doppler). The MSE is related to the estimation of the aggregated curve.