Multivariate regression estimation of continuous-time processes from sampled data: local polynomial fitting approach

Let (Y,X)={Y(t),X(t),-∞<t<∞} be real-valued continuous-time jointly stationary processes and let (t_j) be a renewal point processes on (0,∞), with a finite mean rate, independent of (Y,X). We consider the estimation of regression function r(x₀, x₁,...,x_m-1; τ₁,...,τ_m) of ψ(Y(τ_m)) given (X(0)=x₀, X(τ₁)=x₁,...,X(τ_m-1)=_x-1 ) for arbitrary lags 0<τ₁<...< τ_m on the basis of the discrete-time observations {Y(t_j),X(t_j),t_j)_j=1ⁿ . We estimate the regression function and all its partial derivatives up to a total order p⩾1 using high-order local polynomial fitting. We establish the weak consistency of such estimates along with rates of convergence. We also establish the joint asymptotic normality of the estimates for the regression function and all its partial derivatives up to a total order p⩾1 and provide explicit expressions for the bias and covariance matrix (of the asymptotically normal distribution)