Abstract: Motivated by quantifying the monotonic relationship between gray matter (GM) volume and age in the older population, this study proposes a constrained nonparametric estimation and statistical inference for the monotone mean function of functional/longitudinal data. Under some mild conditions, we systematically investigate the asymptotic properties of the proposed estimators, using a general weighting scheme that includes an equal weight per observation (OBS) and an equal weight per subject (SUBJ) as special cases. Most existing methods without a structural constraint can handle sparse or dense data only. Thus, a subjective choice between the two types may lead to erroneous conclusions from statistical inferences. Our proposed method and theories adapt to sparse and dense cases on a unified platform under a monotonic constraint. The asymptotic results enable us to categorize functional/longitudinal data into three data types (i.e., sparse, dense, and ultra-dense), based on the relative order of the number of repeated measurements relative to the total number of subjects. Simulation studies are conducted to examine the finite-sample performance of the estimating and statistical inference procedures. Our analysis of GM volume data, obtained from the Alzheimer's Disease Neuroimaging Initiative study, confirms the accuracy and rationality of the constrained estimators in characterizing cerebellar GM volume with increasing age.

Key words and phrases: Asymptotic normality, isotone regression, kernel smoothing, monotonicity constraint, nonparametric estimation, sparse and dense functional/longitudinal data, weighting schemes.