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Local Box-Cox Transformation in time varying coefficient models with longitudinal data

*Mohammed Rahim Uddin Chowdhury, The George Washington University 
Colin Wu, NHLBI 

Keywords: local box-cox transformation, conditional distributions, local polynomials, longitudinal data, time-dependent parameters, time-varying parametric models, two-step smoothing.

Nonparametric estimation and inferences of conditional distribution functions with longitudinal data have important applications in biomedical studies, such as epidemiological studies and longitudinal clinical trials. Estimation approaches without any local transformation and without any structural assumptions on variable of interest may lead to inadequate and numerically unstable estimators in practice. We propose in this paper a nonparametric approach based on local time-varying box-cox transformed parametric models for estimating the conditional distribution functions with a longitudinal sample. Our model assumes that the conditional distribution of the outcome variable at each given time point can be approximated by a parametric model, but the parameters are smooth function of time. Our estimation is based on a two-step smoothing method(also known as smoothing later approach), in which we first obtain the raw estimators of the conditional distribution functions at a set of disjoint time points, and then compute the final estimators at any time by smoothing the raw estimators. We have also shown by a real data example that smoothing first approach is not applicable with local box-cox transformation. Asymptotic properties, including the asymptotic biases, variances and mean squared errors, have been derived for the local polynomial smoothed estimators. Asymptotic distribution of the raw estimators of the conditional distribution functions has been derived. Applications of our two-step estimation method have been demonstrated through a large epidemiological study of childhood growth and blood pressure. Finite sample properties of our procedures are investigated through a simulation study.