With the high penetration of distributed energy resources in active distribution networks(ADNs), forecast errors from renewables and loads pose significant risks of bilateral violations, including overvoltage/undervoltage and line overloads. To address this challenge, this paper proposes a KDE-DRCCO model that integrates kernel density estimation (KDE) with distributionally robust chance-constrained optimization (DRCCO). Leveraging the radial topology of ADNs, a linearized Distflow model is adopted to establish a linear dependency between decision variables and uncertainties. A data-driven KDE method is employed to capture the forecast error distributions, which enables an exact convex reformulation of the bilateral chance constraints on voltages and line flows using conditional value-at-risk (CVaR) approximation and duality theory. Compared with traditional moment-based ambiguity set approaches, the proposed model significantly reduces the economic loss caused by over-conservatism, achieving a better balance between operational economy and reliability. Simulations on modified IEEE 33-bus and modified IEEE 123-bus systems demonstrate that the proposed model effectively reduces total operating costs while ensuring system security and robustness.