RATES OF CONVERGENCE IN SEMI‐PARAMETRIC MODELLING OF LONGITUDINAL DATA

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Abstract
Summary: We consider the problem of semi‐parametric regression modelling when the data consist of a collection of short time series for which measurements within series are correlated. The objective is to estimate a regression function of the form E[Y(t) |x] =x'ß+μ(t), where μ(.) is an arbitrary, smooth function of timet, andxis a vector of explanatory variables which may or may not vary witht.For the non‐parametric part of the estimation we use a kernel estimator with fixed bandwidthh.Whenhis chosen without reference to the data we give exact expressions for the bias and variance of the estimators for β and μ(t) and an asymptotic analysis of the case in which the number of series tends to infinity whilst the number of measurements per series is held fixed. We also report the results of a small‐scale simulation study to indicate the extent to which the theoretical results continue to hold whenhis chosen by a data‐based cross‐validation method.

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