Two step estimations for a single-index varying-coefficient model with longitudinal data

In this paper, we propose a two step estimation procedure to improve estimation efficiency of the index coefficients and unknown functions. Specifically, in the first step, we obtain the initial estimators by ignoring the possible correlations between repeated measures. Based on the modified Cholesky decomposition, we apply the least squares technique to estimate the autoregressive coefficients and innovation variance, and then obtain the estimated within-subject covariance matrix. In the second step, we construct the centralized generalized estimating equations to obtain more efficient estimators of index coefficients. Based on the estimated index coefficients, we can obtain more efficient estimators of unknown functions by employing the weighted least squares approach. Simulation studies and progesterone data have proved that the two step estimators are more efficient in practice.