Optimal asymptotic MSE of kernel regression estimate for continuous time processes with missing at random response

Abstract Based on an incomplete observed sample ( X t , Y t , ζ t ) 0 ≤ t ≤ T , where ζ t = 1 if Y t is observed at time t and ζ t = 0 otherwise, the asymptotic mean square error of the regression estimator m T ( x ) , x ∈ R d , is showed to satisfy the following inequality: lim T → ∞ T 4 ∕ ( d + 4 ) E ( m T ( x ) − m ( x ) ) 2 c where c is an explicit constant. A simulation study is performed to illustrate the effect of the sampling mesh as well as the missing at random mechanism on the consistency of the regression estimator for regular and irregular continuous time processes.