Nonparametric Estimation in a Nonlinear Time-Varying Generalized Regression Model with Locally Stationary Covariate and ARCH-Errors

This paper focuses on a nonparametric estimation in nonlinear time-varying generalized regression model of order $$d>1$$ , considering locally stationary covariates and autoregressive conditional heteroskedasticity (ARCH) errors. We establish the uniform almost sure convergence rate for the conditional mean and variance functions, under assumptions of local stationarity and local ergodicity. The results are derived without assuming any specific usual type of mixing conditions or physical dependence measure on the data, making them applicable to a diverse range of dependent processes, including those satisfying strong mixing conditions.