Partially linear models with first-order autoregressive symmetric errors

In this paper we discuss estimation and diagnostic procedures for partially linear models with first-order autoregressive [AR(1)] symmetric errors. The symmetric class includes all symmetric continuous distributions, particularly distributions with heavier and lighter tails than the normal ones, such as Student-t, power exponential and logistic, among others. Estimation is performed by maximum penalized likelihood and by using natural cubic splines. We derive the penalized score functions and the penalized Fisher information matrices for the parameters in the model. A reweighted iterative process based on the back-fitting algorithm is derived for the parameter estimation and the inference is based on the penalized Fisher information matrix. We discuss the effective degrees of freedom estimation and procedures for selecting the smoothing parameter. A small simulation study is performed for assessing the empirical distribution of the parameter estimates obtained from partially linear models with AR(1) errors. Residual analysis and derivation of conformal normal curvatures of local influence for some perturbation schemes are also given. Finally, a real data set is analyzed under partially linear models with AR(1) symmetric errors.