Gaussian assumptions of non-Gaussian noises hinder the improvement of state estimation accuracy. In this paper, an asymmetric generalized Gaussian distribution (AGGD), as a unified representation of various unimodal distributions, is applied to formulate the non-Gaussian forecasting-aided state estimation problem. To address the problem, an improved particle filter is proposed, which integrates a near-optimal AGGD proposal function and an AGGD sampling method into the typical particle filter. The AGGD proposal function can approximate the target distribution of state variables to greatly alleviate particle degeneracy and promote precise estimation, through considering both state transitions and latest measurements. For rapid particle generation from the AGGD proposal function, an efficient inverse cumulative distribution function (CDF) sampling method is employed based on the derived approximation of inverse CDF of AGGD. Numerical simulations are carried out on a modified balanced IEEE 123-bus test system. The results validate that the proposed method outperforms other popular state estimation methods in terms of accuracy and robustness, whether in Gaussian, non-Gaussian, or abnormal measurement errors.