Local asymptotic behavior of regression splines for marginal semiparametric models with longitudinal data

Abstract

In this paper, we study the local asymptotic behavior of the regression spline estimator in the framework of marginal semiparametric model. Similarly to Zhu, Fung and He (2008), we give explicit expression for the asymptotic bias of regression spline estimator for nonparametric function f . Our results also show that the asymptotic bias of the regression spline estimator does not depend on the working covariance matrix, which distinguishes the regression splines from the smoothing splines and the seemingly unrelated kernel. To understand the local bias result of the regression spline estimator, we show that the regression spline estimator can be obtained iteratively by applying the standard weighted least squares regression spline estimator to pseudo-observations. At each iteration, the bias of the estimator is unchanged and only the variance is updated.

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“…Compared to P‐splines, while the smoothing splines and P‐splines show similar estimated curves (Berry et al., 2002), the smoothing splines have the lowest mean squared error (MSE) among other smoothing functions due to the interpolant properties when the smoothing parameters are fixed (Keele, 2008). Qin & Zhu (2009) empirically compared the performance of smoothing splines with P‐splines, which favored smoothing splines in terms of bias and MSE. Moreover, smoothing splines is free from the knot selection problem.…”

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“…Without the presence of measurement error, smoothing splines have the lowest mean squared error (MSE) among other smoothing functions due to the interpolant properties when the smoothing parameters are fixed (Keele, 2008, p. 73). Qin and Zhu (2009) empirically compared the performance of smoothing splines with P‐splines, which favored smoothing splines in terms of bias and MSE. Moreover, smoothing splines is free from the knot selection problem.…”