  We study strong (pathwise) approximation of Cox-Ingersoll-Ross processes. We
propose a Milstein-type scheme that is suitably truncated close to zero, where
the diffusion coefficient fails to be locally Lipschitz continuous. For this
scheme we prove polynomial convergence rates for the full parameter range
including the accessible boundary regime. The error criterion is given by the
maximal $L_p$-distance of the solution and its approximation on a compact
interval. In the particular case of a squared Bessel process of dimension
$\delta>0$ the polynomial convergence rate is given by $\min(1,\delta)/(2p)$.
