Chaotic time series can well reflect the nonlinearity and non-stationarity of real environment changes. The traditional kernel adaptive filter (KAF) with second-order statistical characteristics often suffers from performance degeneration dramatically for predicting chaotic time series containing noises and outliers. In order to improve the robustness of adaptive filtering in the presence of impulsive noise, a nonlinear similarity measure named Cauchy kernel loss (CKL) is proposed, and the global convexity of CKL is guaranteed by the half-quadratic (HQ) method. To improve the convergence rate of stochastic gradient descent and avoid a local optimization simultaneously, the conjugate gradient (CG) method is used to optimize CKL. Furthermore, to address the issue of kernel matrix network growth, the Nystrom sparse strategy is adopted to approximate the kernel matrix and then the probability density rank-based quantization (PRQ) is used to improve the approximation accuracy. To this end, a novel Nystrom Cauchy kernel conjugate gradient with PRQ (NCKCG-PRQ) algorithm is proposed for prediction of chaotic time series in this paper. Simulations on prediction of synthetic and real-world chaotic time series validate the advantages of the proposed algorithm in terms of filtering accuracy, robustness and computational storage complexity.