OPTIMAL ASYMPTOTIC MEAN-SQUARE ERROR FOR KERNEL DENSITY AND REGRESSION ESTIMATES UNDER MIXING

Under mild conditions the mean square error of the kernel density estimate for a stationary process is OF(n-4/5). More precisely this convergence rate is reached if f is-an-element-of C2(b), sup(t greater-than-or-equal-to 1)\\f(X0,Xt)-f X f\\infinity and alpha(sigma(X0), sigma(X(t))) = OF(t-2) with standard notations). Analogous results are given for the kernel regression estimate. We get uniform results:(~)[GRAPHICS](~)where C, J and M are specified.