On estimation of covariance function for functional data with detection limits

In many studies on disease progression, biomarkers are restricted by detection limits, hence informatively missing. Current approaches ignore the problem by just filling in the value of the detection limit for the missing observations for the estimation of the mean and covariance function, which yield inaccurate estimation. Inspired by our recent work [Liu and Houwing-Duistermaat (2022), 'Fast Estimators for the Mean Function for Functional Data with Detection Limits', Stat, e467.] in which novel estimators for mean function for data subject to detection limit are proposed, in this paper, we will propose a novel estimator for the covariance function for sparse and dense data subject to a detection limit. We will derive the asymptotic properties of the estimator. We will compare our method to the standard method, which ignores the detection limit, via simulations. We will illustrate the new approach by analysing biomarker data subject to a detection limit. In contrast to the standard method, our method appeared to provide more accurate estimates of the covariance. Moreover its computation time is small.

Automated summary

In many disease progression studies, biomarkers can't be accurately detected, leading to missing information. The current approach fills in the detection limits for missing observations when estimating the mean and covariance function, resulting in inaccurate estimation. This paper proposes a novel estimator for the covariance function of sparse and dense data subject to a detection limit, inspired by recent work on estimators for mean function under detection limits. The asymptotic properties of the estimator will be derived and compared to the standard method through simulations. Analysis of biomarker data subject to a detection limit shows that the proposed method provides more accurate covariance estimates with shorter computation time compared to the standard method.