Nonparametric estimation of density, regression and dependence coefficients

Consider a strictly stationary time series Z_t taking values in R^q. Let Z₁,..., Z_n+k be consecutive observations of Z_t, where k, n are positive integers. Assume the existence of a function r satisfying r(z₁,..., z_k) = E(φ(Z_k+1) | (Z₁,... , Z_k) = (z₁,... , z_k)), where φ is a continuous real-valued function which is not necessarily bounded. The main problem under consideration is that of nonparametrically estimating r(z₁,..., z_k). Kernel types estimates of marginal densities and of the function r are investigated. Under general conditions, strong consistency of the estimates are established. The estimates can be chosen to achieve the optimal rate of convergence (n^-1 log n)^1/(2+d) in L_∞ norm restricted to compact sets. The series Z_t is assumed to satisfy a weak dependence condition reminiscent of the absolute regularity condition. The results on the density estimates are employed to construct consistent estimates of the dependence coefficients and their rates of decay.