Kernel and smoothing methods for nonparametric function and curve estimation have been particularly successful in "standard" settings, where function values are observed subject to independent errors. However, when aspects of the function are known parametrically, or where the sampling scheme has significant structure, it can be quite difficult to adapt standard methods in such a way that they retain good statistical performance and continue to enjoy easy computability and good numerical properties. In particular, when using local linear modeling it is often awkward to both respect the sampling scheme and produce an estimator with good variance properties, without resorting to iterative methods: a good case in point is longitudinal and clustered data. In this paper we suggest a simple approach to overcoming these problems. Using a histospline technique we convert a problem in the continuum to one that is governed by only a finite number of parameters, and which is often explicitly solvable. The simple expedient of running a local linear smoother through the histospline produces a function estimator which achieves optimal nonparametric properties, and the "raw" histospline-based estimator of the semiparametric component itself attains optimal semiparametric performance. The function estimator can be used in its own right or as the starting value for an iterative scheme based on a different approach to inference.
Longitudinal Data Analysis and Time Series | Statistical Models
Carroll, Raymond J.; Hall, Peter; Apanasovich, Tatiyana V.; and Lin, Xihong, "Histospline Method in Nonparametric Regression Models with Application to Clustered/Longitudinal Data" (September 2003). The University of Michigan Department of Biostatistics Working Paper Series. Working Paper 13.