Confidence bands for Horvitz–Thompson estimators using sampled noisy functional data

Abstract

When collections of functional data are too large to be exhaustively observed, survey sampling techniques provide an effective way to estimate global quantities such as the population mean function. Assuming functional data are collected from a finite population according to a probabilistic sampling scheme, with the measurements being discrete in time and noisy, we propose to first smooth the sampled trajectories with local polynomials and then estimate the mean function with a Horvitz-Thompson estimator. Under mild conditions on the population size, observation times, regularity of the trajectories, sampling scheme, and smoothing bandwidth, we prove a Central Limit theorem in the space of continuous functions. We also establish the uniform consistency of a covariance function estimator and apply the former results to build confidence bands for the mean function. The bands attain nominal coverage and are obtained through Gaussian process simulations conditional on the estimated covariance function. To select the bandwidth, we propose a cross-validation method that accounts for the sampling weights. A simulation study assesses the performance of our approach and highlights the influence of the sampling scheme and bandwidth choice.

Uniform convergence and asymptotic confidence bands for model-assisted estimators of the mean of sampled functional data

“…As in Cardot et al (2012a), it is possible to deduce from the previous proposition that the chosen value c α = c α ( γ MA,d ) provides asymptotically the desired coverage since it satisfies…”

Estimating with kernel smoothers the mean of functional data in a finite population setting. A note on variance estimation in presence of partially observed trajectories

“…For example, it is shown in a similar context with a small simulation study in Cardot et al (2013a) that linear interpolation can outperform kernel smoothing, even if the noise level is rather high, if the value of the bandwidth is chosen by a classical cross-validation performed curve by curve. This individual procedure leads to oversmoothing (see also Hart and Wehrly, 1993), so that the bias of the resulting mean estimator is much larger than its variance.…”

“…This individual procedure leads to oversmoothing (see also Hart and Wehrly, 1993), so that the bias of the resulting mean estimator is much larger than its variance. As in Cardot et al (2013a), we suggest to use a modified cross-validation in order to choose the value of the bandwidth. This modified criterion takes account of the sampling design as well as the non response process, the bandwidth value is chosen to minimize…”

Functional calibration estimation by the maximum entropy on the mean principle

“…A depth overview on functional data analysis can be found in Ramsay and Silverman [24], Ramsay and Silverman [25] and Horváth and Kokoszka [16]. Functional versions of the Horvitz-Thompson estimator have been proposed recently by Cardot and Josserand [2] and Cardot et al [3] for the cases of error free and noisy functional data, respectively.…”