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

Abstract

Let (Y,X)={Y(t),X(t),-/spl infin/<t</spl infin/} be real-valued continuous-time jointly stationary processes and let (t/sub j/) be a renewal point processes on (0,/spl infin/), with a finite mean rate, independent of (Y,X). We consider the estimation of regression function r(x/sub 0/, x/sub 1/,...,x/sub m-1/; /spl tau//sub 1/,...,/spl tau//sub m/) of /spl psi/(Y(/spl tau//sub m/)) given (X(0)=x/sub 0/, X(/spl tau//sub 1/)=x/sub 1/,...,X(/spl tau//sub m-1/)=/sub x-1/) for arbitrary lags 0</spl tau//sub 1/<...< /spl tau//sub m/ on the basis of the discrete-time observations {Y(t/sub j/),X(t/sub j/),t/sub j/)/sub j=1//sup n/. We estimate the regression function and all its partial derivatives up to a total order p/spl ges/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/spl ges/1 and provide explicit expressions for the bias and covariance matrix (of the asymptotically normal distribution).