Fused kernel-spline smoothing for repeatedly measured outcomes in a generalized partially linear model with functional single index

TL;DR

This paper introduces a novel modeling approach combining kernel and spline methods within a generalized partially linear functional single index model for repeatedly measured outcomes, accounting for complex covariance structures.

Contribution

It develops a new estimation framework that fuses kernel and spline techniques, providing valid inference for both continuous and discrete outcomes in a longitudinal setting.

Findings

Derived large sample properties for the combined estimation method.

Established different convergence rates for each model component.

Validated the approach's applicability to various outcome types.

Abstract

We propose a generalized partially linear functional single index risk score model for repeatedly measured outcomes where the index itself is a function of time. We fuse the nonparametric kernel method and regression spline method, and modify the generalized estimating equation to facilitate estimation and inference. We use local smoothing kernel to estimate the unspecified coefficient functions of time, and use B-splines to estimate the unspecified function of the single index component. The covariance structure is taken into account via a working model, which provides valid estimation and inference procedure whether or not it captures the true covariance. The estimation method is applicable to both continuous and discrete outcomes. We derive large sample properties of the estimation procedure and show a different convergence rate for each component of the model. The asymptotic properties when the kernel and regression spline methods are combined in a nested fashion has not been studied prior to this work, even in the independent data case.